passivity based control of euler-lagrange systems

Passivity�based Control of Euler�Lagrange

Systems�

Mechanical� Electrical and Electromechanical

Applications

Romeo Ortega�Antonio Lor��a�

Per J� Nicklasson�Hebertt Sira�Ram��rez�

To Amparo with all my love�Romeo�

To Lena with r � �� sin��to Mari my sister with my deepest admiration�

To�no�

To my parents�Per Johan�

To Jos�e Humberto Ocariz E� with respect and a�ection�to Mar��a Elena and Mar��a Gabriela with all my love�

Hebertt�

Preface

By its own de�nition the �nal purpose of control is to control something� In fact�the foundational developments of Huygens� Maxwell� Routh� Minorsky� Nyquist andBlack �to name a few were motivated by realworld applications� In the hands ofmathematicians such as Wiener� Bellman� Lefschetz� Kalman and Pontryagin �again�to name just a few control theory developed in the �� s and �� s as a branch of ap�plied mathematics� independent of its potential application to engineering problems�Some tenuous arguments were typically invoked to provide some practical motiva�tion to the research on this socalled mathematical control theory� For instance� thestudy of the triple �A� B� C was rationalized as the study of the linearization of anarbitrary nonlinear system an argument that had a grain of truth� By the end ofthe �� s a fairly complete body of knowledge for general linear systems includingpowerful techniques of controller synthesis had been completed� Some spectacu�lar applications of this theory to practical situations that �tted the linear systemsparadigm were reported�

The attempt to mimic the developments of linear systems theory in the generalnonlinear case enticed many researchers� Extensions to a fairly general class of non�linear systems of the basic concepts of controllability� observability� and realizabilitywere crowned with great success� The controller synthesis problem proved to be�however� much more elusive� Despite some signi�cant progress� to date� general tech�niques for stabilization of nonlinear systems are available only for special classes ofnonlinear systems� This is� of course� due to the daunting complexity of the behaviourof nonlinear dynamic systems which puts a serious question mark on the interest ofaiming at a monolithic synthesis theory� On the other hand� new technological de�velopments had created engineering problems where certain wellde�ned nonlineare�ects had to be taken into account� Unfortunately� the theory developed for gen�eral nonlinear systems could not successfully deal with them� basically because the�admissible structures� were determined by analytical considerations� which do notnecessarily match the physical constraints� It became apparent that to solve thesenew problems� the ��nd an application for my theory� approach had to be abandoned�and a new theory tailored for the application had to be worked out�

The material reported in this book is an attempt in this direction� Namely� westart from a wellde�ned class of systems to be controlled and try to develop a theory

vii

viii Preface

best suited for them� As the title suggests� the class we consider covers a very broadspectrum �EulerLagrange systems with mechanical� electrical and electromechanicalapplications however� detailed analysis is presented only for robots� AC machines andpower converters� We have found that this set of applications is su�ciently general�it has at least kept us busy for the last � years�

Di�erent considerations and techniques are used to solve the various problemshowever� in all cases we strongly rely on the information provided by the variationalmodeling and� in particular� concentrate our attention on the energy and dissipationfunctions that de�ne the dynamics of the system� A second unifying thread to allthe applications is the fundamental concept of passivity� Finally� a recurrent themethroughout our work is the notion of interconnection that appears� either in the formof a feedback decomposition instrumental for the developments� or as a frameworkfor focusing on the relevant parts of a model�

An important feature of the proposed controller design approach is that it isbased on the inputoutput property of passivity� hence it will typically not requirethe measurement of the full state to achieve the control objectives� Consequently�throughout the book we give particular emphasis to �more realistic� but far morechallenging outputfeedback strategies�

The book is organized in the following way� In Chapter � we present �rst abrief introduction that explains the background of the book and elaborates uponits three keywords� EulerLagrange �EL systems� passivity and applications� Thenotion of passivitybased control �PBC is explained in detail also in this chapter�underscoring its conceptual advantages� The main background material pertainingto EL systems is introduced in Chapter �� In particular we mathematically describethe class of systems that we study throughout the book� exhibit some fundamentalinputoutput and Lyapunov stability properties� as well as some basic features oftheir interconnection� We also give in this section the models of some examples ofphysical systems that will be considered in the book�

The remaining of the book is divided in three parts devoted to mechanical� elec�trical and electromechanical systems� respectively� The �rst part addresses a class ofmechanical systems� of which a prototypical example are the robot manipulators� butit is not restricted to them� For instance� we consider also applications to simple mod�els of marine vessels and rotational translational actuators� The results concerningmechanical systems are organized into set point regulation �Chapter �� trajectorytracking �Chapter � and adaptive disturbance attenuation� with application to fric�tion compensation �Chapter �� The theoretical results are illustrated with realisticsimulation results� In this part� as well as in other sections of the book� we carry outcomparative studies of the performance obtained by PBC with those achievable withother schemes� In particular� for robots with �exible joints� we compare in Chapter� PBC with schemes based on backstepping and cascaded systems�

The second part of the book is dedicated to electrical systems� in particular DC�

Preface ix

DC power converters� In Chapter � the EL model is derived and the control relevantproperties are presented� We present both� a switched model that describes the exactbehaviour of the system with a switching input� and an approximate model for thepulse width modulator controlled converters� While in the �rst model we have todeal with a hybrid system �with inputs or �� in the latter model� which is valid forsu�ciently high sampling frequencies� the control input is the duty ratio which is acontinuous function ranging in the interval � � �� Besides the standard EL modeling�we also present a rather novel� and apparently more natural� Hamiltonian model thatfollows as a particular case of the extended Hamiltonian models proposed by Maschkeand van der Schaft�

Chapter � is devoted to control of DC�DC power converters� We present� ofcourse� PBC for the average models� To deal with hybrid models we also introducethe concept of PBC with sliding modes� We show that combining this two strategieswe can reduce the energy consumption� a wellknown important drawback of slidingmode control� Adaptive versions of these schemes� that estimate online the loadresistance are also derived� An exhaustive experimental study� where various linearand nonlinear schemes are compared� is also presented�

In the third part of the book we consider electromechanical systems� To handlethis more challenging problem we introduce a feedback decomposition of the sys�tem into passive subsystems� This decomposition naturally suggests a nestedloopcontroller structure� whose basic idea is presented in a motivating levitated systemexample in Chapter � This simple example helps us also to clearly exhibit the con�nections between PBC� backstepping and feedback linearization� In Chapters ��we carry out a detailed study of nonlinear control of AC motors� The torque trackingproblem is �rst solved for the generalized machine model in Chapter � As an o�spin of our analysis we obtain a systems invertibility interpretation of the well�knowncondition of BlondelPark transformability of the machine�

The next two chapters� �� and �� are devoted to voltagefed and currentfedinduction machines� respectively� For the voltagefed case we present� besides thenestedloop scheme� a PBC with total energy shaping� Connections with the industrystandard �eld oriented control and feedback linearization are thoroughly discussed�These connections are further explored for currentfed machines in Chapter �� First�we establish the fundamental result that� for this class of machines� PBC exactly re�duces to �eld oriented control� Then� we prove theoretically and experimentally thatPBC outperforms feedback linearization control� The robustness of PBC� as well assome simple tuning rules are also given� Finally� motivated by practical considera�tions� a globally stable discretetime version of PBC is derived� Both chapters containextensive experimental evidence�

At last� in Chapter �� we study the problem of electromechanical systems withnonlinear mechanical dynamics� The motivating example for this study is the controlof robots with AC drives� for which we give a complete theoretical answer� The

x Preface

chapter clearly illustrates how PBC� as applied to EulerLagrange models� yields amodular design which e�ectively exploits the features of the interconnections� In asimulation study we compare our PBC with a backstepping design showing� onceagain� the superiority of PBC�

Background material on passivity� variational modeling and vector calculus areincluded in Appendices A� B and C� respectively�

The book is primarily aimed at graduate students and researchers in control theorywho are interested in engineering applications� It contains� however� new theoreticalresults whose interest goes beyond the speci�c applications� therefore it might be use�ful also to more theoretically oriented readers� The book is written with the convictionthat to deal with modern engineering applications� control has to reevaluate its roleas a component of an interdisciplinary endeavor� A lot of emphasis is consequentlygiven to modeling aspects� analysis of current engineering practice and experimentalwork� For these reasons it may be also of interest for students and researchers� aswell as practitioning engineers� involved in more practical aspects of robotics� powerelectronics and motor control� For this audience the book may provide a source toenhance their theoretical understanding of some wellknown concepts and to establishbridges with modern control theoretic concepts�

We have adopted the format of theoremproofremark� which may give the er�roneous impression that it is a �theoretical� book� this is done only for ease of pre�sentation� Although most of the results in this book are new� they are presented ata level accessible to audiences with a standard undergraduate background in controltheory and a basic understanding of nonlinear systems theory� In order to favourthe �readability� of our book we have moved some of the most �technical� proofs toAppendix D�

The material contained in the book summarizes the experience of the authors oncontrol engineering applications over the last � years� It builds upon the PhD thesesof the second and fourth author as well as collaborative research among all of us� andwith several other researchers� Numerous colleagues and collaborators contributeddirectly and indirectly� and in various ways to this book�

The �rst author is particularly indebted to his former PhD students� G� Espinosaand R� Kelly triggered his interest in the areas of electrical machines and robotics�respectively� we have since kept an intensive and very productive research collabo�ration� G� Escobar� K� Kim and D� Taoutaou carried out some of the experimentalwork on converters and electrical machines� He has also enjoyed a long scienti�c col�laboration with L� Praly who always provided insightful remarks and motivation tohis work� Many useful scienti�c exchanges have been carried out over the years withH� Nijmeijer� M� Spong and A� J� van der Schaft� while Henk and Arjan motivatedhim to improve his theoretical background� Mark always found the threshold neces�sary to make a robot turn� He would like to thank all his coauthors from whom helearned the importance of collaborative work� Finally� he wants to express his deep

xi

gratitude to the french CNRS� which provides his researchers with working conditionsunparalleled by any other institution in the world�

The second author wishes to acknowledge specially the collaboration with hisformer undergraduate�school teacher R� Kelly who earlier introduced him to robotcontrol and Lyapunov theory� The enthusiasm of Rafael on these topics increased themotivation of the second author to pursue a doctoral degree in the �eld� During hisdoctoral research period he was also enriched with the advice and collaboration of H�Nijmeijer and L� Praly� The author wishes to express as well his deepest gratitudeto his �ancee and collaborator E� Panteley for her fundamental moral support in thisproject and for helping with the �gures of Chapter �� Last but not least� the work ofthe second author has been sponsored by the institutions he has been a�liated to inthe past � years� in chronological order� CONACyT� Mexico� University of Twente�The Netherlands� University of Trondheim� Norway� and University of California atSanta Barbara� USA�

The third author wants to thank Research Director Peter Singstad� SINTEF Elec�tronics and Cybernetics� Automatic Control� for supporting parts of this project ��nancially�

The fourth author is indebted to his colleagues and students of the Control Sys�tems Department of the Universidad de Los Andes �ULA in M�erida �Venezuela forthe continuous support over the years in many academic endeavors� Special thanksand recognition are due to his former student� Dr� Orestes Llanes�Santiago� for hiscreative enthusiasm and hard work in the area of switched power converters� Visits toR� Ortega� since �� have been generously funded by the Programme de Coopera�tion Postgradu�e �PCP� by the National Council for Scienti�c Research of Venezuela�CONICIT� as well as by the Centre National de la Recherche Scienti�que �CNRSof France� Thanks are due to Professor Marisol Delgado� of the Universidad Sim�onBol��var� who has acted as a highly e�cient PCP Coordinator in Venezuela� Overthe years� the author has bene�ted from countless motivational discussions with hisfriend Professor Michel Fliess of the Laboratoire des Signaux et Syst�emes �CNRS�France� His experience and vision has been decisively helpful in many of the author�sresearch undertakings�

R� Ortega� A� Lor��a� P� J� Nicklasson� H� Sira�Ram��rez�May ��

Contents

Notation xxxi

� Introduction �

� From control engineering to mathematical control theory and back � � �

A route towards applications � � � � � � � � � � � � � � � � � � � � � � � !

! Why EulerLagrange systems" � � � � � � � � � � � � � � � � � � � � � � #

# On the role of interconnection � � � � � � � � � � � � � � � � � � � � � � $

� Why passivity" � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� What is passivitybased control" � � � � � � � � � � � � � � � � � � � � �

$ Some historical remarks � � � � � � � � � � � � � � � � � � � � � � � � � �

$�� EulerLagrange systems and nonlinear dynamics � � � � � � � � �

$� Passivity and feedback stabilization � � � � � � � � � � � � � � � �

� Euler�Lagrange systems ��

� The EulerLagrange equations � � � � � � � � � � � � � � � � � � � � � � ��

Inputoutput properties � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Passivity of EL systems � � � � � � � � � � � � � � � � � � � � �

� Passivity of the error dynamics � � � � � � � � � � � � � � � � �

�! Other properties and assumptions � � � � � � � � � � � � � � � � #

�# Passive subsystems decomposition � � � � � � � � � � � � � � � � �

�� An EL structure�preserving interconnection � � � � � � � � � � �

! Lyapunov stability properties � � � � � � � � � � � � � � � � � � � � � � $

!�� Fully�damped systems � � � � � � � � � � � � � � � � � � � � � � $

!� Underdamped systems � � � � � � � � � � � � � � � � � � � � � � �

# Examples � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � !

xiii

xiv CONTENTS

#�� A rotational%translational proof mass actuator � � � � � � � � � !

#� Levitated ball � � � � � � � � � � � � � � � � � � � � � � � � � � � !

#�! Flexible joints robots � � � � � � � � � � � � � � � � � � � � � � � !#

#�# The Du�ng system � � � � � � � � � � � � � � � � � � � � � � � � !�

#�� A marine surface vessel � � � � � � � � � � � � � � � � � � � � � � !�

� Concluding remarks � � � � � � � � � � � � � � � � � � � � � � � � � � � � !$

I Mechanical Systems ��

� Set�point regulation ��

� State feedback control of fully�actuated systems � � � � � � � � � � � � #

�� A basic result� The PD controller � � � � � � � � � � � � � � � � #

�� An introductory example � � � � � � � � � � � � � � � � � � � � � ##

��! Physical interpretation and literature review � � � � � � � � � � #�

Output feedback stabilization of underactuated systems � � � � � � � � #�

�� Literature review � � � � � � � � � � � � � � � � � � � � � � � � � #�

� Problem formulation � � � � � � � � � � � � � � � � � � � � � � � #�

�! EulerLagrange controllers � � � � � � � � � � � � � � � � � � � � #�

�# Examples � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

! Bounded output feedback regulation � � � � � � � � � � � � � � � � � � ��

!�� Literature review � � � � � � � � � � � � � � � � � � � � � � � � � ��

!� Problem formulation � � � � � � � � � � � � � � � � � � � � � � � ��

!�! Globally stabilizing saturated EL controllers � � � � � � � � � � �!

!�# Examples � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

# Set�point regulation under parameter uncertainty � � � � � � � � � � � $�

#�� Literature review � � � � � � � � � � � � � � � � � � � � � � � � � $�

#� Adaptive control � � � � � � � � � � � � � � � � � � � � � � � � � $$

#�! Linear PID control � � � � � � � � � � � � � � � � � � � � � � � � $�

#�# Nonlinear PID control � � � � � � � � � � � � � � � � � � � � � � �

#�� Output feedback regulation� The PI�D controller � � � � � � � ��

� Concluding remarks � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

CONTENTS xv

� Trajectory tracking control �

� State feedback control of fully�actuated systems � � � � � � � � � � � � �#

�� The PD& controller � � � � � � � � � � � � � � � � � � � � � � � � ��

�� The Slotine and Li controller � � � � � � � � � � � � � � � � � � � ��

Adaptive trajectory tracking � � � � � � � � � � � � � � � � � � � � � � � �$

�� Adaptive controller of Slotine and Li � � � � � � � � � � � � � � �$

� A robust adaptive controller � � � � � � � � � � � � � � � � � � � ��

! State feedback of underactuated systems � � � � � � � � � � � � � � � � �

!�� Model and problem formulation � � � � � � � � � � � � � � � � � �

!� Literature review � � � � � � � � � � � � � � � � � � � � � � � � � � �

!�! A passivitybased controller � � � � � � � � � � � � � � � � � � � �

!�# Comparison with backstepping and cascaded designs � � � � � � #

!�� A controller without jerk measurements � � � � � � � � � � � � � � �

# Output feedback of fully�actuated systems � � � � � � � � � � � � � � � � �

#�� Semiglobal tracking control of robot manipulators � � � � � � � � �

#� Discussion on global tracking � � � � � � � � � � � � � � � � � � ��

� Simulation results � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Concluding remarks � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��!

� Adaptive disturbance attenuation Friction compensation ��

� Adaptive friction compensation � � � � � � � � � � � � � � � � � � � � � ��

�� The LuGre friction model � � � � � � � � � � � � � � � � � � � � ��$

�� DC motor with friction � � � � � � � � � � � � � � � � � � � � � � ��

��! Robot manipulator � � � � � � � � � � � � � � � � � � � � � � � � �

��# Simulations � � � � � � � � � � � � � � � � � � � � � � � � � � � � � #

State�space passi�able systems with disturbances � � � � � � � � � � � � $

�� Background � � � � � � � � � � � � � � � � � � � � � � � � � � � � � $

� A theorem for passi�able a�ne nonlinear systems � � � � � � � � �

! Concluding remarks � � � � � � � � � � � � � � � � � � � � � � � � � � � � �!�

xvi CONTENTS

II Electrical systems ��

� Modeling of switched DC�to�DC power converters ��

� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �!�

Lagrangian modeling � � � � � � � � � � � � � � � � � � � � � � � � � � � �!$

�� Modeling of switched networks � � � � � � � � � � � � � � � � � � �!$

� A variational argument � � � � � � � � � � � � � � � � � � � � � � �!�

�! General Lagrangian model� Passivity property � � � � � � � � � �#

�# Examples � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �#�

! Hamiltonian modeling � � � � � � � � � � � � � � � � � � � � � � � � � � ��$

!�� Constitutive elements � � � � � � � � � � � � � � � � � � � � � � � ��

!� LC circuits � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

!�! Examples � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

# Average models of PWM regulated converters � � � � � � � � � � � � � ��

#�� General issues about pulse�width�modulation � � � � � � � � � � ��

#� Examples � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �$�

#�! Some structural properties � � � � � � � � � � � � � � � � � � � � �$�

� Conclusions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Passivity�based control of DC�to�DC power converters ��

� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

PBC of stabilizing duty ratio � � � � � � � � � � � � � � � � � � � � � � ��

�� The Boost converter � � � � � � � � � � � � � � � � � � � � � � � ��!

� The Buckboost converter � � � � � � � � � � � � � � � � � � � � ��$

�! Simulation results � � � � � � � � � � � � � � � � � � � � � � � � � ��

! Passivity based sliding mode stabilization � � � � � � � � � � � � � � � ��

!�� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

!� Sliding mode control of the Boost converter � � � � � � � � � � ��

!�! Passivity�based sliding controller � � � � � � � � � � � � � � � � ��

# Adaptive stabilization � � � � � � � � � � � � � � � � � � � � � � � � � � �

#�� Controller design � � � � � � � � � � � � � � � � � � � � � � � � � �

#� Simulation results � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Experimental comparison of several nonlinear controllers � � � � � � � �!

CONTENTS xvii

�� Feedback control laws � � � � � � � � � � � � � � � � � � � � � � �!

�� Experimental con�guration � � � � � � � � � � � � � � � � � � � ��

��! Experimental results � � � � � � � � � � � � � � � � � � � � � � � �

��# Conclusions � � � � � � � � � � � � � � � � � � � � � � � � � � � � !�

III Electromechanical systems �

Nested�loop passivity�based control An illustrative example ��

� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ##

�� Model and control problem � � � � � � � � � � � � � � � � � � � � #�

Passivitybased control with total energy�shaping � � � � � � � � � � � #�

! Nestedloop passivitybased control � � � � � � � � � � � � � � � � � � � #$

!�� Control structure � � � � � � � � � � � � � � � � � � � � � � � � � #�

!� Passivitybased controller design � � � � � � � � � � � � � � � � #�

# Outputfeedback passivitybased control � � � � � � � � � � � � � � � � �!

� Comparison with feedback linearization and backstepping � � � � � � � �#

�� Feedbacklinearization control � � � � � � � � � � � � � � � � � � ��

�� Integrator backstepping control � � � � � � � � � � � � � � � � � ��

��! Comparison of the schemes � � � � � � � � � � � � � � � � � � � � �$

��# Simulation results � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Conclusions and further research � � � � � � � � � � � � � � � � �

Generalized AC motor ��

� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� AC motors � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Review of previous work � � � � � � � � � � � � � � � � � � � � � ��

��! Outline of the rest of this chapter � � � � � � � � � � � � � � � � $�

Lagrangian model and control problem � � � � � � � � � � � � � � � � � �

�� The EulerLagrange equations for AC machines � � � � � � � � ��

� Control problem formulation � � � � � � � � � � � � � � � � � � � �!

�! Remarks to the model � � � � � � � � � � � � � � � � � � � � � � �#

�# Examples � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �$

xviii CONTENTS

! A passivity�based approach for controller design � � � � � � � � � � � � ��

!�� Passive subsystems feedback decomposition � � � � � � � � � � ��

!� Design procedure � � � � � � � � � � � � � � � � � � � � � � � � � ��

# A globally stable torque tracking controller � � � � � � � � � � � � � � � ��

#�� Strict passi�ability via damping injection � � � � � � � � � � � � �

#� Current tracking via energy�shaping � � � � � � � � � � � � � � � �

#�! From current tracking to torque tracking � � � � � � � � � � � � �#

#�# PBC for electric machines � � � � � � � � � � � � � � � � � � � � �$

� PBC of underactuated electrical machines revisited � � � � � � � � � � !

�� Realization of the PBC via BP transformability � � � � � � � � !

�� A geometric perspective � � � � � � � � � � � � � � � � � � � � � ! #

� Examples � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! �

$ Conclusions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! $

$�� Summary � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! $

$� Open issues � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! �

�� Voltage�fed induction motors ��

� Induction motor model � � � � � � � � � � � � � � � � � � � � � � � � � !�

�� Dynamic equations � � � � � � � � � � � � � � � � � � � � � � � � !�

�� Some control properties of the model � � � � � � � � � � � � � � !�!

��! Coordinate transformations � � � � � � � � � � � � � � � � � � � !��

��# Remarks to the model � � � � � � � � � � � � � � � � � � � � � � !��

�� Concluding remarks � � � � � � � � � � � � � � � � � � � � � � � � !

Problem formulation � � � � � � � � � � � � � � � � � � � � � � � � � � � !

! A nestedloop PBC � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! �

!�� A systems �inversion� perspective of the torque tracking PBC ! !

!� Observer�less PBC for induction motors � � � � � � � � � � � � ! $

!�! Remarks to the controller � � � � � � � � � � � � � � � � � � � � !!�

!�# Integral action in stator currents � � � � � � � � � � � � � � � � !!!

!�� Adaptation of stator parameters � � � � � � � � � � � � � � � � � !!#

!�� A fundamental obstacle for rotor resistance adaptation � � � � !!�

!�$ A dq�implementation � � � � � � � � � � � � � � � � � � � � � � � !!$

CONTENTS xix

!�� De�nitions of desired rotor �ux norm � � � � � � � � � � � � � � !!�

!�� Simulation results � � � � � � � � � � � � � � � � � � � � � � � � � !#

# A PBC with total energyshaping � � � � � � � � � � � � � � � � � � � � !#

#�� Factorization of workless forces � � � � � � � � � � � � � � � � � !#!

#� Problem formulation � � � � � � � � � � � � � � � � � � � � � � � !##

#�! Ideal case with full state feedback � � � � � � � � � � � � � � � !##

#�# Observer�based PBC for induction motors � � � � � � � � � � � !#�

#�� Remarks to the controller � � � � � � � � � � � � � � � � � � � � !#�

#�� A dq�implementation � � � � � � � � � � � � � � � � � � � � � � � !#�

#�$ Simulation results � � � � � � � � � � � � � � � � � � � � � � � � � !��

#�� Concluding remarks � � � � � � � � � � � � � � � � � � � � � � � � !�!

� Field�oriented control and feedback linearization � � � � � � � � � � � � !�!

�� Rationale of �eldoriented control � � � � � � � � � � � � � � � � !�#

�� State estimation or reference values � � � � � � � � � � � � � � � !�$

��! Shortcomings of FOC � � � � � � � � � � � � � � � � � � � � � � � !��

��# Feedback linearization � � � � � � � � � � � � � � � � � � � � � � !��

� Experimental results � � � � � � � � � � � � � � � � � � � � � � � � � � � !�!

�� Experimental setup � � � � � � � � � � � � � � � � � � � � � � � � !�!

�� Outline of experiments � � � � � � � � � � � � � � � � � � � � � � !��

��! Observer�less control � � � � � � � � � � � � � � � � � � � � � � � !$

��# Observer�based control � � � � � � � � � � � � � � � � � � � � � � !$�

�� Comparison with FOC � � � � � � � � � � � � � � � � � � � � � � !$�

�� Concluding remarks � � � � � � � � � � � � � � � � � � � � � � � � !$�

�� Current�fed induction motors ��

� Model of the currentfed induction motor � � � � � � � � � � � � � � � � !�!

Field orientation and feedback linearization � � � � � � � � � � � � � � � !��

�� Direct �eldoriented control � � � � � � � � � � � � � � � � � � � !��

� Indirect �eld�oriented control � � � � � � � � � � � � � � � � � � !��

�! Observerbased feedbacklinearizing control � � � � � � � � � � !�$

�# Remarks to OBFL and FOC � � � � � � � � � � � � � � � � � � � !�

! Passivitybased control of currentfed machines � � � � � � � � � � � � !�

xx CONTENTS

!�� PBC is downward compatible with FOC � � � � � � � � � � � � !�

!� Stability of indirect FOC for known parameters � � � � � � � � !�!

# Experimental comparison of PBC and feedback linearization � � � � � !�#

#�� Experimental setup � � � � � � � � � � � � � � � � � � � � � � � � !��

#� Selection of �ux reference in experiments � � � � � � � � � � � � !��

#�! Speed tracking performance � � � � � � � � � � � � � � � � � � � !��

#�# Robustness and disturbance attenuation � � � � � � � � � � � � # �

#�� Conclusion � � � � � � � � � � � � � � � � � � � � � � � � � � � � � #

� Robust stability of PBC � � � � � � � � � � � � � � � � � � � � � � � � � # !

�� Global boundedness � � � � � � � � � � � � � � � � � � � � � � � � # #

�� Coordinate changes and uniqueness of equilibrium � � � � � � � # �

��! Local asymptotic stability � � � � � � � � � � � � � � � � � � � � # �

��# Global exponential stability � � � � � � � � � � � � � � � � � � � #�

� O�line tuning of PBC � � � � � � � � � � � � � � � � � � � � � � � � � � #��

�� Problem formulation � � � � � � � � � � � � � � � � � � � � � � � #��

�� Change of coordinates � � � � � � � � � � � � � � � � � � � � � � #�$

��! Local stability � � � � � � � � � � � � � � � � � � � � � � � � � � � #��

��# A performance evaluation method based on passivity � � � � � #

�� Illustrative examples � � � � � � � � � � � � � � � � � � � � � � � # �

$ Discretetime implementation of PBC � � � � � � � � � � � � � � � � � # �

$�� The exact discretetime model of the induction motor � � � � � #!�

$� Analysis of discretetime PBC � � � � � � � � � � � � � � � � � � #!

$�! A new discrete�time control algorithm � � � � � � � � � � � � � #!!

$�# Discussion of discretetime controller � � � � � � � � � � � � � � #!�

$�� Experimental results � � � � � � � � � � � � � � � � � � � � � � � #!�

� Conclusions and further research � � � � � � � � � � � � � � � � � � � � #!�

�� Feedback interconnected systems Robots with AC drives ��

� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ##

�� Cascaded systems � � � � � � � � � � � � � � � � � � � � � � � � � ##

�� Robots with AC drives � � � � � � � � � � � � � � � � � � � � � � ##�

General problem formulation � � � � � � � � � � � � � � � � � � � � � � � ##�

CONTENTS xxi

! Assumptions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ##�

!�� Realizability of the controller � � � � � � � � � � � � � � � � � � ##�

!� Other assumptions � � � � � � � � � � � � � � � � � � � � � � � � #�

# Problem solution � � � � � � � � � � � � � � � � � � � � � � � � � � � � � #��

#�� Proof of Theorem � �$ � � � � � � � � � � � � � � � � � � � � � � #��

� Application to robots with AC drives � � � � � � � � � � � � � � � � � � #��

�� Model � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � #��

�� Global tracking controller � � � � � � � � � � � � � � � � � � � � #�$

� Simulation results � � � � � � � � � � � � � � � � � � � � � � � � � � � � � #��

$ Concluding remarks � � � � � � � � � � � � � � � � � � � � � � � � � � � � #�#

�� Other applications and current research ��

� Other applications � � � � � � � � � � � � � � � � � � � � � � � � � � � � #��

Current research � � � � � � � � � � � � � � � � � � � � � � � � � � � � � #��

�� Power electronics � � � � � � � � � � � � � � � � � � � � � � � � � #��

� Power systems � � � � � � � � � � � � � � � � � � � � � � � � � � � #$

�! Generation of storage functions for forced EL systems � � � � � #$

�# Performance � � � � � � � � � � � � � � � � � � � � � � � � � � � � #$�

A Dissipativity and passivity ��

� Circuit example � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � #$�

L� and L�e spaces � � � � � � � � � � � � � � � � � � � � � � � � � � � � #$$

! Passivity and �nitegain stability � � � � � � � � � � � � � � � � � � � � #$$

# Feedback systems � � � � � � � � � � � � � � � � � � � � � � � � � � � � � #$�

� Internal stability and passivity � � � � � � � � � � � � � � � � � � � � � � #�

� The KalmanYakubovichPopov lemma � � � � � � � � � � � � � � � � #��

B Derivation of the Euler�Lagrange equations ��

� Generalized coordinates and velocities � � � � � � � � � � � � � � � � � #�!

Hamilton�s principle � � � � � � � � � � � � � � � � � � � � � � � � � � � #�$

! From Hamilton�s principle to the EL equations � � � � � � � � � � � � � #��

# EL equations for non�conservative systems � � � � � � � � � � � � � � � #��

� List of generalized variables � � � � � � � � � � � � � � � � � � � � � � � #��

xxii CONTENTS

� Hamiltonian formulation � � � � � � � � � � � � � � � � � � � � � � � � � #��

C Background material ��

D Proofs ��

� Proofs for the PI�D controller � � � � � � � � � � � � � � � � � � � � � � #��

�� Properties of the storage H��q� 'q� � � � � � � � � � � � � � � � � #��

�� Lyapunov stability of the PI�D � � � � � � � � � � � � � � � � � #�$

Proof of positive de�niteness of f��qp de�ned in �!�#! � � � � � � � � � #��

! The BP transformation � � � � � � � � � � � � � � � � � � � � � � � � � � �

!�� Proof of Proposition ��

!� A Lemma on the BP Transformation � � � � � � � � � � � � � � �

# Proof of Eqs� �� #� and �� $$ � � � � � � � � � � � � � � � � � � � � � !

#�� A theorem on positivity of a block matrix � � � � � � � � � � � � !

#� Proof of Eq� �� $$ � � � � � � � � � � � � � � � � � � � � � � � � !

#�! Proof of Eq� �� #� � � � � � � � � � � � � � � � � � � � � � � � � �

� Derivation of Eqs� �� and �� $

�� Derivation of Eq� �� $

�� Derivation of Eq� ��

� Boundedness of all signals for indirect FOC � � � � � � � � � � � � � � ��

�� Proof of Proposition ��

Bibliography ��

Index ��

List of Figures

�� Elastic pendulum regulated with a PBC� � � � � � � � � � � � � � � � $

�� Feedback decomposition of an EL system� � � � � � � � � � � � � � � � �

� Feedback interconnection of two EL systems� � � � � � � � � � � � � � $

�! Rotational%translational proof mass actuator� � � � � � � � � � � � � � !�

�# Ball in a vertical magnetic �eld� � � � � � � � � � � � � � � � � � � � � !

�� Ideal model of a �exible joint� � � � � � � � � � � � � � � � � � � � � � � !#

!�� Simple pendulum� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ##

!� Physical interpretation of a PD plus gravity compensation controller� #$

!�! EL closed loop system� � � � � � � � � � � � � � � � � � � � � � � � � � �

!�# EL Controller of �#�B� � � � � � � � � � � � � � � � � � � � � � � � � � �

!�� EL Controller of �#�B��

!�� Transient behaviour for translational and angular positions� � � � � � ��

!�$ Applied control and controller state� � � � � � � � � � � � � � � � � � � $

!�� Translational and rotational responses to an external disturbance� � � $�

!�� Applied control and controller state for an external disturbance� � � � $�

!�� Translational and rotational responses for qp�� $

!�� Applied control and controller state for qp�� $!

!�� EL Controller of Subsection �#�B� � � � � � � � � � � � � � � � � � � $#

!��! Saturated EL controller without cancelation of g�q� � � � � � � � � � $�

!��# PD plus gravity compensation� � � � � � � � � � � � � � � � � � � � � � $�

!�� Passivity interpretation of PID Control� � � � � � � � � � � � � � � � � �

!�� The EL controller of Kelly ��

!��$ PI�D Controller� block diagram� � � � � � � � � � � � � � � � � � � � � ��

xxiii

xxiv LIST OF FIGURES

!�� PI�D control� First link position error� � � � � � � � � � � � � � � � � � �

!�� PID control under noisy velocity measurement� � � � � � � � � � � � � �

#�� Output feedback of rigid�joint robots� � � � � � � � � � � � � � � � � � ��

#� State feedback of �exible�joint robots� � � � � � � � � � � � � � � � � � ��

�� Reference position q� and rotor angle position q without friction com�pensation� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Reference speed 'q� and rotor angle speed 'q without friction compen�sation� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

��! Reference position q� and rotor angle position q with adaptive frictioncompensation� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

��# Reference speed 'q� and rotor angle speed 'q with adaptive friction com�pensation� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Position tracking error q � q��

�� Control signal u� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

��$ Estimated parameters (��

�� Friction force F � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Position tracking error q � q� for the controller of �#�� $

�� The Boost converter circuit� � � � � � � � � � � � � � � � � � � � � � � � �#�

�� The Buckboost converter circuit� � � � � � � � � � � � � � � � � � � � �#$

��! The �Cuk converter circuit� � � � � � � � � � � � � � � � � � � � � � � � �#�

��# The Boostboost converter circuit� � � � � � � � � � � � � � � � � � � � �#�

�� Boost converter model with parasitic components� � � � � � � � � � � ��#

�� An ideal transformer circuit� � � � � � � � � � � � � � � � � � � � � � � ��

��$ Boost converter model with conjugate switches� � � � � � � � � � � � � ��

�� Cuk converter model with conjugate switches� � � � � � � � � � � � � � ��!

�� Boost converter model with a diode� � � � � � � � � � � � � � � � � � � ��

�� The Flyback converter circuit� � � � � � � � � � � � � � � � � � � � � � ��

�� Equivalent circuit of the average PWM model of the Boost convertercircuit� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �$#

�� Ideal transformer representing the average PWM switch position func�tion� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �$�

LIST OF FIGURES xxv

��! Equivalent circuit of the average PWM model of the Buckboost con�verter circuit� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �$�

��# Zero�dynamics of Boost converter corresponding to constant averageoutput voltage� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �$$

�� Zero�dynamics of Boost converter corresponding to constant averageinput current� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �$�

$�� PWM feedback control scheme for indirect PBC output voltage reg�ulation of DC�to�DC power converters� � � � � � � � � � � � � � � � � � ��$

$� Closed loop performance of PBC in a stochastically perturbed Boostconverter model including parasitics� � � � � � � � � � � � � � � � � � � ��

$�! Robustness test of controller performance to sudden load variations� ��

$�# Typical sliding �current�mode� controlled state responses for the Boostconverter� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��#

$�� Weigthed integral square stabilization error behavior for the Boostconverter� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��$

$�� SM&PBC scheme for regulation of the Boost converter� � � � � � � � �

$�$ Controller and plant state responses for di�erent controller initial con�ditions� �� x�d� � x�d� � � � � �� x�d� � x�d� � �� x�d� � x�d� � �!�� !$��

$�� Comparison of total stored stabilization error energy for traditionaland SM&PBC Boost converter� � � � � � � � � � � � � � � � � � � � � !

$�� Closed loop state response of the passivity�based controlled perturbed�Boost� converter� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

$�� Performance evaluation of the indirect adaptive controller in a per�turbed average �Boost� converter� � � � � � � � � � � � � � � � � � � � �

$�� Experimental setup� � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

$�� Boost circuit card � � � � � � � � � � � � � � � � � � � � � � � � � � � �

$��! Open loop step response � � � � � � � � � � � � � � � � � � � � � � � � �

$��# Step responses for LAC � � � � � � � � � � � � � � � � � � � � � � � � � !

$�� Step responses for FLC � � � � � � � � � � � � � � � � � � � � � � � � � #

$�� Step responses for PBC � � � � � � � � � � � � � � � � � � � � � � � � � �

$��$ Step response for SMC � � � � � � � � � � � � � � � � � � � � � � � � � �

$�� Step responses for SM&PBC � � � � � � � � � � � � � � � � � � � � � � $

$�� Frequency responses to periodic reference voltage� �Vd�t �� x��t � �

xxvi LIST OF FIGURES

$� Time responses to a periodic reference signal Vd�t � � � � � � � � � � �

$� � Response to a pulse disturbance w�t � � � � � � � � � � � � � � � � � � !

$� Open loop frequency response� Magnitude Bode plots of w �� x� � � !

$� ! Openloop response to a step change in the output resistance � � � � !#

$� # Output voltage behaviour for a disturbance in the output resistance !�

$� � Response to a pulse disturbance in the output resistance for adaptivePBC � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � !�

$� � Response to a disturbance in the output resistance for LAC & integralterm � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � !$

$� $ Response to a disturbance in the output resistance for PBC & integralterm � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � !�

�� Feedback decomposition� � � � � � � � � � � � � � � � � � � � � � � � � #�

�� Nestedloop controller structure� � � � � � � � � � � � � � � � � � � � � #�

��! Linear controller� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

��# Feedback�linearization controller� � � � � � � � � � � � � � � � � � � � � ��

�� Passivity�based controller without integral term� � � � � � � � � � � � ��

�� Passivity�based controller with integral term� � � � � � � � � � � � � � ��

��$ Passivity�based controller with integral term plus damping injection� �

�� Integrator backstepping�based controller� � � � � � � � � � � � � � � � �

�� Cross�section and schematic circuits of generalized electric machine� �

�� Passive subsystems decomposition of generalized electric machine� � ��

� �� Nested�loop control con�guration� � � � � � � � � � � � � � � � � � � � !

� � Connection with system inversion� � � � � � � � � � � � � � � � � � � � ! $

� �! References for speed and �ux norm� � � � � � � � � � � � � � � � � � � !#

� �# Tracking errors for speed and �ux norm� � � � � � � � � � � � � � � � � !#�

� �� !� Currents and voltages ia� ib� ua� ub� � � � � � � � � � � � � � � � � � !#�

� �� Tracking errors for �ux norm and speed� Observer�based controller� � !��

� �$ Components of estimation error '(qr � 'qr and error between the realspeed 'qm and the internal speed 'qmd from �� Observer�basedcontroller� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � !��

� �� !� Voltages and currents ua� ub� ia� ib� Observer�based controller� � � !�

� �� Block diagram of experimental setup� � � � � � � � � � � � � � � � � � !�#

LIST OF FIGURES xxvii

� �� Main block diagram for C�code generation from SimulinkTM� � � � � !��

� �� Example of SimulinkTMblock diagram for controller� � � � � � � � � � !��

� �� Inverter�motor con�guration� � � � � � � � � � � � � � � � � � � � � � � !��

� ��! Speed regulation%�ux tracking without integral action in currents� � !$�

� ��# Speed regulation%�ux tracking with �upper two �gures� and without'� in the controller� � � � � � � � � � � � � � � � � � � � � � � � � � � � !$

� �� Speed tracking%�ux regulation at low speed �� rpm� � � � � � � � !$!

� �� Speed regulation with load torque disturbance� L � �� Nm for t � �� s� 'qm� � � rpm� Error between desired torque and measuredtorque is shown in lower right plot� � � � � � � � � � � � � � � � � � � � !$!

� ��$ Position control� Passivity�based controller� � � � Wb� � � � � � � !$#

� �� E�ect of desired dynamics in controller for high sampling period�Tsampl � � �s� Lower two plots are result from an experiment

with only integral action and the nonlinear damping term in the con�troller� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � !$�

� �� Speed and �ux regulation� Observer�based controller� Estimates ofelectrical quantities denoted by �� !$�

� � Comparison of the passivity�based controller �PBC with an imple�mentation of FOC� Speed regulation%�ux tracking� � � � � � � � � � � !$$

� � � Comparison with FOC� Speed tracking%�ux regulation� � � � � � � � !$�

� � Comparison with FOC� Speed tracking%�ux regulation� Rr � ��RNr � !$�

� � ! Comparison with FOC� Speed regulation%�ux tracking� Rr � ��RNr !$�

�� Currentfed induction motor with indirect FOC� Motor model pre�sented in a frame of reference rotating with rotor �electrical speed� � !��

�� Simulation illustrating instability of feedback linearizing control� �aIdeal case with k� � � � �b�f parameters perturbed� and k� � � �k� � �� k� � $� k� � � and k� � � � � � � � � � � � � � � � � � � � � !��

��! Experimental setup� � � � � � � � � � � � � � � � � � � � � � � � � � � � !��

��# Results for a periodic square wave with amplitude changing between rpm and � rpm� Sampling time ! ms� KP � � � � KI � � � !��

�� a and �b show results with PBC for di�erent sampling times� �cand �d show results with OBFL for di�erent values of KP � � � � � � #

�� Variation of (Rr�� )� �a show results of PBC� �b show results ofOBFL� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � # �

xxviii LIST OF FIGURES

��$ �a and �b show speed and desired current amplitude for the PBC��c and �d show the same quantities for the OBFL� � � � � � � � � � #

�� Inputoutput description of closedloop system� � � � � � � � � � � � � # #

�� Root locus of the system equilibria for (Rr � !Rr� � � � � � � � � � � � # �

�� Root locus of the system equilibria for (Rr � !Rr� � � � � � � � � � � � # �

�� Decomposition of the closedloop system� � � � � � � � � � � � � � � � # �

�� Manifold of local stabilityinstability� � � � � � � � � � � � � � � � � � # �

��! Simulation showing the stabilityinstability boundary� Speed in �rad%s�versus time� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � # �

��# Simulation showing the improvement of performance� Speed in �rad%s�versus time� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � # $

�� Block diagram of experimental setup� � � � � � � � � � � � � � � � � � # �

�� Experimental instability with KP � �� KI � $� (Rr � ! * and'qm� � �� rpm� � � � � � � � � � � � � � � � � � � � � � � � � � � � � # �

��$ Experimental performance improvement with KP � �� 'qm� � �� rpm and� �a KI � $� �bKI � �� cKI � � � � � � � � � � � � #!

�� Improvement of �ux tracking with new discrete controller� � � � � � � #!$

� �� Feedback interconnected system� � � � � � � � � � � � � � � � � � � � � ##!

� � Cascaded �nestedloop control con�guration� � � � � � � � � � � � � � ##!

� �! Position errors ��ymi� i � �� #��

� �# Motor torques i� i � �� #�

� �� Stator voltages u for the two motors�� #�

� �� Stator currents 'qs for the two motors� � � � � � � � � � � � � � � � � � #�!

A�� RLC network � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � #$�

A� Feedback interconnection of passive systems � � � � � � � � � � � � � � #$�

B�� The path of motion according to Hamilton�s principle� � � � � � � � � #�$

List of Tables

!�� TORA parameters� � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� An EL approach for the modeling of the Buck�boost converter� � � � �#�

�� An EL approach for the modeling of the �Cuk converter� EL parameters��

��! An EL approach for the modeling of the �Cuk converter� Dynamicequations� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��# A Lagrangian approach to modeling of the multivariable Boost�boostconverter� EL parameters� � � � � � � � � � � � � � � � � � � � � � � � � ��

�� A Lagrangian approach to modeling of the multivariable Boost�boostconverter� Dynamic equations� � � � � � � � � � � � � � � � � � � � � � ��!

$�� Parameters for the power converters experimental setup� � � � � � � �

$� Comparison of cut�o� frequency ranges� � � � � � � � � � � � � � � � � �

$�! Comparison of ampli�cation ratios� � � � � � � � � � � � � � � � � � � � !�

$�# Comparison of gain and cut�o� frequency ranges� � � � � � � � � � � � !�

� �� De�nitions of '�a � a� � � � � � � � � � � � � � � � � � � � � � � � � � � !�$

�� Motor parameters� � � � � � � � � � � � � � � � � � � � � � � � � � � � � !��

�� Computation time� � � � � � � � � � � � � � � � � � � � � � � � � � � � � !�$

��! Motor parameters� � � � � � � � � � � � � � � � � � � � � � � � � � � � � # �

��# Motor parameters� � � � � � � � � � � � � � � � � � � � � � � � � � � � � #!�

� �� Comparison of computational requirements� � � � � � � � � � � � � � � #�#

B�� De�nition of variables� � � � � � � � � � � � � � � � � � � � � � � � � � � #�

xxix

xxx LIST OF TABLES

Notations� acronyms� and

numbering

Most commonly used mathematical symbols�

IR Field of real numbers�IRn Linear space of real vectors of dimension n�IRn�m Ring of matrices with n rows and m columns and elements in IR�IR�� Field of nonnegative real numbers�In The identity matrix of dimension n�t Time� t � IR��Ln� Space of n�dimensional square integrable functions�

Ln�e Extended space of n�dimensional square integrable functions�

h�j�i Inner product in Ln� �

h�j�iT Truncated inner product�k � k�T The truncated L� norm�� Mapping from a domain into a range� Also �tends to�� Mapping of two elements into their image�� de�ned as��dzd�

Derivative of z � f��ddt�� '�� Total time derivative�

p�� d

dtDi�erentiation operator�

��

Di�erentiation operator with respect to ��

n The set of integers �� n��

Matrices�

A usual convention undertaken in this book �though not exhaustivelyis to use UPPERCAPS for matrices and lowercaps for vectors and scalars�

kxk The Euclidean norm of x � IRn�kAk with A � IRn�m� Induced �norm�jaj Absolute value of the scalar a�

xxxi

xxxii Notation

Kp Frequently used �e�g� Part I to denote a �proportional� gain�Kd Frequently used �e�g� Part I to denote a �derivative� gain�Ki Frequently used �e�g� Part I to denote an �integral gain��diagfg or diag� Diagonal operator� transforms a vector into a diagonal matrix�� Inverse operator�Aij or aij ij�th element of the matrix A�km Positive constant corresponding to a square matrix K� such that

km kxk� x�Kx for all x � IRn� Correspondingly kpm for Kp�kM Positive constant corresponding to a square matrix K� such that

kM kxk� � x�Kx for all x � IRn� Correspondingly kpM for Kp��K Smallest eigenvalue of K��K Largest eigenvalue of K�� Transpose operator�

Euler�Lagrange systems�

q Vector of generalized positions�q� Denotes the desired reference value for q imposed by the designer�

hence an external signal�qd Denotes a �desired� value of certain internal signals produced by

the passivity�based�control design�'q Vector of generalized velocities�� Denotes an error between two quantities� typically a state variable

and its desired value� Notice that the desired value can be denotedusing � or d depending on the nature of the reference�

� Output of of the �dirty derivatives� �lter�(� Denotes the estimate of a parameter �or vector of parameters �� The estimation error (� � ��T �q� 'q Kinetic energy�V�q Potential energy�F� 'q Rayleigh dissipation function�M Inputs matrix�H Energy or storage function�D�q� D�q Inertia matrix�Dm Moment of inertia�De�qm Inductance matrix�C�q� 'q� C�q� 'q Coriolis and centrifugal forces matrix�g�q Potential forces vector�Q External generalized forces�Rm Mechanical viscous damping constant�u� up Control inputs�

�'e� ��m Denote electrical and mechanical variables respectively�

Notation xxxiii

qp Generalized positions vector of an EL plant�Tp�qp� 'qp Kinetic energy of a plant�Vp�qp Potential energy of a plant�Fp� 'qp Rayleigh dissipation function of a plant�Mp Matrix which maps the inputs to an EL plant�s coordinates�qc Generalized positions of an EL controller�Tc�qc� 'qc Controller�s kinetic energy�Vc�qc� qp Controller�s potential energy�Fc� 'qc Controller�s Rayleigh dissipation function�� Flux linkage vector�'qe Current vector�e �� Matrix exponential function�

J The matrix

� ��

��

e J �� The rotation matrix

�cos�� sin��sin�� cos��

��

Frequent acronymsAC Alternate current�BP BlondelPark�DC Direct current�EL Euler�Lagrange�FET Field e�ect transistor�FLC Feedback linearization control�FOC Field�oriented�control�GUAS Global uniform asymptotic stability�GAS Global asymptotic stability�IBC Integrator backstepping control�ISP Input Strictly Passive�KYP Kalman�Yakubovich�Popov�LC Inductor�Capacitor�LQG Linear quadratic Gaussian�OBFL Observerbased feedback linearizing�OSP Output Strictly Passive�PBC Passivity�based control�PD Proportional derivative�PID Proportional integral derivative�PI�D Proportional double�integral derivative�PWM Pulse�width modulation�RLC Resistor�Inductor�Capacitor�SM Sliding mode�TORA Translational�rotational actuator�

xxxiv Notation

Numbering of chapters� sections� equations� theorems� propositions�properties� assumptions� examples� de�nitions� facts�corollaries� lemmas and remarks

This book contains three parts labeled I�III and � chapters labeled �� Thechapters are structured in sections labeled in each chapter as �� etc� The subsectionnumbers are relative to the section which they belong to that is� �� etc� The sub�subsections are labeled A�� B�� C�� etc� Below sub�subsections we use the numberingA�� A� �� etc� Only sections and subsections are listed in the table of contents�

When citing a section or subsection belonging to a di�erent chapter� we use thebold font for the chapter number� Thus� the reader should not be confused forinstance� between Section of Chapter �� cited �� in Chapter �� from Subsection of Section �� cited in Chapter �� as ��

The equation numbers are relative to the corresponding chapter� for instance�equation �!�� is the ��th equation of Chapter �� Theorems� propositions� lemmas�de�nitions� examples� etc succeed each other in the numbering� which is relative tothe chapter number� For instance� Remark !� follows after Proposition !�� in thethird chapter�

Properties and assumptions are labeled as P� and A� respectively where thenumber + is relative to the chapter� For instance P�� corresponds to Property ��which is de�ned in Chapter ��

Figure and table counters are also relative to the chapter number�

Chapter �

Introduction

�A control theorist�s �rst instinct in the face of a new problemis to �nd a way to use the tools he knows� rather that acommitment to understand the underlying phenomenon� Thisis not the failure of individuals but the failure of our professionto foster the development of experimental control science� In away� we have become the prisoners of our rich inheritance andpast successes��

Y� C� Ho ��

The �nal objective of the research reported in this book is to contribute to thedevelopment of a system theoretic framework for control of nonlinear systems thatincorporates at a fundamental level the systems physical structure and provides so�lutions to practical engineering problems� This is� of course� a very ambitious andsomehow imprecise objective� To help delineate what we really want to accomplishwe underscore the three major keywords of our work� Euler�Lagrange �EL systems�passivity and applications� The �rst keyword mathematically de�nes the class of sys�tems that we study� the second one the main physical property that we focus on� whilethe last one is our �nal objective� In this chapter we will develop upon this threekeywords to explain the background and the contents of the book� and to motivateour approach�

� From control engineering to mathematical con�

trol theory and back

The early developments in control theory �from the ��! s to the early �� s weremotivated by technological problems ranging from feedback ampli�er design to spaceapplications� Actually� the fundamental notion of feedback was �rst introduced by

�

� Ch� �� Introduction

Black in his celebrated feedback ampli�er� Experiments were usually carried outto establish insightful frequency response models� which via an intrinsic averagingprocess� captured the e�ect of uncertainties and in this way led to robust designs�The mathematical framework that was suited for modeling the plant� to analyzeit� and to synthesize controllers was based on inputoutput ideas with Fourier andLaplace transforms being the key tools� This viewpoint took a major shift with theintroduction of statespace ideas� where the mapping between inputs and outputstakes place via the transformation of the internal state of the system� The naturalmathematical framework in the statespace approach became then the di�erentialequations�

The mathematical advantages of the statespace formalism were rapidly exploitedin the �� s and ��$ s resulting in the introduction of many new fundamental con�cepts� notably those of controllability� observability and optimality� and the develop�ment of powerful techniques for controller synthesis �identi�cation� adaptive control�digital control� among others� The theoretical research expanded very rapidly� withthe control �eld attracting many of the best students who indulged themselves inbeautiful mathematical problems� Unfortunately� with this paradigm shift controldrifted away from its design vocation and became more mathematically driven thandriven by its original aim of providing the tools to cope with nonlinearity and uncer�tainty in practical engineering problems��

The research in the ��$ s and early �� s seemed to be motivated more by trendyacademic problems� mathematical abstraction� and an overemphasis on certain givenmethodologies than by engineering needs� In particular� mathematical modeling� acentral ingredient for successful control design� was essentially relegated� Also� theinterest in control experiments faded out in view of the conventional wisdom that itwas scienti�cally pointless to build an experiment to test a mathematical statementthat was selfconsistent and provably correct�

By the mid�� s the widening gap between theory and practice did not passunnoticed to funding agencies and industries which� via economic pressure� forcedthe control community to reevaluate its research directions and put more emphasis onpractical applications� This trend has not reduced the importance of control theory�since new technological developments have created di�cult engineering challengesthat require the use of sophisticated theoretical concepts� It has� however� underscoredthe role of control as a component of an interdisciplinary approach needed to solvethe new practical problems�� The control specialist has to interact then with peoplefrom other engineering �elds� The establishment of such communication is� of course�a long process and it is di�cult to decide where to start and which route to take�

�A brief respite to this scenario resulted from the revival of robustness issues in the early ��s�In particular� the elegant formulation and solution of what is called theH� problem and the researchon structured uncertainties�

�A successful story that illustrates the advantages of synergetic collaboration is the exciting new�eld of mechatronics�

� A route towards applications �

� A route towards applications

The work reported in this book� which summarizes our experience in control engi�neering applications over the last � years� has roughly proceeded along the followingpath�

� Since the main skills of control engineers concern the study of dynamical sys�tems� the �rst step is to give back to modeling the central role that it deservesand focus on structural aspects of some speci�c systems typically nonlinearthat can be exploited for control system design�

� The next step is then to try to mathematically formalize current engineeringpractice� which is usually developed from physical considerations and practicalexperience� as opposed to theoretical analysis� This is a fundamental step� whoseimportance can hardly be overestimated� It provides a solid systemtheoreticfoundation to existing control strategies which enhances their understandingallowing us to estimate their achievable performance and paves the way forsubsequent improvements�

� Based on our understanding of the model� the mathematical rationalization ofcurrent engineering practice and new analytical considerations� the third step isto propose new controllers� In some fortunate and some times serendipitouscases these may be viewed as �upgrades� of the existing schemes� facilitating inthis way the transfer of this knowledge to practitioners�

� The �nal� and sine qua non step is to test the theoretical developments inrealistic simulations or experiments and confront the achievable performancewith standard controllers� Even though in all steps the interaction with thespecialist of the application �eld is essential� it is perhaps in the latter one thatit is particularly crucial�

The book describes two instances in which the payo� for abiding to these guide�lines was particularly rewarding� The �rst one is in robotics� to which the �rst partof the book is devoted� First� a careful study of the robot dynamics� and in particu�lar the establishment of its passivity� provided the theoretical underpinnings for theindustry standard PD control� Later developments� exploiting suitable passive sub�systems decompositions� proved that non�measurable velocities could be replaced byits approximate derivative again� a standard engineering practice without a�ectingits stability properties� Simulation results showed later that a control input beyondthe physical saturation constraints was sometimes required to achieve fast motions�Hence� it became necessary to study the e�ect of saturations� which turned out tobe possible with a simple and natural addition to our design toolbox� As a resultof further analysis it was possible to prove that adding a double integral action tothe PD controller would not just remove the need for velocity measurements� but


also enhance its robustness making it insensitive to uncertainty in the payload� Theoverall result of our research was to provide a nice and natural upgrade� with guar�anteed stability properties� of the classical PID control with saturated inputs� Themathematical machinery we had set up for the stability analysis could now be usedfor tuning and performance improvement�

The second case study concerns AC electric machines� �described in Part III of thebook� which in spite of their complex dynamics are shown to de�ne passive maps� Aninsightful decomposition of the machine dynamics into passive subsystems opened thegate for a rigorous theoretical analysis� in particular its proof of global asymptoticstability� of the industry standard �eldoriented control� A �rst upgrade to thiscontroller was then proposed for the case when� due to performance considerations�the model of the machine has to be re�ned to include the stator dynamics� Themodularity of our control design was exploited at an even higher level when we hadto take into account a nonlinear model of the load to enhance performance� Thiswas the case of controlling fast moving robots with AC drives� We proved that wecould cascade the control law for the machine with an outerloop controller� designedupon the same passivity principles� for the robot dynamics� Extensive experimentalevidence proved later the validity of our theoretical considerations� Furthermore�as an outcome of our stability analysis� we provided some simple tuning rules tocommission and retune either the �eldoriented controller or our upgraded version�

There are� of course� alternative routes towards control engineering applicationsto the one delineated above� For instance� instead of our modelbased starting pointother research groups have developed nonmodelbased techniques� This approachis motivated by the massive availability of data either from extensive simulations orfrom experiments� and centers around the use of powerful �data�tters� for modelingand control purposes� the best wellknown example being neural nets and fuzzy sets�Even though at this stage it is not clear how the reported analytical results uses prop�erties particular to neural nets� it is indeed the case that unveiling the complicatedmechanisms of operation of neural nets requires a deep understanding of nonlineardynamics� the natural realm of application of control theory� Although this approachdoes not seem to enhance a synergetic collaboration with specialist of other areas�it is fair to say that� in some sense� it fosters the development of an experimentalcontrol science� and certainly has a pro�table role to play in a control practitioner�stoolbox�

� Why Euler�Lagrange systems�

We are interested in this book on controller design for dynamical systems where thenonlinear components cannot be neglected� For instance� motion control of robotsactuated by AC drives where besides the intrinsic nonlinear operation of the ACmotors we have that a simple double integrator model of the mechanics cannot capture

� Why Euler�Lagrange systems� �

the behaviour of the robot under fast motions� It is widely recognized� at least asfar back as Poincar�e� that nonlinear systems can exhibit extremely complex dynamicbehaviour� This renders futile the quest of a monolithic control theory applicable forall systems� we must therefore specialize the class of systems under consideration inone way or another��

We have proposed above to single out from the outset a class of physical systemsand concentrate our e�orts upon them� This should be contrasted with the researchon� what we have called� mathematical control theory� where one takes o� from aclass of systems whose solution is known� e�g� LTI systems� and start to enlargethe class including some special nonlinearities �or structures for which the availableanalysis and design tools can be suitably extended� A typical example of this scenariois the absolute stability problem consisting of an LTI system in feedback with a staticnonlinearity� One way to study this paradigm is to add to the standard quadraticLyapunov function of the LTI system an integral term that takes care of the presenceof the nonlinearity� leading to a Lur�ePostnikov Lyapunov functional�

The �nal aim of this type of research is to build up tools and procedures to treat amore or less general paradigm� the �nal aim typically being 'x � f�x� u� y � h�x� u�Needless to say that this fundamental research can provide some stepping stones in ourroute towards applications� it is particularly useful to de�ne achievable performances�However� a fundamental obstacle for the application of the resulting techniques tophysical systems is that it is hard� if not impossible� to incorporate a posteriorithe natural structures imposed by the systems physical character� This stems fromthe fact that the design techniques are applicable only to systems with particularstructures which are de�ned only by mathematical considerations� hence there is noapparent reason why a physical system belongs to the admissible class�

It is sometimes possible to �force� the system into the required form� via changesof coordinates or nonlinearity cancelations� it is in any case somehow distressing thatto control a physical system we have to start by �destroying� its structure� Theextreme case where the physical structure of the system is neglected is� of course�feedback linearization� where the aim of the control is to render the system linearin closed loop� Sliding mode control lies somewhere in between in so far as once arelative degree one minimum phase output is constructed the remaining dynamicscan be swamped via highgain �relay feedback�

It is the authors� belief that to develop a practically meaningful nonlinear con�trol theory we should start by considering a practically meaningful class of systems�whose physical structure should be taken into account� from the outset� in the de�sign procedure� It is in the de�nition of this class that EL systems enter into thepicture� The most important reason for singling out the study of EL systems is that

�In this respect it is interesting to mention the concept of atness �� which identi�es a classof systems for which trajectory planning becomes trivial� The stabilization itself is then carried outalong the reference trajectory with a lower hierarchy standard control loop�


they describe the behaviour of a large class of contemporary engineering systems�specially some which are intractable with linear control tools� EL systems are theoutcome of a powerful modeling technique the variational method whose startingpoint is the de�nition of the energy functions in terms of sets of generalized variables�typically positions and charges for mechanical and electrical systems� respectively�which leads to the de�nition of the Lagrangian function� The equations of motionare then derived invoking wellknown principles of analytical dynamics� in particularthe fundamental Hamilton�s principle� which roughly speaking states that the systemmoves along trajectories that minimize the integral of the Lagrangian��

The variational modeling method is one of the most powerful techniques of dy�namics� As we will show throughout the book this method is particularly suited forour purposes for the following reasons�

� Given that the modeling problem is formulated in terms of energy quantities thevariational method allows us to treat� without the need of any special bookkeep�ing� systems of �mixed nature�� which often appear in engineering applications�e�g�� having both electrical and mechanical components�

� It automatically provides us with the storage and dissipation functions of thesystem� These are the cornerstones of the design technique that we advocate inthis book which is based on energy dissipation�

� Since the modeling is based on some kind of network representation and energy�ow� it is compatible with one of the important viewpoints of systems theorythat complicated systems are best thought of as being interconnections of sim�pler subsystems� each one of them being characterized by its energy functions�This aggregation procedure underscores the role of the interconnections betweenthe subsystems helping us to think in terms of the structure of the system andto realize that sometimes the pattern of the interconnections is more importantthan the detailed behaviour of the components�

� We will show that the EL structure is preserved under feedback interconnection�That is� the interconnection of two EL systems is still an EL system� �As wewill explain below� this invariance property is also enjoyed by passive systems�which is the second key component of our approach�

Before closing this subsection a word on notation is in order� As explained inSection �� throughout the book we use the name EL systems because we choose torepresent their dynamics with the EL equations of motion� Our choice stems fromthe fact that in this representation it is easier to reveal some structural propertiesinstrumental for our controller design� They can alternatively be represented with

�Putting it in another way� The possible trajectories of a dynamical system are precisely thosethat minimize a suitable action integral� In the words of Legendre� �Ours is the best of all worlds��

� On the role of interconnection �

Hamilton�s equation and called Hamiltonian systems� A more generic� but in otherrespects somehow confusing name encountered in the literature is mechanical systems�

On the role of interconnection

The features above will be particularly relevant for our research where interconnectionis a recurrent theme� At a conceptual level� we follow Willems� behavioral perspectiveand think of control as the establishment of a suitable interconnection of the systemwith its environment� For instance� we will show in Section �� ! that the outcome ofour design procedure� which relies upon energy dissipation considerations� as appliedto the set point regulation of a �exible pendulum is the interconnection of the systemwith a virtual pendulum as depicted in Fig� �� That is� the controller consists ofa rigid pendulum �of unitary mass without gravity forces that we attach to a plane�with inclination � with a spring �of sti�ness coe�cient K� and a damper �withdamping coe�cient Rc� Then� we attach a spring �of sti�ness coe�cient K� betweenthis virtual pendulum and the �rst link angle in such a way that the virtual pendulum�pulls� the actual pendulum to the desired equilibrium �q�� The position of the planeis selected so that in steadystate the springs store the excess of potential energy� thedamper is added to ensure that energy is dissipated and asymptotic convergence ishenceforth achieved� To take into account saturation in the input torque� which wasmentioned in Section � we simply make the controller springs nonlinear�

��

q c

q *

δ( )

q q

p1

p2

K 2

Rc

K 1m=1

D

Figure �� Elastic pendulum regulated with a PBC�

Cela va sans dire that this controller has an obvious physical interpretation� afact whose importance can hardly be overestimated� Its relevance pertains not justto aesthetical considerations� but more importantly to its impact on the controllertuning stage� Commissioning of a nonlinear controller is a di�cult task� which hasunfortunately been overlooked in most theoretical developments� A �nal� but not lessimportant� advantage of a physically interpretable design is that it clari�es the role ofthe sensors and actuators� For instance� in the present example we assumed available

Ch� �� Introduction

the measurement of the actuated side of the link� it is easy to modify the design ifwe add to it �or replace it by measurement of the unactuated side��

Interconnection appears also in the book when� in the quest to achieve the de�sired performance� we propose to iterate at the modeling and control levels� Moreprecisely� a �rst design is made on the basis of a simple model �say the purely rigidmechanical dynamics� if performance is below par we re�ne the model �for instance�include �exible modes or the electrical dynamics of the actuators and cascade thenew controller with the previous one�

To be able to carry out this interlaced modelingcontrol procedure we e�ectivelyexploit the aggregation property of EL models described above� We also require thatthe controller design should be based upon a property that remains invariant uponinterconnection� it is at this stage that we invoke the fundamental physical propertyof passivity�

Why passivity�

Passive systems are a class of dynamical systems in which the energy exchanged withthe environment plays a central role� In passive systems the rate at which the energy�ows into the system is not less than the increase in storage� In other words� a passivesystem cannot store more energy than is supplied to it from the outside� with thedi�erence being the dissipated energy� See Appendix A for a basic circuittheoreticintroduction to passivity� and the formal de�nition� as well as various properties ofpassive systems that will be used throughout the book� Let us simply point out herethat in the LTI case passive systems are minimum phase and have relative degree notgreater than one� i�e� their frequency response satis�es IRefH�j g � � The ground�breaking result of �#�� furthermore shows that a nonlinear system can be renderedpassive via statefeedback if and only if �the nonlinear analog of these two conditionshold�

It is clear from the energy interpretation of passivity given above that it is inti�mately related with the physics of the system� and in particular its stability properties�For instance� viewing a feedback interconnection as a process of energy exchange� itis not surprising to learn that passivity is invariant under negative feedback intercon�nection� In other words� the feedback interconnection of two passive systems is stillpassive� see Theorem A�!� If the overall energy balance is positive� in the sense thatthe energy generated by one subsystem is dissipated by the other one� the closed loopwill be furthermore stable as stated in Proposition A�� As an immediate corollary

�The interplay of actuators� sensors and dissipation has traditionally being a basic building blockfor structural control� active vibration suppression and control of large space structures�

�It should be mentioned here that� even though our de�nition is given in terms of inputs andoutputs� invoking the behavioral framework of Willems it is possible to de�ne this concept withoutreference to inputs and outputs�

� Why passivity�

we have that passive systems are �easy to control�� for instance� with a simple con�stant gain� which can actually be made arbitrarily large recall the relative degreeand minimum phase properties of passive systems mentioned above� This propertytogether with the clean characterization of passi�able systems reported in �#�� ex�plains the interest of passivity as the basic building block for control of nonlinearsystems�

One �nal� but not less important property of passivity is that it is a propertywhich is independent of the notion of the state� hence as we will see throughout thebook� statefeedback �which we know is unrealistic in most applications will not bea prerequisite to achieve the control objectives�

Summarizing� Passivity �and the closely related property of �nite gain stabilityprovide a natural generalization �to the nonlinear timevarying case of the fact thatstability of an LTI feedback system depends on the amount of gain and phase shiftinjected into the loop� Furthermore� and perhaps more importantly� the measures ofsignal ampli�cation �the operator gain and signal shift �its passivity can be in someinstances associated to physical quantities� These fundamental properties motivatesus to take passivity as the basic building block of our constructions�

A more precise evaluation of the concepts above can be made for EL systems�namely�

� We prove that EL systems de�ne passive maps with storage function their totalenergy� This is the most fundamental property for our work since it identi�esthe output which is �easy� to control and provides a storage function that willtypically motivate the desired closedloop storage function�

� In most problems we need a stronger socalled �skewsymmetry� property of ELsystems that essentially says that the dynamics of EL systems can be treatedas a set of linear double integrators� provided a suitable factorization of the�workless forces is used� This factorization is essential in all our developments�

� Invoking the property of invariance under feedback interconnection of the ELstructure we will� in some control tasks� view the action of the controller asan EL�structure�preserving interconnection� In this way the properties of theclosedloop are still captured by the energy and dissipation functions that char�acterize an EL system� Furthermore� these are simply the sum of the corre�sponding functions of the plant and the controller�

� Under some reasonable assumptions we prove that EL systems can be decom�posed into the feedback interconnection of two passive subsystems� This decom�position is instrumental for the interlaced modelingcontrol procedure describedabove�

�� Ch� �� Introduction

All these properties are used in the sequel to design what we call passivitybasedcontrol �PBC schemes�

Before closing this section it is convenient to clarify that the fact that EL systemsare passive� �or can even be �treated as double integrators� does not make theproblem of control of EL system trivial� First of all� the output with respect to whichthe system is passive will in general not be the signal that we want to control� Forinstance� in mechanical systems it is velocity� and we are often interested in positioncontrol� In electrical machines it will be current� which is nonlinearly related totorque� which is the output we want to control�

Secondly� imposing a desired behaviour to the passive output will only be a pre�liminary step of the control design� we have in addition to de�ne the shape of thissignal that will ensure the output we want to control behaves as desired� e�g�� thecurrents that will generate the desired torque� In this second step of the design weare confronted with the complexity of the nonlinear dynamics which can be dauntingeven restricted to EL systems� It su�ces to say that� as pointed out in Section � ofChapter �� and thoroughly discussed in �$!�� the dynamics of a simplied model ofthe induction machine coincides �up to a linear drift term with the nonholonomicintegrator of Brockett� which has enticed control theory researchers for many years�

� What is passivity�based control�

The term PBC was �rst introduced in � � � to de�ne a controller methodology whoseaim is to render the closedloop passive� This objective seemed very natural withinthe context of adaptive control of robot manipulators� since as shown in that paperthe robot dynamics de�nes a passive map� and it had been known since the earlywork of Landau ��#�� that parameter estimators are also passive� �See Section $� fora brief history of passivity and stabilization of nonlinear systems�

The PBC approach presented in this book may be viewed as an extension of theby now wellknown energyshaping plus damping injection technique introduced tosolve statefeedback set point regulation problems in fully actuated robotic systemsby Takegaki and Arimoto in � �� For this particular problem we can concentrateour attention on the potential energy and the dissipation functions and proceed alongtwo basic stages� First an energy shaping stage where we modify the potential energyof the system in such a way that the �new� potential energy function has a globaland unique minimum in the desired equilibrium� �It is clear that� similarly to thechoice of a Lyapunov function� no systematic procedure exists to select the desiredpotential energy� The basic di�erence is that inspiration comes now at least forEL systems from physical considerations� Second� a damping injection stage wherewe now modify the dissipation function to ensure asymptotic stability� In the case

�See the pendulum example of Chapter � for a detailed discussion of these points�

� What is passivity�based control� ��

considered in � �� the assumption of full actuation allows us to assign any arbitrarypotential energy function� while the availability of the fullstate trivializes the task ofdamping injection� The controller resulting from this technique is a simple PD law�

Viewed from the broader PBC perspective pursued in this book the energy shapingstage accomplishes a passivation objective with a desired storage function that consistsof the original kinetic energy and the new desired potential energy� The dampinginjection reinforces this property to output strict passivity� Finally� Lyapunov stabilityfollows from the inputoutput stability of the output strictly passive map providedsome dissipation propagation �i�e�� detectability conditions are met�

This basic �energy shaping plus damping injection methodology is extended inthis book in several directions� �rst� we consider output feedback and underactuatedsystems� second� we address also tracking problems� �nally� all these more realisticcases are considered for a broader class of EL systems which contains� besides me�chanical� electrical and electromechanical systems� In carrying out these extensionsseveral additional fundamental di�culties which require nontrivial modi�cationsappear�

� When the state is not available for measurement the damping must be addedvia a dynamic extension�

� In underactuated cases the systems potential energy cannot be cancelled� but ithas to be now dominated� If furthermore we have to modify the kinetic energy�the controller will �rst be de�ned implicitly and an additional �inversion� stepwill be needed to obtain an explicit realization�

� In all cases� besides regulation of mechanical systems� the kinetic energy playsa role in the control task� it must therefore also be �shaped�� In terms ofpassivation this is tantamount to saying that the desired storage function thatwe want to assign to the closedloop cannot be simply chosen as the sum of thesystems kinetic energy and a new potential energy as in � �� We will typicallyselect the desired storage function from the consideration of an error dynamicswhere� thanks to a suitable factorization of the system �workless forces� weobtain linearity with respect to the error signal�

To solve all these new challenging problems we often make appeal to feedbackinterconnection decompositions and strongly rely on the invariance properties of ELsystems and passivity discussed above�


� Some historical remarks

�� Euler Lagrange systems and nonlinear dynamics

Nonlinear dynamics in the guise of planetary motions� has some claim to be the mostancient of scienti�c problems� The celebrated names of Newton� Kepler� Cartan�Hamilton and Dirac �to mention a few are attached to the development of classicalmechanics as a primary motivation for the study of dynamical systems and their prop�erties� As usual for many other important developments in nonlinear control theory�credit goes to Brockett for the introduction of controlled Hamiltonian systems withinputs and outputs�� Controllability and observability theories were later developedby van der Schaft� Parallels with the linear systems theory were obtained in real�ization theory� where the �existence and uniqueness� of realizations has been provedconclusively for suitably �smooth� systems by Crouch and coworkers� The specialnature of mechanical systems for control purposes was �rst exploited in robotics�whose history is reviewed in Remark �� and Section ��! of Chapter �� The studyof mechanical control systems has also provided some interesting case studies� Forinstance� the spacecraft attitude control problem is an excellent test bed for con�trollability of nonlinear control systems� Another more recent example is the classof mechanical control systems� whose underlying plant is characterized as �nonholo�nomic�� which clearly exhibited limits on the extent to which intuition about linearsystems can be transfered to the nonlinear domain� It also illustrated the practicale�ect of Lie brackets in the synthesis of controllers for nonlinear systems and the useof �atness for trajectory planning�

�� Passivity and feedback stabilization

The concept of passivity has played a prominent role in many areas of systems theoryfor many years now� See� for instance� the introduction to Chapter ! of � �� foran authoritative answer to the question Why are passive �positive� operators impor�tant � One of the early connections between passivity and stability is due to Youla� �� who proved that a passive network in closedloop with a resistive element is L�

stable� meaning by this that �nite energy inputs will be mapped into �nite energyoutputs� There are also many scholar books that cover the subject of passivity� ormore generally inputoutput theory� and contain detailed descriptions of its history�We recommend in particular the seminal books � �� and �� for encyclopedic cover�age� Viewed from a more recent perspective we have � $ � and � !$�� See also � !�for a brief history of this topic�

Interestingly� one fundamental early connection between passivity and stability

�See �� for a more detailed account on controlled mechanical systems� including the referencescited below�

�� Passivity and feedback stabilization ��

in nonlinear systems was obtained via optimal control in ��$!� This connectionhas a surprising� and somehow little well known history which is described in � !��and we repeat here for ease of reference� In �� the authors solved an inverseoptimal regulator problem for nonlinear systems and establish as a corollary thatoptimal systems de�ne passive maps� This is the nonlinear extension of the celebratedKalman�s inverse optimal control paper�

A far reaching implication of this result is that passivity provides a criterion fordeciding the optimality or otherwise of a feedback loop� An o�spin of this funda�mental result is the nonlinear KalmanYakubovichPopov �KYP lemma of �� !��which has triggered so much interest in the recent years� To the best of the authors�knowledge �� is the �rst paper where the important concepts of stabilization� ex�istence of Lyapunov functions and optimality are shown to be closely connected viapassivity� An early� and quite modest� attempt to explore these connections in anadaptive stabilization problem was reported in �� in � �� The recent book � !$�further investigates applications of inverse optimality and passivity for stabilizationof nonlinear systems�

In recent years passivity� and more speci�cally feedback passivation� has beenused to reformulate� in an elegant and unifying manner� the fundamental problem offeedback stabilization of nonlinear systems in �#�� The history of this result �whichonce again is retraced in � !� is intertwined with the history of backstepping� awidely popular stabilization technique for nonlinear systems that we also discuss inthis book� We refer the reader to Chapter ! of ��# � for a detailed description of thehistory of backstepping and to the authoritative survey paper �� for an overviewon nonlinear stabilization� For the sake of completeness we give here some elementsmissing in ��# �� As pointed out in ��$� backstepping has also roots in the work onadaptive control of Feuer and Morse in ��$�� The term �integrator backstepping�had not been coined at that time but all the elements of the technique� including thenonlinear damping� may be found there� In the last section of this important paper itis explained how the prototypical problem� whose solution triggered the reemergenceof backstepping in nonlinear stabilization problems� can be solved with the techniquesadvanced by Feuer and Morse�

One �nal apostille in this brief historical review is that� to the best of our knowl�edge� the �rst attempt to use feedback passivation for stabilization was made in � ��and � � �� where the work of �� and the nonlinear KYP lemma of �� !� are usedas design tools for adaptive stabilization of non�feedback linearizable� but passi�able�nonlinear systems�

Chapter �

Euler�Lagrange systems

�All happy families �linear systems� are alike� every unhappyfamily �nonlinear system� is unhappy �nonlinear� in its own way�

L� Tolstoi

It has been argued in the Introduction that a good starting point to develop apractically meaningful nonlinear control theory is to specialize the class of systemsunder consideration� The main reason being� of course� that the vast array of nonlin�ear systems renders futile the quest of a monolithic theory applicable for all systems�In particular� it de�es the approach of mimicking the� by now fairly complete� lineartheory� Specializing the systems� on the other hand� introduces additional constraintsand structure� which may enable otherwise intractable problems to be answered� Inthis chapter we describe the class of systems that we will consider throughout thebook and which we call EulerLagrange �EL systems� The most important reasonfor singling out the study of EL systems is that they capture a large class of contem�porary engineering problems� specially some which are intractable with linear controltools� Finally� by restricting ourselves to systems with physical constraints we believewe can contribute to reverse the tide of ��nd a plant for my controller� which stillpermeates most of the research on control of general nonlinear systems�

What is an EL system" To answer this question we will borrow inspiration fromthe de�nition of adaptive control quoted in the seminal book of ,Astr-om and Wit�tenmark ��!� �i�e�� An adaptive system is a system that has been designed with anadaptive viewpoint�� Hence we will say� An EL system is a system whose motion isdescribed by the EL equations� The logical question which arises next is �What arethe EL equations"� From a purely mathematical viewpoint they are a set of nonlinearordinary di�erential equations with a certain speci�c structure� A far more interestingquestion is �Where do they come from"� In contrast to the �rst two questions� the an�swer to the latter is far from simple and involves principles of minimization� calculusof variations and other tools from analytical dynamics� For the purposes of this bookthe EL equations are important because they are the outcome of a powerful modeling

��

�� Ch� � Euler�Lagrange systems

technique the variational method which describes the behaviour of a large classof physical systems� There are several excellent textbooks on variational modelingboth for mechanical �� $� and electromechanical systems ��$�� We refer theinterested reader to these books to prove further� For the sake of self�containment wehave summarized in Appendix B some of the relevant elements of this fundamentaltheory�

We thus start this chapter by describing the EL equations and introducing thenotation used throughout the remaining parts� EL systems models are� roughly speak�ing� obtained from the minimization of an energy function� It is therefore expectedthat they enjoy some energy dissipation properties� in particular that they de�nepassive maps� that one can pro�tably use for the controller design� These� as well assome other interconnection features and Lyapunov stability properties of EL systemsare also reviewed in this chapter�

� The Euler�Lagrange equations

In modeling physical systems with lumped parameters two basic approaches have beentypically used� derivation of the equations of motion using forces laws or applicationof variational principles to selected energy functions� For simple systems having onlyelements of the �same nature� the �rst approach is usually su�cient� For instance�for purely mechanical or electrical systems� Newton�s second law and Kirchho��s lawsrespectively� yield the desired equations� This method can still be applied for systemshaving �mixed natures�� e�g�� having both electrical and mechanical portions� Inthis case� the forces of interaction can be obtained invoking the method of arbitrarydisplacements and conservation of energy� This approach requires a lot of insight� andmuch bookkeeping must be done in complicated problems� In order to develop in asystematic manner the equilibrium equations a more general formulation is required�this is the aim of the variational approach� �See Appendix B�

The common link between the di�erent subsystems is that all of them transformenergy� Therefore� it seems natural to formulate the modeling problem in terms ofenergy quantities� The starting point of the variational approach to modeling is thede�nition of the energy functions in terms of sets of generalized variables �typicallypositions and charges for mechanical and electrical systems� respectively� this proce�dure leads to the introduction of the Lagrangian function� The equations of motionare then derived invoking wellknown principles of analytical dynamics� in particularthe fundamental Hamilton principle� which roughly speaking states that the systemmoves along trajectories that minimize the integral of the Lagrangian� The varia�tional modeling method is one of the most powerful techniques of dynamics� As wewill show throughout the book this method is particularly suited for PBC since itunderscores the role of the interconnections between the subsystems and provides uswith the storage and dissipation functions� which are the cornerstones of the PBC

The Euler�Lagrange equations ��

design technique�

As a �rst step towards the development of PBC� we have reviewed in AppendixB some of the relevant elements of analytical dynamics� and highlighted the role ofenergy and Rayleigh dissipation functions in the model development� Hamilton�s prin�ciple and its application for the derivation of the equations of motion for dynamicalsystems are also presented there�

We have discussed in Appendix B that the resulting dynamics can be described bythe EL or the Hamiltonian equations of motion� In this book we favor the Lagrangianrepresentation� Although� for the purposes of the material contained in this book�there is no essential di�erence between the two approaches� we believe that in theEL equations it is easier to reveal some structural aspects of the workless forces �e�g��Coriolis terms in robots manipulators� which is somehow obscured in the Hamilto�nian model� As we will see later a suitable factorization of the workless forces is anessential step for the application of the PBC methodology in most tasks� includingtrajectory tracking of mechanical systems and regulation or tracking for electrical andelectromechanical systems� See also � �� for further motivation of this choice�

We have shown in Appendix B that an n degrees of freedom dynamical systemwith generalized coordinates q � IRn and external forces Q � IRn� is described by theEL equations

d

dt

��L� 'q

�q� 'q

�� L

�q�q� 'q � Q� � ��

where

L�q� 'q �� T �q� 'q� V�q � �

is the Lagrangian function� T �q� 'q is the kinetic energy �or coenergy function whichwe assume to be of the form

T �q� 'q ��

'q�D�q 'q� � �!

where D�q � IRn�n is the generalized inertia matrix that satis�es D�q � D��q � �and V�q is the potential function which is assumed to be bounded from below thatis� there exists a c � IR such that V�q � c for all q � IRn�

Throughout the book we will consider �unless otherwise speci�ed three types ofexternal forces� the action of controls� dissipation and the interaction of the systemwith its environment� We assume controls enter linearly� as Mu � IRn� where M �IRn�nu is a constant matrix and u � IRnu is the control vector� Dissipative forces

�In Part II of the book we will study a class of electrical systems where the control does not enteradditively�

� Ch� � Euler�Lagrange systems

are of the form ��F� �q� 'q� where F� 'q is the Rayleigh dissipation function which by

de�nition satis�es

'q��F� 'q

� 'q � � �#

In summary we have the external forces

Q � ��F� 'q

� 'q &Q� &Mu � ��

where Q� is an external signal that models the e�ect of disturbances�

As explained in the introduction� in PBC the control objective is achieved byimposing to the closedloop dynamics a certain passivity property� which in its turnreduces to assigning some desired storage and dissipation functions� It comes naturalthen to de�ne EL systems in the following� admittedly redundant� way

De�nition �� EL equations and EL parameters�� The EL equations of mo�tion

d

dt

��L� 'q

�q� 'q

�� L

�q�q� 'q &

�F� 'q

� 'q �Mu&Q� � � ��

with �� de�ne an EL system which is characterized by its EL parame�ters �

fT �q� 'q�V�q�F� 'q�M� Q�g�

The use of the EL parameters as de�ned by the quintuple above captures a fairlygeneral notation which we use throughout this book� However� when clear from thecontext� we may use the more compact notation fT �q� 'q�V�q�F� 'q�Mg for systemsfor which Q� � or fT �q� 'q�V�q�F� 'qg in the case when only dissipative forcesa�ect the EL system�

The matrix M is a full column rank matrix relating the external inputs to thegeneralized coordinates� We �nd thus convenient to distinguish two classes of ELsystems according to the structure of this matrix�

De�nition �� Underactuated EL systems�� An EL system is fully�actuated ifit has equal number of degrees of freedom than available control inputs �that isif n � nu� e�g� if M � In � Otherwise� if nu � n we say that the system isunderactuated� In the latter case� q can be partitioned into non�actuated M�q andactuated components Mq� where M� denotes the perpendicular complement of M�

A second classi�cation that we �nd convenient to introduce at this point� involvesthe presence of damping� We can thus distinguish two classes of systems�

� Input�output properties �

De�nition �� Underdamped and fully�damped systems�� The EL system�� is said to be fully�damped� if the Rayleigh dissipation function satis�es

'q��F� 'q

� 'q �nXi �

�i 'q�i

with �i � for all i � n�� f�� ng� It is� on the other hand� underdamped if

i � n such that �i � �

Remark �� Generality of the model� In most of the practical cases we willassume that the Rayleigh dissipation function is quadratic �this models� e�g�� linearfriction or constant resistances as

F� 'q��

�

'q�R 'q

with R � R� � and diagonal� Fully damped and underdamped EL systems corre�spond to R being positive de�nite or only positive semide�nite� respectively� Also�to simplify the presentation we will assume in the sequel a special structure of M� Itwill become clear later that these assumptions are not essential for the basic devel�opments�

� Input�output properties

As discussed in the introduction� the inputoutput approach to systems analysisprovides a natural generalization �to the nonlinear timevarying case of the fact thatstability of an LTI feedback system depends on the amount of gain and phase shiftinjected into the loop� Furthermore� and perhaps more importantly� the measuresof signal ampli�cation �the operator gain and signal shift �its passivity can be insome instances associated to physical quantities� The inputoutput approach is alsoconsistent with one of the important viewpoints of control theory that complicatedsystems are best thought of as being interconnections of simpler subsystems� Thisaggregation procedure has two important implications� On one hand� it help us tothink in terms of the structure of the system and to realize that more often than notthe pattern of the interconnections is more important than the detailed behaviourof the components� which can be characterized by an inputoutput property� e�g�its passivity� On the other hand� it yields a designoriented methodology since itallows us to isolate the controller as a �free� subsystem� As we will show in the bookboth features of the inputoutput approach are clearly illustrated when applied toEL systems�

�This class of systems is sometimes called pervasively damped in analytical mechanics and struc�tural vibration�


In this important section we establish the main tools that will be used for our PBCdesign� They concern not just the passivity of EL systems� but more importantly theirinterconnection properties� In the present book we e�ectively exploit these propertiesof EL systems for the design of PBC in three di�erent ways�

� We prove that EL systems de�ne passive maps� This is the most fundamentalproperty for PBC since it identi�es the output which is �easy� to control andprovides a storage function that will typically motivate the desired closedloopstorage function� Actually� we will prove a stronger socalled �skewsymmetry�property that essentially says that the dynamics of EL systems can be treatedas a set of linear double integrators� The proof of these properties has aninteresting intermingled story which is brie�y reviewed in Remark �� and inSubsection ��!�

� To simplify the controller design we introduce a decomposition of the EL systemdynamics into the feedback interconnection of two passive subsystems� We giveconditions under which this decomposition is possible� these include some casesof practical interest� in particular in electromechanical systems� This propertywas �rst established in �$�� and� as shown in Chapter � is instrumental for thesolution of the global tracking problem of electrical machines�

� We view the action of the controller as an EL�structure�preserving intercon�nection� In this way the properties of the closedloop are still captured by theenergy and dissipation functions� which are simply the sum of the correspondingfunctions of the plant and the controller� This property� which was �rst reportedin � �� will be exploited in Chapter � to design PBC� which are themselvesEL systems� for output feedback regulation of mechanical systems�

� Passivity of EL systems

Passive systems are a class of dynamical systems in which the energy exchangedwith the environment plays a central role� As we have seen in Appendix A in passivesystems the rate at which the energy �ows into the system is not less than the increasein storage� In other words� a passive system cannot store more energy than it issupplied to it from the outside� with the di�erence being the dissipated energy� Theproposition below proves that EL systems de�ne passive maps�

Proposition �� The EL system �� with Q� � de�nes a passive operator . �u �� M� 'q with storage function the systems total energy� H�q� 'q� That is�

hu j M� 'qiT � H�q�T � 'q�T ��H�q� � 'q� � � �$

�Recall from Appendix A that a storage function need only be positive semide�nite and boundedfrom below� thus not necessarily zero for zero argument�

�� Passivity of EL systems ��

for all T � and all u � Lm�e� Further� this property is strengthened to output strict

passivity �OSP if the system is fully damped� In this case

hu j M� 'qiT � �kM� 'qk��T &H�q�T � 'q�T ��H�q� � 'q� � � ��

for some � � and all u � Lm�e� �

Proof� The property can be established taking the time derivative of the Lagrangianfunction L�q� 'q� where for simplicity in the notation we drop the arguments�

dLdt

�

��L�q

��dq

dt&

��L� 'q

��d 'q

dt� ��

and using the EL equations � �� to write

�L�q

�d

dt

��L� 'q

��Q�

so � �� can be rewritten as

dLdt

�

��L� 'q

�� d 'q

dt&

d

dt

��L� 'q

��'q � 'q�Q

then� reordering the terms above and using � �� it follows that

d

dt

��L� 'q

��'q � L

�� 'q�

�Mu� �F

� 'q

��

Now� notice that the term in parenthesis on the left hand side coincides with thesystems total energy� which we denote by H�q� 'q� that is�

�L�q� 'q� 'q

��'q � L�q� 'q � T �q� 'q & V�q �� H�q� 'q

Integrating � �� from to T we establish the key energy balance equation

H�q�T � 'q�T ��H�q� � 'q� �� z stored energy

&

Z T

�

'q��F� 'q

� 'qds� �z

dissipated

�

Z T

�

'q�Muds� �z supplied

��

Now� observe that� since V�q is bounded from below by c� and T �q� 'q � we havethat H�q� 'q � c� Finally� the Rayleigh dissipation function satis�es � �# hence � �$follows�

If the system is fully damped it follows immediately from De�nition �! and � ��

that � �� holds with �� minif�ig

kMk� �

�


Remark �� The energy balance equation � �� reveals several interesting proper�ties of EL systems� First� if we set u � we see that energy is not increasing� hence thetrivial equilibrium of the unforced system is stable in the sense of Lyapunov� Needlessto say that these considerations constituted the starting point of Lyapunov�s originalwork� Second� stability is also preserved if we now �x the output M 'q to zero� hencere�ecting the fact that the system is minimum phase�� Thirdly� we see that dampingcan be easily added along the actuated channels if 'q is measurable� Notice� however�that the operator u �� M� 'q may be output strictly passive even if energy is not dis�sipated �in all directions�� Namely� it is enough to ensure 'q� �F� �q�

� �q� �kM� 'qk�� This

property will be used in Chapter � to solve output feedback stabilization problems�

� Passivity of the error dynamics

The passivity property described above is su�cient to solve regulation tasks in me�chanical systems� where the PBC only needs to modify the potential energy and thedissipation function� However� to study tracking problems or treat electrical or elec�tromechanical systems� we need a stronger property� The main reason� that will beexplained in detail later� is that in these cases a desired behaviour should be imposed�not only on q� but on 'q as well� which in its turn translates into the need for modifyingthe kinetic energy�

Leaving aside the problem of regulation in mechanical systems �which is basicallythe only case that has been documented in the literature� and we will only very brie�yreview it here throughout the book we will see that a �rst step in PBC may be toachieve a closedloop dynamics of the form

D�q 's& �C�q� 'q &Kd�q� 'q�s � � ��

where s denotes an error signal that we want to drive to zero� Kd�q� 'q � K�d �q� 'q �

is a damping injection matrix� and C�q� 'q is a matrix� univocally de�ned by D�q�which satis�es

'D�q � C�q� 'q & C��q� 'q� � ��!

Notice that � ��! is equivalent to the skewsymmetry property�

z�� 'D�q� C�q� 'q�z � � � z � IRn

The motivation for aiming at � �� stems from the following key lemma from � � ��

Lemma �� The di�erential equation

D�q 's& �C�q� 'q &Kd�q� 'q�s � /

�Actually stability and inverse stability hold true for general passive systems��As explained in �� no matter how C�q� �q� is de�ned in C�q� �q� �q it is always true that �q� �D�q��

�C�q� �q�� q � �� This� however� does not imply ��

�� Passivity of the error dynamics ��

where D�q and Kd�q� 'q are positive de�nite and C�q� 'q satis�es �� de�nes anoutput strictly passive operator .d � / �� s� Consequently� if / � we have s � L��

Proof� The proof follows using the storage function

Hd ��

s�D�qs � � � ��#

By di�erentiating � ��# with respect to time and using the skew�symmetry of 'D�q� C�q� 'q we obtain that 'Hd �Kd�q� 'qs

� & /�s� The OSP property follows byintegrating on both sides of this inequality from to T � The second part of the prooffollows from the fact that OSP systems are L�stable �see Corollary A�� Henceforth�with / � we have s � L��

The skewsymmetry property � ��! is essential for the lemma above� The lemmabelow� �rst published in � � � �but implicit in the work of ��!!�� establishes this fact�

Lemma �� Let the ik�th entry of the matrixC�q� 'q �called in the robotics literaturethe �Coriolis and centrifugal forces� matrix be given by

Cik�q� 'q �nXj

cijk�q 'qj�

where

cijk�q��

�

��dik�q

�qj&�djk�q

�qi� �dij�q

�qk

�are the so called Christo�el symbols of the �rst kind� Then� �� holds� �

Using the special factorization described above the EL equations � �� can bewritten in the equivalent form

D�q-q & C�q� 'q 'q & g�q &�F� 'q

� 'q�Mu&Q� � ��

where

g�q��

�V�q

�q

which is called in robotics the gravity forces�

Remark �� The passivity property presented in Proposition �� and the �skewsymmetry� property� although clearly closely related� are fundamentally di�erent�Attaching the basic principle of passivity to the dynamics is a far reaching systemstheoretic concept that goes beyond the mathematical convenience of a �suitable fac�torization�� The establishment of passivity is at the core of the whole PBC methodol�ogy� which started with its application in robot control� To the best of our knowledge�


the property was �rst explicitly expressed in terms of passivity in the conference ver�sion of �� #� in �� for the case when the potential energy is absent� The completeproof with the potential energy term appeared �rst in � � �� The quali�er �PBC� wascoined in the latter paper� and has since enjoyed widespread popularity� On the otherhand� the �skewsymmetry� property was �rst obtained by ��!!�� Both propertieshave played a fundamental role in the development of energy shaping plus dampinginjection controller designs� which have a parallel interesting history� We refer thereader to Subsection ��! for further historical precisions�

�� Other properties and assumptions

In this book we focus our attention on those systems for which the following propertiesand assumptions hold

P�� The system � �� can be parameterized as

D�q-q & C�q� 'q 'q & g�q &�F� 'q

� 'q� 0�q� 'q� -q��

where � � IRp is a vector of constant parameters and 0�q� 'q� -q � IRq�n is calledregressor matrix�

A�� The matrix D�q is symmetric positive de�nite and there exist some positiveconstants dm and dM such that

dmI � D�q � dMI � ��$

A�� There exists some positive constants kg and kv such that

kg � supqIRn

��V�q�q�

� ��

kv � supqIRn

�V�q�q

� ��

P�� The matrix C�x� y is bounded in x and linear in y� then for all z � IRn

C�x� yz � C�x� zy � �

kC�x� yk kckyk� kc � � � � �

As a matter of fact� � � is a direct consequence of the de�nition of C�q� 'q�Also� inequality � � � follows using � ��$ and the de�nition of Christo�el symbols�

�� Passive subsystems decomposition ��

�� Passive subsystems decomposition

In some practical cases the generalized inertia matrix is block diagonal� expressingthe fact that the angular part of the kinetic energy of each subblock is due onlyto its own rotation� See Section #�! for a discussion of this point in robots with�exible joints� and Chapter for its application in electrical machines� For this classof systems we have the following nice property�

Proposition �� Assume the Lagrangian �� can be decomposed in the form

L�q� 'q � Le�qe� 'qe� qm & Lm�qm� 'qm

where q�� q�e � q

�m�� with qe � IRne and qm � IRnm� Then� the EL system �� can be

represented as the negative feedback interconnection of two passive subsystems �asshown in Fig� ��

.e �

�Qe

� 'qm

��

�'qe

�

.m � � &Qm �� 'qm

with storage functions Le�qe� 'qe� qm and Lm�qm� 'qm� respectively� where

��

�Le

�qm�qe� 'qe� qm

is the subsystems coupling signal� and Q�� Q�

e � Q�m�� with Qe � IRne� Qm � IRnm �

�

�

��

�

�

� �

�

.e

.m

Qe 'qe

Qm'qm

Figure �� Feedback decomposition of an EL system�


Proof� Splitting � � into its components we get

d

dt

��Le

� 'qe

�� Le

�qe� Qe � �

d

dt

��Lm

� 'qm

�� Lm

�qm� Qm & � � !

Evaluating the total time derivative of Le results in

'Le �

��Le

�qe

��'qe &

��Le

� 'qe

��-qe &

��Le

�qm

��'qm � � #

Noting that

��Le

� 'qe

��-qe �

d

dt

��Le

� 'qe

��'qe

�� d

dt

��Le

� 'qe

��'qe

inserting this into � � #� using � � and rearranging the terms� it follows that

d

dtHe � Q�

e 'qe � 'q�m

where� similarly to the proof of Proposition �� we have de�ned

He�qe� 'qe� qm��

��Le

� 'qe

��'qe � Le

which is the total energy of the subsystem .e� Passivity of .e follows� as done above�integrating from to T�

A similar procedure can be used to establish the passivity of .m� using the energy

function Hm � �L�m� �qm

'qm � Lm� �

�� An EL structure�preserving interconnection

We have explained in the Introduction that the aim of PBC is to achieve a desiredpassive map in closedloop� Now� we have seen in Proposition �� that EL systemsde�ne passive operators� while in Proposition A�� of Appendix A we mentioned thatpassivity is invariant under feedback interconnection� Putting all these pieces togethermotivates us to look for our PBC among the class of EL systems� Interestingly enough�if we restrict ourselves to this class we can de�ne a controller interconnection that notonly preserves the EL structure �and hence closedloop passivity� but furthermore and probably more signi�cantly the new storage and energy dissipation functions areobtained by adding up the corresponding functions of the plant and the controller� Aswe will see later� this essentially trivializes the energy shaping and damping injectionstages of the passivitybased design in regulation tasks for mechanical systems�

� Lyapunov stability properties ��

Proposition �� Interconnected EL systems�� Consider two EL systems .p �fTp�qp� 'qp� Vp�qp� Fp� 'qp� Mpg and .c fTc�qc� 'qc� Vc�qc� qp�Fc� 'qcg with general�ized coordinates qp � IRnp and qc � IRnc� respectively� �notice that the potentialenergy of .c depends on qp � Interconnect the systems via

Mpu � ��Vc�qc� qp�qp

where u is the input of the subsystem .p� See Fig� �� Under these conditions�the closed�loop system is an EL system . � fT �q� 'q� V�q� F� 'qg� with generalized

coordinates q�� q�p � q

�c �� and EL parameters

T �q� 'q � Tc�qc� 'qc & Tp�qp� 'qp V�q � Vc�qc� qp & Vp�qp�

F � Fc� 'qc & Fp� 'qp�

�

qp

.p

.c

��V�qc�qp��qp

Figure � � Feedback interconnection of two EL systems�

� Lyapunov stability properties

In this section� we stress other properties of EL systems which are related to theirstability in the sense of Lyapunov� For the sake of clarity� we separate the results forfully�damped and underdamped systems�

�� Fully�damped systems

The proposition below is a well�known result that establishes conditions for internalstability of fully damped EL systems� It was stated by Joseph L� La Grange �c� �$��and later proved by Dirichlet� The proof is typically established invoking La SalleKrasovskii�s lemma� To underscore the role of passivity and detectability we give

�See �� for an interesting historical review on the important question of stability of mechanicalsystems including several converse versions of this theorem�


below a proof that combines the strict passivity property established in Proposition �� with Proposition A��

Proposition �� GAS with full damping�� The equilibria of a fully dampedunforced EL system �i�e�� with u � Q� � are �q� 'q � �1q� � where 1q is the solutionof

�V�q�q

� � � � �

The equilibrium is stable if 1q is a strict local minimum of the potential energy functionV�q� Furthermore� if V�q is proper �for instance if V�q satis�es the conditions ofLemma C�� and the minimum is unique� then this equilibrium is GAS� �

Proof� The existence of the equilibrium follows immediately writing the EL equa�tions with u � Q� �

D�q-q &N�q� 'q &�V�q�q

&�F� 'q

� 'q� � � �

where in order to simplify the notation we de�ned

N�q� 'q��

d

dtfD�qg � �T �q� 'q

�q� � $

which satis�es N�q� � � hence noting that �F� �q�� q

j �q � � it follows that the equi�

libria 1q are the solutions of � � ��

We will now verify the conditions of �ii of Proposition A�� We have shownalready in Proposition �� that� for fully damped EL systems� the map u �� 'q is OSPwith storage function the total energy� In view of the assumption on the potentialenergy� the total energy is bounded from below� and we can simply add a constantto make it positive de�nite� To prove stability it only remains to show that the ELsystem is zerostate detectable actually� it will be shown to be zerostate observable�This follows immediately setting 'q � in � � � and noting that this implies that�V�q��q

� � Finally� GAS results from the assumptions of uniqueness of the equilibrium

and that V�q is proper� �

�� Underdamped systems

In the proposition below we show that global asymptotic stability of a unique equilib�rium point can still be ensured even when the system is not fully damped providedthe inertia matrix D�q has a certain block diagonal structure� and the dissipation is

�An obvious change of coordinates is needed to shift the equilibrium to the origin�

�� Underdamped systems �

suitably propagated� Even though other� more general� versions of this theorem canbe easily proved� we give here the one that will be needed in the sequel� It is inter�esting to remark that� as far as we know� the �rst paper which established su�cientconditions for asymptotic stability of underdamped EL systems � �� was publishedmore than !� years ago�

Our motivation to present this result stems from its application for PBC with ELsystems which will be presented in Chapter �� For this case it is natural to partitionq as

qc�� Inc�q� qp

�� Inp �q� n � np & nc� � � �

to distinguish the damped and undamped coordinates� Our motivation to use thesubindices ��c and ��p� which suggest controller and plant� respectively� will becomeclear in Chapter ��

Proposition �� GAS with partial damping�Consider an unforced underdamped EL system with the coordinate partition �� The equilibrium � 'q� q � � � 1q is GAS if the potential energy function is proper andhas a global and unique minimum at q � 1q� and if

�i D�q��

�Dp�qp

Dc�qc

�� where Dp�qp � IRnp�np and Dc�qc � IRnc�nc�

�ii 'q� �F� �q�� q

� �k 'qck� for some � � �

�iii For each qc� the function �V�q��qc

� has only isolated zeros in qp� �

Proof� First� proceeding as is Proposition �� it clearly follows that the map u �� 'qcis OSP�

Now� as in Proposition �� we write the EL equations � �� with u � Q� � byexploiting the block diagonal structure of D�q� in the form

Dp�qp-qp &Np�qp� 'qp &�V�q�qp

� � � �

Dc�qc-qc &Nc�qc� 'qc &�F� 'q

� 'qc&�V�q�qc

� � �!

where Nc�qc� 'qc� Np�qp� 'qp are suitably de�ned vectors of the form � � $� The equi�libria are determined by the critical points of the potential energy function V�q� thatis� the solutions of �V�q�

�q� �

The stability proof is carried out� again� using Proposition A�� Hence� we needto verify only zerostate detectability with respect to the output 'qc� To this end� weset 'qc � � From qc � const in � �! and �iii it follows that qp is also constant�Since q � 1q is the unique global equilibrium of � � �� ! � zerostate detectabilityfollows� �


Remark �� It is interesting to notice that if we rewrite � � �� ! as a losslesssystem

Dp�qp-qp &Np�qp� 'qp &�V�q�qp

�

Dc�qc-qc &Nc�qc� 'qc &�V�q�qc

� u

in closedloop with the dissipation term

y ��F� 'q

� 'qcu � �y

we can prove Proposition ��! invoking zerostate observability properties and thefollowing interesting result�

Proposition �� Corollary �� of �� Consider a lossless system with a pos�itive de�nite and proper storage function H� The output feedback law

u � �ky� k � �

renders the origin of the closed loop system globally asymptotically stable if and onlyif the system is zero�state observable� �

Examples

In Appendix B we have shown that the Lagrangian formalism provides a very con�venient way of setting up the equations of motion� One simply has to write T andV in generalized coordinates and use � �� to derive the equations of motion� In thissection we give some examples of dynamical systems whose equations of motion canbe derived� using the EL equations� from their corresponding EL parameters�

�� A rotational�translational proof mass actuator

A translational oscillator with an attached eccentric rotational proof mass actuator�TORA is shown in Fig� �!� It consists of a cart of mass M connected by a linearspring with sti�ness k to a �xed wall� This system model was recently proposed as abenchmark problem for nonlinear system design� See �#�� for further details�

The cart has only one�dimensional motion parallel to the spring axis� The proofmass actuator attached to the cart has mass m and moment of inertia I around itscenter of mass� The latter is located at a distance l from its rotational axis� The

�� A rotational�translational proof mass actuator ��

gravitational forces are neglected because the motion occurs in an horizontal plane�The control torque applied to the proof mass is denoted by u�

ql

Im

Mk

f

q1

2

u

Figure �!� Rotational%translational proof mass actuator�

Let q� be the translational position of the cart and q� the angular position of theproof mass� where q� � is perpendicular to the motion of the cart� and q� � �

is aligned with the positive q� direction� The generalized coordinates are then q ��q�� q�� IR�� the kinetic and potential energy functions are de�ned as

T �q�� 'q ��

'q�D�q� 'q � 'q�

�M &m �ml cos�q�

�ml cos�q� I &ml�

�'q

V�q� � �

kq��

Applying the EL equations � �� with Q� � yields the model

D�q�

�-q�-q�

�&

� �ml 'q�� sin�q� 'q�

�&

�kq�

��

� u

�which can be written in compact notation as

D�q�-q & C�q�� 'q� 'q & g�q� �Mu � �!�

where M � � � �� g�q�� kq�� The system is clearly underactuated� with

actuated coordinate q�� and undamped� Further� it de�nes a passive operator u �� 'q��

Notice that� via the de�nition of the matrix

C�q�� 'q� �

� �ml 'q� sin�q�

�we have introduced the factorization of the vector� �ml 'q� sin�q� 'q�

�� C�q�� 'q� 'q

discussed in Proposition �� As pointed out there this factorization is uniquelyde�ned by the inertia matrix D�q� and constitutes a fundamental step� which revealsthe workless forces of the system� for all our further developments�


�� Levitated ball

u

i

y

g

m

λ

Figure �#� Ball in a vertical magnetic �eld�

Consider the system of Fig� �#� consisting of an iron ball in a vertical magnetic �eldcreated by a single electromagnet� We denote by qm the position of the ball measuredwith respect to the nominal position� with the qm�axis oriented upwards� and with 'qethe electric current in the inductance� To apply the variational modeling technique�we de�ne �rst the generalized coordinates� which in this case are q � �qe� qm�

�� If weassume linearity of the magnetic circuit and neglect all fringing �elds� we obtain themagnetic �eld coenergy and the mechanical kinetic energy as

Te�qm� 'qe � �

L�qm 'q

�e � Tm� 'qm � �

m 'q�m

where� L�qm is the inductance and m � the mass of the ball� A suitable approxi�mation for the former� in the open domain qm � �� c� where c� � is the nominalair gap� is given by

L�qm �c�

c� � qm

where c� is some positive constant that depends on the number of coil turns� airpermeability and the crosssectional area of the electromagnet� To simplify the pre�sentation in the sequel we will assume c� � c� � ��

We must also derive the potential energy which is given as V�qm � mg�� qm�The Rayleigh dissipation function is F� 'qe � �

�Re 'q

�e � where Re � is the electrical

resistance� The control u is the input voltage� therefore M � ��

�See Examples �� and �� of �� for a detailed derivation of this model�

�� Levitated ball ��

The resulting Lagrangian is

L�qm� 'qe� 'qm � Te�qm� 'qe & Tm� 'qm� V�qm � �

'q�D�qm 'q � V�qm

where we have introduced the generalized inertia matrix

D�qm��

�L�qm

m

�

Applying the EL equations � �� with Q� � yields��

��qm�

-qe &�

��qm��'qm 'qe &Re 'qe � u

m-qm � ��

��qm��

'q�e �mg �

These equations look messy� and as we have discussed above to reveal its fundamentalstructural features it is convenient to write them in compact form

D�qm-q & C� 'qe� qm� 'qm 'q &R 'q &G �Mu

where we have de�ned

C� 'qe� qm� 'qm��

��

�� qm�

�'qm 'qe� 'qe

�� R

��

�Re

�� G

��

�

�mg

�

This system is also underactuated� with actuated coordinate qe� and underdampedwith qe the damped coordinate� It de�nes a passive operator u �� 'qe� as expected ifwe view the system as a two port with a terminal voltage u and an input current 'qe�We will see in the following chapter that for the PBC design it is sometimes moreconvenient to use other passivity properties�

Notice we have again introduced a particular factorization of the vector

�

�� qm�

�'qe 'qm��

�'q�e

�� C� 'qe� qm� 'qm 'q

Admittedly� it is di�cult at this point to convince the reader of the advantage ofusing the variational approach to model this extremely simple system which can beeasily obtained from basic physical laws as�

'� � �Re�� qm�& um-qm � �

�� mg

where �� L�qm 'qe is the �ux in the inductance� The advantage will become apparent

when we will come to the controller design�


�� Flexible joints robots

�i� ��th joint

ith motorspringrotational

ith joint

Figure �� Ideal model of a �exible joint�

Some robotic manipulators have �exible joints� for instance� manipulators where har�monic drives� elastic bands� or motors with long shafts are used� The joint �exibilityphenomenon can be modeled by a rotational spring� see e�g� �� !� and referencestherein� as it is illustrated in Fig� ��

Flexible joint robots are underactuated EL systems with generalized coordinates

q�� q�� q

�� q�� q� � IR

n� being the link and motor shaft angles respectively� The

control variables are the torques at the shafts� thus m � n�and M �

� � j Im��The kinetic and potential energies of a �exible joint robot are given by see � ��

T �q�� 'q��

�

'q�D�q� 'q� V�q ��

q�Kq & Vg�q� � �!

where

K ��

�K �K�K K

�� D�q�

��

�D��q� D��q�D�

��q� J

�� !!

with D��q� of the form

D��q� �

��

d��q�� d��q�� q�� d�m�q�� q��m�� d��q�� d�m�q�� q��m��

��

� � ��

� � �

��

� �!#

and q��j� for j � ��m�� is the j�th component of the vector q�� Matrix D�q� �D��q� � is the robot inertia matrix� J � IRm�m is a diagonal matrix of actuatorinertias re�ected to the link side� K is a diagonal matrix containing the joint sti�nesscoe�cients� and Vg�q� is the potential energy due to the gravitational forces�

�� The Du�ng system ��

Assuming no internal damping� that is� F� 'q � we obtain the dynamic equationsof the �exible joint robot�

D�q�-q & C�q�� 'q 'q & G�q� &Kq �Mu � �!�

where G�q� �� g�p��q��

� � �Vg�q��q�

� Once again we have factored the second lefthand term in a suitable manner de�ning the Coriolis matrix�

C�q�� 'q ��

�C��q�� 'q� & C �

��q�� 'q� C��q�� 'q�C��q�� 'q�

�� !�

�C��i�j�q�� 'q��

�

�'q��D��j�i

�q�&��D��

i

� 'q�'q�

�� !$

where ��i�j� ��i denote the �i� j�th term and i�th row of a matrix respectively� Whenthe angular part of the kinetic energy of each rotor can be considered due only to itsown rotation then we obtain the simpli�ed model of � �!��

Dl�q�-q� & C�q�� 'q� 'q� & g�q� � K�q� � q�

J -q� &K�q� � q� � u�� !�

In the case where �exibility is negligible �K � � it is shown in � �!� that themodel � �!� reduces to the wellknown rigid robot model

D�q�-q� & C�q�� 'q� 'q� & g�q� � u � �!�

where D�q�� Dl�q� & J �

�� The Du�ng system

Consider a mechanical system with generalized coordinates q � IR and EL parameters

T � 'q ��

'q�

V�q ��

p�q

� &�

#p�

F� 'q ��

p� 'q

�

M � ��

where p� is a nonnegative constant and p�� p� are real� The direct evaluation ofthe Lagrangian equations using the EL parameters de�ned above yields the Du�ngequation

-q & p� 'q & p�q & p�q� � u � �#


where u is external force� If we set u � Q cos t to the right hand side of � �# weobtain the well known periodically�forced Du�ng equation which borrows the namefrom his creator� who used it in �� to study the dynamics of a pendulum movingin a viscous medium�

Notice that the Du�ng equation is similar to a common mass�spring�damper sys�tem except from the term p�q

�� As a matter of fact� the Du�ng equation also modelsthe motion of a mass�spring�damper system where the spring induces a restoring forcewhich obeys Hooks law �F � �kq only for small displacements� For large displace�ments however� the restoring force is given by the expression F � ��q�� such springis called �hardening�since after a certain limit a small displacement induces a largerestoring force�

�� A marine surface vessel

One of the simplest models of a marine surface vessel is the Lagrangian type model

M '� &R� � u& J��b � �#�

'� � J� � �#

whereM � IR�� is the ship�s mass �positive de�nite matrix�D is a damping constantmatrix� not necessarily positive de�nite but bounded� � is the vector of position andorientation of the vessel with respect to a �xed frame and � is the vector of velocitiesreferred to a mobile frame attached to the ship� The term b represents the in�uenceof external forces acting on the ship due to environmental disturbances�

The Jacobian J is an orthogonal rotation matrix� hence full rank� This propertyis fundamental since we can rewrite the equations � �#�� # in the familiar form

� � �� Let us write with an abuse of notation 'J�� d

dt�J�� Derivate � �# once

with respect to time to obtain '� � 'J� '� & J�-�� Then substituting this in � �#� andpremultiplying on both sides of the equality by J�� we may write�

D�q-q & C�q� 'q 'q & 1R�q 'q & g�q � u � �#!

where we have de�ned q��

D�q�� J�qMJ��q

C�q� 'q�� J�qM 'J��q� 'q

g�q�� b

u�� J�q

Further� since J�q is orthogonal then � �#! possesses the properties described inSection �!�

� Concluding remarks ��

It is interesting to remark that even though there is no gravitational energy as inthe case of a manipulator� one can think of the bias b as derived from of a conservativepotential� in other words� the energy of the weather perturbations�

Finally� we point out that the model � �#�� # is most appropriate for setpointregulation purposes� However� for trajectory tracking control additional dynamicsmust be taken into account� such as the added mass of the water as the ship moveson the surface� For further detail on this and other Lagrangian models of ships werefer the reader to ��

Concluding remarks

In this chapter we have de�ned EL systems� which is the class that we will considerthroughout the book� We have also recalled some important properties of EL sys�tems that will be used in the sequel in our control design� These properties can besummarized as follows�

� EL systems are characterized by their EL parameters� Kinetic energy� Potentialenergy� Rayleigh dissipation function� Inputs matrix and Disturbance signal�

� EL systems de�ne passive operators with storage function the total energy�

� The EL dynamics can be parameterized so as to generate a linear error dynamicswhich is output strictly passive �OSP�

� The feedback interconnection of two EL systems yields an EL system� TheEL parameters of the resulting system are simply the addition of those of bothsubsystems�

� Under suitable conditions� EL systems can be decomposed as a feedback inter�connection of passive subsystems�

� The stable equilibria of an EL system correspond to the minima of its potentialenergy function�

� EL systems are asymptotically stable if they have a suitable damping�

Part I

Mechanical Systems

!�

Chapter �

Set�point regulation

In the previous chapter we underlined several fundamental properties of EL systems�In particular we saw that the equilibria of an EL plant are determined by the crit�ical points of its potential energy function� moreover the equilibrium is unique andglobally stable if this function has a global and unique minimum� We also saw thatthis equilibrium is asymptotically stable if suitable damping is present in the sys�tem� These two fundamental properties motivated Takegaki and Arimoto in � �� toformulate the problem of set point regulation of robots in two steps� �rst an energyshaping stage where we modify the potential energy of the system in such a way thatthe �new� potential energy function has a global and unique minimum in the desiredequilibrium� Second� a damping injection stage where we now modify the Rayleighdissipation function� This seminal contribution contained the �rst clear expositionof the use of energy functions in robotics� �See Subsection ��! for a brief reviewof the literature� It generated a lot of interest in the robotics community since itrigorously established that computationally simple control laws� derived from energyconsiderations� could accomplish rather sophisticated tasks�

Of course� the key property that underlies the success of this procedure is the pas�sivity of the EL system �Proposition �� Viewed from the broader PBC perspectivepursued in this book the energy shaping stage accomplishes a passivation objectivewith a storage function that contains the desired potential energy� The damping in�jection reinforces this property to output strict passivity� Finally� Lyapunov stabilityfollows from the inputoutput L� stability of the output strictly passive map providedsome detectability �i�e�� dissipation propagation conditions are met� This two stageprocedure will be used throughout the book with the fundamental di�erences that intasks� other than regulation of mechanical systems� we must also shape the kineticenergy of the system and an additional �system inversion� step will also be required�

In this chapter we study the problem of set�point regulation of mechanical systemsdescribed by the EL equations � �� We start by recalling the fundamental resultof � �� for fully�actuated systems with full state measurement� To underscore the

#�

�� Ch� !� Set�point regulation

main steps of this technique we carefully explain the basic steps of energy shapingplus damping injection in the classical pendulum example� For the application ofthis methodology there are two structural obstacles related with the actuators andsensors of the EL system� On one hand� if the system is not fully�actuated it is clearfrom � �� that we cannot assign an arbitrary potential energy function� On theother hand� it follows from the energy balance equation �and it has been pointed outalready in Remark �� that damping can be easily injected if the generalized velocitiesare available for measurement� but as shown in this chapter a dynamic extension isneeded otherwise� Cost considerations and the fact that velocity measurements areoften contaminated with noise are two clear motivations to look for regulators withoutvelocity feedback� Also� for some applications the assumption of full actuation is notrealistic� For instance� in robot manipulators with fast motions the existence of joint�exibilities signi�cantly a�ects the dynamic behaviour�

The main contribution of this chapter is the development of a class of globallystable output feedback PBC for underactuated mechanical systems� To solve thisproblem we exploit two additional properties of EL systems established in the previouschapter� First� the fact �pointed out in Proposition ��! that� for asymptotic stability�the damping needs not be pervasive �i�e�� the EL system need not be fully�damped�it su�ces that it propagates through the whole system coordinates� Second� the keyproperty that the feedback interconnection of two EL systems yields an EL systemwith added EL parameters �Proposition �� Since the latter property essentiallytrivializes the design we are motivated to choose the controllers as EL systems� thatis� the dynamics of our controllers is described by the EL equations�

This contribution is further extended to the case of constrained inputs� we thenidentify a subclass of EL systems which can be controlled by EL controllers whichyield control inputs satisfying an a priori imposed bound�

Finally� we explore the setpoint control problem of EL systems with uncertainpotential energy knowledge� We give particular importance to PID control and showhow this popular control law may be interpreted as an interconnection of passivesystems� In the case of unmeasurable velocities we add a second integral action to thePID� yielding a PI�D controller� which we also present from a passivity perspective�

We illustrate the results with the example of robots with �exible joints and theTORA system and conclude the chapter with some numerical simulations�

� State feedback control of fully�actuated systems

� A basic result� The PD controller

For the sake of clarity let us recall here the seminal result of � �� It concernsthe global asymptotic stabilization via energy shaping plus damping injection of the

� A basic result� The PD controller ��

equilibrium �q� 'q�� q�� with q� a constant vector� for the EL system � ��

Proposition �� Consider an n�degrees of freedom fully�actuated EL system withno internal damping nor external forces described by �� which for convenience ofpresentation we repeat below with Q� �

D�q-q & C�q� 'q 'q & g�q � u

where q � IRn and u � IRn is the vector of control inputs� and D�q� C�q� 'q and g�qare de�ned in Chapter �� hence satisfy the properties of Section ��

Let the state�feedback control law be given as

u � ��Vc�q

�q� �Fc

� 'q� 'q

and let us assume the following�

A�� The function Vc�q is such that the potential energy of the closedloop system

Vd�q �� V�q & Vc�qhas a unique global minimum at q � q� �a constant and is radially unbounded�with respect to q � q��

A�� The dissipation function Fc� 'q satis�es

�Fc

� 'q� � and 'q�

�Fc

� 'q� 'q � � � 'q ��

Under these conditions� the equilibrium �q� 'q�� q�� is globally asymptoticallystable� �

Proof� The proof follows verbatim from Proposition �� considering as storagefunction the total energy of the closedloop

Hd�q� 'q � T �q� 'q & Vd�q� Vd�q�where the last term is included to enforce Vd�q��

Remark �� If Vd�q has a unique local minimum at q � q� we can conclude onlylocal asymptotic stability�

Remark �� The state feedback controller of Proposition !�� has been extended byTomei to the case of underactuated mechanical systems �in particular robots with�exible joints in the important paper � �� The result in that paper is establishedinvoking Lyapunov arguments� In � �� where we presented a solution to the outputfeedback problem� we give a passivity interpretation to Tomei�s controller�

�In particular� the EL parameters are fT �q� �q��V�q�� Ing� where as explained after De�nition�� when clear from the context� an EL parameter which is equal to zero will be simply omittedfrom the list� in this case Q� � ��


� An introductory example

We illustrate here the application of the proposition above with the simple pendulumshown in Fig� !�� Although this system has been exhaustively studied in manynonlinear systems texts� we believe it is still an ideal example to put forth the basicprinciples of PBC� We will therefore go through it in some detail�

The total �kinetic & potential energy of the simple pendulum is

H ��

ml� 'q�� z T �q� �q�

&mgl�� cos�q� �z V�q�

where q � IR and g is the gravity acceleration� We assume torque as the controlinput u� hence in the absence of friction the EL parameters of such system arefT �q� 'q� V�q� � �g� Using the EL equations we can easily derive the dynamics

ml�-q & g�q � u �!��

where g�q� which we call the gravitational force� is the force derived from the potentialenergy� that is�

g�q��

�V�q�q

� mgl sin�q

q *

��

q m

k

k

d

p

Figure !�� Simple pendulum�

Now let us calculate the equilibria of the unforced system �!�� i�e� with u � �As discussed in the previous chapter� and is clear from �!�� the positions of equilibriacorrespond to the critical points of the potential energy function� that is� the solutionsof the equation�

�V�q�q

� � mgl sin�q � �

Hence the equilibria are �q� 'q�� i�� i � � � � �� Next� taking thesecond partial derivative of V�q with respect to q yields

��V�q�q�

� mgl cos�q�

�� An introductory example ��

which is positive for q � and negative for q � �� It is clear then that the origincorresponds to a minimum of the potential energy function� hence recalling Proposi�tion �� we conclude that �q� 'q�� is a stable equilibrium� On the other hand�q � � is a local maximum� and it can be shown that this is an unstable equilibrium�

Our design problem is to stabilize the pendulum at a constant equilibrium �q� 'q�� q�� As suggested by the proposition above we will seek to modify the potentialenergy and the Rayleigh dissipation function of the system� leaving untouched thekinetic energy� since it plays no role on the stability properties of the equilibrium�That is we want the closedloop system to be an EL system with EL parametersfT �q� 'q� Vd�q� Fd� 'qg�

Since we know that a minimum of the potential energy corresponds to a stableequilibrium point� the �new� potential energy function should have a global andunique minimum at the desired position� A natural candidate is then

Vd�q � �

kp�q

� �!�

where kp � and �q�� q � q�� It is trivial to see that this function satis�es condition

A�� of Proposition !�� To make this stable equilibrium attractive we choose thedesired Rayleigh dissipation function Fd� 'q � �

�kd 'q

�� kd � � which clearly veri�esA�� of the proposition� These choices lead to the control

u ��

�q�V�q� Vd�q� �z

uES

� �Fc

� 'q� 'q� �z

uDI

� g�q� kp�q � kd 'q �!�!

Using Proposition �� global asymptotic stability follows� Remark that the controllaw consists of two terms taking care of the energy shaping and the damping injection�respectively� This nice nestedloop structure of the control will be encountered in allsubsequent PBC�

The PD control law above is one of the simplest one can obtain� however it has thedrawback that besides the computational charge that it represents to compute on linethe term g�q� it is widely believed that dominating instead of cancelling the nonlinearterm g�q enhances the robustness of the system vis�a�vis parametric uncertainties��

Thus� we consider the desired potential energy function � ��

Vd�q � V�q & �

kp�q � ��q�� !�#

�Although this is very hard to prove in general� interested readers are referred to �� and Section�� where� for an induction motor example� this claim is theoretically proven and experimentallyillustrated�


where ��q� is a constant chosen to assign a global and unique minimum at q � q� toVd�q� Its computation then proceeds by evaluating

�Vd�q

�q � g�q & kp�q � ��q��

upon setting the latter to zero at q � q� we get

��q� � q� � �

kpg�q��

To ensure that this critical point is a global and unique minimum we calculate

��Vd�q�

�q ��g

�q�q & kp�

We now invoke AssumptionA�� of Chapter � and choose kp � kg where kg is de�ned

by � �� so that ��Vd�q�

�q � � � for all q � IRn�

The energy shaping part of the control law is given by

uES ��

�q�V�q� Vd�q � �kp�q & g�q� �!��

and adding the same damping as above we �nd the well known PD plus precompen�sated gravity controller�

u � �kp�q � kd 'q & g�q�� !��

As before� the closed loop system �!�� !�� is a fully�damped Euler�Lagrangesystem with EL parameters fT �q� 'q�Vd�q�Fd� 'qg� therefore� by virtue of Proposition �� the equilibrium point q � q� is globally asymptotically stable�

Remark �� This pendulum example also illustrates an interesting connection be�tween the energy�shaping condition of Proposition �� and the zero�state detectabil�ity condition of �� Notice that the closed loop �!�� with �!�� is a lossless system�By de�ning the output y � 'q it is also zero�state detectable if kp � kg �energy shap�ing� Hence the feedback �kd 'q� kd � globally asymptotically stabilizes the closedloop system �!�� !��

�� Physical interpretation and literature review

The two PBCs above have clear physical interpretations depicted in Figs� !�� and Fig�!� � The proportional gain kp can be regarded as the sti�ness constant of a linearspring connecting the pendulum to a virtual line� In the �rst controller �!�! gravityforces are cancelled therefore the spring will not store energy at the equilibrium� and

�� Physical interpretation and literature review ��

we can set the virtual line at the angle q�� On the other hand� for the PBC �!��the spring must store the energy due to the gravitational forces and the plane shouldbe at an angle ��q�� The damping injection term in both cases represents a viscousdamper with gain kd which introduces viscous friction to attain asymptotic stability�We can hardly overestimate the importance of having a direct physical interpretationof the PBC action� Its relevance pertains not just to aesthetical considerations� butmore importantly to its impact on the controller tuning stage� Commissioning anonlinear controller is a di�cult task� which has unfortunately been overlooked inmost theoretical developments� it is rendered quite transparent to the practicingengineer by PBC� At a more fundamental level�

� PBC underscores the role of control as the establishment of suitable interconnectionsof the system with its environment�

This is perhaps the main asset of PBC� which explains its great success in appli�cations�

q *

q *

��

k

k

d

p

q= mg

δ( )

Figure !� � Physical interpretation of a PD plus gravity compensation controller�

It is interesting to note that� simultaneously and independently of the work of Ari�moto and coworkers in robot control� Jonckeere suggested in ��!� also the utilizationof energy shaping plus damping injection ideas for controller design of a broader classof EL systems which included electrical and electromechanical systems�� It shouldbe noted that while Arimoto and coworkers presented their developments with theHamiltonian formulation of the dynamics� Jonckeere used instead the EL formalism�Subsequent to the publication of � �� and ��!�� independent derivations of the sameresult were reported by van der Schaft � $ � and Koditschek ��!!�� As pointed out inRemark �� in the latter work the important �skewsymmetry� property of Propo�sition �� was �rst reported� We refer the reader to ��!�� for an interesting reviewof this circle of ideas that spans� in a tour de force� from the times of Lagrange andLord Kelvin to the late � s�

We conclude this section by singling out three main drawbacks of the utilization ofthe PD controller of Proposition !�� which stymie its utilization in some applications�

�This very interesting paper� which is the outgrowth of Jonckeere�s �� PhD thesis in Toulouse�passed relatively unnoticed but strongly inuenced the work of the �rst author�

� Ch� !� Set�point regulation

These motivate all the further developments of the present chapter�

�� Measurement of the generalized velocities 'q� and full actuation are required toadd the necessary damping�

� No amplitude constraints are imposed on the control input�

!� The potential energy function V�q is supposed to be exactly known�

Each of these drawbacks being of indisputable practical importance� we devote tothem the next three sections� respectively�

� Output feedback stabilization of underactuated

systems

In order to put the material of this section in perspective we �rst brie�y review theliterature�

� Literature review

Speed measurement increases cost and imposes constraints on the achievable band�width because of the presence of noise� This has motivated the researchers to look forregulators which avoid velocity measurements� Linear control laws that obviate theneed of this signal preserving GAS have been recently �and independently proposedin �!� � # � � !�� In the �rst three papers it is shown that velocity can be replacedby approximate di�erentiation� In �!� a linear PBC that shapes both the kinetic andthe potential energy� and adds damping injection is presented� It must be remarkedthat in this paper the solution was given also for underactuated EL systems� e�g��exible�joint robots� Thus extending� to the output feedback case� the controller of� �� Finding a common feature to all of these controllers and extending the resultsto the more general frame of underactuated EL systems� motivated us to look for anew methodology for output feedback regulation of underactuated EL systems� Thisresearch culminated in the de�nition of the EL controllers in � �� Other researche�orts related with the material of this section may be found in ��$� �#�� In partic�ular� in ��$�� some connections between PBC and control based on output injectionideas are explored�

� Problem formulation

We consider in this section the underactuated EL plants with no internal dampingwhich is the worst case scenario because� as it will become clear later� damping in the

�� Euler�Lagrange controllers �

plant helps us to relax the conditions for stabilization� That is� we consider plantswith EL parameters

fTp�qp� 'qp�Vp�qp� �MpgSince the PBC of this section will be dynamical �and furthermore also an EL systemwe have added the subindex p for �plant�� The dynamic model has the form � ��which for convenience of presentation we write below

Dp�qp-qp & C�qp� 'qp 'qp & g�qp �Mpu �!�$

where qp � IRn� u � IRm� For ease of presentation and without loss of generality wewill assume that Mp � � � Im�

��

The problem we study in this section is formulated as follows�

De�nition �� Output feedback global stabilization problem� Consider theEL system �� where qp is partitioned as qp � �q�p� � q

�p�� qp� �Mpqp� Assume that

themeasurable outputs are qp� and the regulated outputs are qp� with constant desiredvalue qp�� Then� design a controller qp� �� up that makes the closed loop system GASat an equilibrium point 1q � � 1qp

�� 1q�c �� such that such that qp�� M�

p 1q � �Inp� � �1q�

�� Euler Lagrange controllers

Motivated by the energy shaping plus damping injection technique� and the proper�ties of EL systems described in the previous chapter we will de�ne here a class ofcontrollers which� preserving the EL structure� suitably modi�es the potential en�ergy and dissipation properties of the EL plant� Towards this end� we invoke theEL structure preserving interconnection property of Proposition �� and proposeto consider EL controllers with generalized coordinates qc � IRnc and EL parametersfTc�qc� 'qc�Vc�qc� qp�Fc� 'qcg� We remark that since we are dealing here with a regula�tion and not a tracking problem there are no external inputs to the controller� whichexplains our choice of as the �input matrix� Mc� Thus� the controller dynamics isgiven by

Dc�qc-qc &Nc�qc� 'qc &�Vc�qc� qp

�qc&�Fc� 'qc

� 'qc� � �!��

Notice that the potential energy of the controller depends on the measurable output

qp� � therefore qp� enters into the controller via the term�Vc�qc�qp��

�qc� On the other hand�

following Proposition �� the feedback interconnection between plant and controlleris established by

Mpu � ��Vc�qc� qp�qp�

� �!��


In this way� it follows from Proposition �� that the closed�loop system is still an ELsystem with EL parameters fT �q� 'q�V�q�F� 'qg where

T �q� 'q�� Tp�qp� 'qp & Tc�qc� 'qc� V�q �� Vp�qp & Vc�qc� qp� F� 'q

�� Fc� 'qc

and q � �qp�� q�c �

�� The resulting feedback system is depicted in Fig� !�!� where.p � up �� qp� is an operator de�ned by the dynamic equations �!�$ and operator.c � qp� �� up is de�ned by �!�� !��

The following remarks are in order� � The interconnection constraint �!�� im�poses some clear limitations on the achievable desired potential energy functions� Itis clear that the constraints are removed for fully�actuated systems� From thenew de�nition of F� 'q we see that the dynamic extension we just introduced injectsdamping to the system through the controller dynamics� As it will be seen later thisdamping has to be suitably propagated for asymptotic stability�

qp�

.p

.c

��V�qc�qp��qp�

Figure !�!� EL closed loop system�

We will apply now Proposition ��! to establish conditions for solvability of theproblem using EL controllers� These conditions are summarized in the propositionbelow�

Proposition �� Output feedback stabilization� An EL controller �� with EL parameters

fTc�qc� 'qc�Vc�qc� qp��Fc� 'qcgsolves the global output feedback stabilization problem above if

�i �Energy shapingV�q is proper and has a global and unique minimum at q � 1q� where 1q is suchthat qp�� Inp� j �1q�

�ii �Damping injectionFc� 'qc satis�es

'q�c�Fc� 'qc

� 'qc� �k 'qck�

for some � �

�� Examples ��

�iii �Dissipation propagation

For each trajectory such that qc � const and �Vc�qc�qp��qc

� � we have thatqp � const�

�

Proof� The proof follows from Proposition ��! observing that the closed loopsystem is an underdamped EL system with damped coordinates qc and undampedcoordinates qp� Notice that condition �iii above implies that 'qp �� and �Vc�qc�qp�

�qc�

cannot happen simultaneously� hence this condition implies �iii of Proposition ��!��

Remark �� From Proposition !�� it is clear that the kinetic energy of the con�troller plays no role on the stabilization task� it may however a�ect the transientperformance� In particular� the result applies even in the case when Tc�qc� 'qc � �Furthermore� the conditions on the Rayleigh dissipation function of Proposition !��are satis�ed with Fc� 'qc �

��'q�c Rc 'qc� where Rc � R�c � � Thus� with this choice of

Fc� 'qc and setting Tc�qc� 'qc � we get controllers with dynamics

'qc � �R��c�Vc�qc� qp�

�qc

Mpu � ��Vc�qc� qp�qp�

With an obvious abuse of terminology� and with the purpose of enlarging the classof controllers �and actually to obtain simpler solutions� we will also call these con�trollers EL controllers� As it will become clear later this scheme corresponds to theapproximate di�erentiation �lter� widely used in practical applications�

Remark �� It is interesting to underline that for some underactuated systems onecan instead reshape the kinetic energy for regulation purposes� as shown in � $� wherethe authors introduced the related �controlled Lagrangians� methodology� This alter�native framework allows in principle� to include mechanic systems with nonholonomicconstraints��

�� Examples

We have proven above that for underactuated EL systems global stabilization withPBC controllers is still possible with only output feedback provided a dissipation propa�gation condition is satis�ed� In this section we apply this general result to two physicalsystems� the TORA and the �exible�joint robots� whose dynamical models have beenderived in Sections #�� and #�! of Chapter �� respectively�

�See the URL http��www�cds�caltech�edu�� marsden� for an extensive bibliography�


A The TORA

For ease of reference we recall here from Section ��#�! the EL parameters of the TORAsystem

Tp�qp� 'qp � �

'q�p Dp�qp 'qp �

�

'q�p

�M &m �ml cos�qp�

�ml cos�qp� I &ml�

�'qp

Vp�qp � �

kq�p�� Fp� 'qp � � Mp �

� �

��

This is a very simple EL system that can be globally asymptotically stabilized at anygiven constant equilibrium point� in particular at zero� using Proposition !�� Thisleads to a proportional plus derivative �PD controller

up � �kpqp� � kd 'qp�

with kp� kd � � The proof of GAS is easily established checking conditions A�� andA�� of this proposition� Notice thet the total energy of the closedloop is

H�qp� 'qp � Tp�qp� 'qp & Vp�qp & �

kpq

�p��

The more practically interesting problem of making the zero equilibrium GAS assum�ing that only qp� is available for measurement can also be easily solved by invokingProposition ��!� To this end� we propose an EL controller with EL parameters

Tc�qc� 'qc � � Fc� 'qc �kd ab

'q�c

Vc�qc� qp� � kp q�p� &

kd b

�qc & bqp��

where a� b� kp� kd � � Notice that this corresponds to the �degenerate� case of ELsystem with zero inertia discussed in Remark !�$� and a standard quadratic damping�The �rst term of Vc�qc� qp� corresponds to the potential energy stored by a springconnecting the pendulum to a vertical plane� We will see below that the secondterm of Vc�qc� qp� corresponds to a �spring action� between 'qc and a zero reference�Actually� we will prove that this PBC corresponds to the PD controller above with'qp� replaced by its dirty derivative�

Following Proposition ��! this de�nes our EL controller dynamics as

'qc � �a�qc & bqp� �!��

u � �kpqp� � kd�qc & bqp� �!��

We will now verify the conditions of the proposition� First� the energyshaping con�dition is satis�ed for all kp� kd � because the total potential energy is a quadraticform

V�q �� Vp�qp & Vc�qc � �

q�

�� k

kp & kdkdb

kdkdb

�� q


which is positive de�nite� Second� the damping injection is clearly satis�ed with� � kd� Finally� we check the damping propagation condition �iii� From �!�� and qc � const we have that qp� � const� This� replaced in �!�� implies thatu � const� which in its turn implies that -qp� � const� from the �rst dynamic equationof the TORA� We conclude the proof noting that the second dynamic equation of theTORA yields qp� � const as desired�

Let us now prove that the controller above is a PD controller with approximatedi�erentiation� Let us introduce the notation

�� qc & bqp�

Hence�u � �kpqp� � kd�

and � satis�es'� � �a� & b 'qp�

thus the control can be written as

u � �kpqp� � kd

�bp

p& a

�qp�

proving the claim�

The simplicity of this controller should be contrasted with the derivations reportedin �� see also � !$�� In the latter paper the �rst step is to make a coordinate changethat transforms the system into the cascaded structure required by the backsteppingtechnique� Unfortunately� since this coordinate transformation destroys the physicalstructure of the system� the controller design is not transparent though in some sensesystematic�

In Subsection !�#�A� where we will further assume the input is subject to a satura�tion constraint� we will present some simulation results comparing the two controllers�

B Flexible�joint robots

We will derive here a class of EL PBC solving the output feedback global stabilizationproblem for �exible�joint robots� This provides a uni�ed framework to compare di�er�ent schemes via analysis of their energy dissipation properties� Further� we show thatas particular cases of this class we can obtain the �apparently unrelated controllersof �� and �!�� and extend to the output feedback case the controller of � ��

We use in this section� the dynamic model � �!�� whose EL parameters are givenby � �! � We recall the reader that in our notation� qp� stands for the vector of linkpositions and qp� for the vector of motor shafts angles� The control variables are thetorques at the shafts� We are interested in the set�point control of the link angles to


a constant desired value qp�� and we assume that only the motor shaft angles� qp�are available for measurement�� We also recall that the map u �� 'qp� is passive� aswe have shown in the previous section�

B�� EL controller with a virtual rigid�joints robot

In �!� an EL controller for a �exible�joint robot was designed using the followingphysically motivated approach� First� we design a rigid robot without gravity forcesthat we attach to a hyperplane with a PD control ��a la TakegakiArimoto� Then�we attach some springs between this virtual robot and the motor shaft angles in sucha way that the virtual robot �pulls� the actual robot to the desired equilibrium� Theposition of the hyperplane is selected so that in steadystate the springs store thepotential energy required to place the robot links at the desired angle� This reasoningled us to consider an EL controller with EL parameters

Tc�qc� 'qc � �

k 'qck� � Fc� 'qc �

�

'q�c Rc 'qc� �!��

Vc�qc� qp� � �

f�qc � qp�

�K��qc � qp� & �qc � ��qp��K��qc � ��qp��g� �!��!

That is� the virtual rigid robot is fully�damped Rc � R�c � � has unitary inertiamatrix� is attached to a �xed hyperplane at an angle ��qp�� via a spring of sti�nesscoe�cient K� � K�

� � �as in the pendulum example above� Finally� it is connectedto the motor shaft angles through some additional springs of sti�ness coe�cient K� �K�

� � � See Fig� ��

Using the EL equations � �� we easily derive the controller dynamics

-qc &Rc 'qc &K��qc � � &K��qc � qp� � � �!��#

while the feedback interconnection is given� according to �!�� by

u � ��Vc�qc� qp��qp�

� K��qc � qp�� !��

It is interesting to note that the control signal of this PBC does not have the structureu � uES & uDI� this is because we are also shaping the kinetic energy�

Now� adding up the EL parameters of the robot and the controller we get

T �q� 'q ��

'q�p D�qp 'qp &

�

k 'qck� � F� 'q �

�

'q�c Rc 'qc

V�q � �

qp�Kpqp & Vg�qp� &

�

f�qc � qp�

�K��qc � qp� & �qc � ��K��qc � �g�See �� and �� for globally stabilizing schemes measuring qp� instead�


We proceed to verify now conditions �i and �iii of Proposition !�� the condition�ii being trivially satis�ed with � � ��Rc � �

�i �Energy ShapingFirst� we need to insure that our EL controller �!��#� �!�� shapes the closed loopsystem�s potential energy to make it have a global and unique minimum at the desiredequilibrium point� This step involves the de�nition of ��qp�� and some restrictions

on the various springs� To this end� we �rst calculate �V�q��q

and set it equal to zero atan equilibrium containing qp��

� K �K �K K &K� �K�

�K� K� &K�

�� qp��

1qp�1qc

��&

�� gp��qp��

��

��

K��qp��

��

�!��

We see that by de�ning the constant

��qp�� qp�� & �K�� &K��

� &K�� gp��qp��

we assure that �!�� has a solution of the required form

1q �

�� 1qp�

1qp�1qc

��

�� qp��

qp�� &K��gp��qp��qp�� & �K�� &K��

� gp��qp��

�� !��$

Now� to enforce V�q to have a global and unique minimum� that is

��V�q�q�

�

�� K &

�gp��qp� �

�qp��K

�K K &K� �K�

�K� K� &K�

�� In� �

which happens to hold if ��Ka � kg where

Ka��

�� K �K �K K &K� �K�

�K� K� &K�

��

and kg is de�ned by � �� The authors of �!� observed thatKa accepts the congruencetransformation �

� I I I I I I

��Ka

�� I I I

I I I

��

�� K

K� K�

��

therefore ��Ka � kg if and only if

block�diagfK� K�� K�g � kg

�� I I I I I I

�� I I I

I I I

�� kg

�� I I II I II I !I

��


On the other hand� it can be shown that there exists a permutation matrix P �IR�n��n such that

P

�� I I II I II I !I

��P� � block�diagfEg

where

E��

��

� � !

��

Thus� recalling that permutation matrices are orthogonal� the required condition onK�� K�� K reduces to

K�� K�� K � In1��Ekg�

�iii �Dissipation propagationNotice that

�Fc� 'qc

� 'qc&�Vc�qc� qp�

�qc� Rc 'qc &K��qc � � &K��qc � qp� �!��

then setting qc � const and equating the right hand side of �!�� to zero� it followsthat qp� � const� The proof is completed as done by � �� observing the uppertriangular structure of D��qp� �see eq� �!# in order to conclude that qp� is alsoconstant� Since qp� is constant then we can write the last n di�erential equations of� �!� using � �!! and � �!� as

D��qp�-qp� & C��qp�� 'qp� 'qp� �Kqp� � �Kqp� �K��qc � qp� & gp��qp�d � constant�

�!��

Considering � �!$ and � �!#� the �rst equation of �!�� becomes qp�� constant�Substituting the latter into the second equation of �!�� we get qp�� constant�Proceeding in the same way till the nth di�erential equation of �!�� we can concludethat qp� is constant� Since we proved that q � 1q is the only equilibrium point� thecontrol goal has been achieved�

B�� EL controller with approximate di�erentiation

The PBC that we derive in this section� corresponds to the PD of � �� wherevelocity is replaced by its approximate di�erentiation �or �dirty derivative�� It con�stitutes an extension� to �exible�joint robots� of the controller reported in �� !�� andwas originally proposed in �� To understand the rationale of the controller we re�call that the �exible�joint robot de�nes a passive map u �� 'qp� with storage functionthe total energy Hp � Tp�qp� 'qp & Vp�qp� that is� it satis�es

'Hp � 'q�p�u


Let us close a �rst loop with a proportional energyshaping term

u � uES & uDI � �Kp�qp� � ��qp�� & uDI

where ��qp�� is a constant term that plays the same role as in the previous controllerand Kp is a diagonal positive de�nite matrix� Clearly� we have passivity of the mapuDI �� 'qp� with the new storage function

Hp &�

�qp� � ��qp��

�Kp�qp� � ��qp��

Now� we propose to add damping by feeding back the dirty derivative of qp� � that is

uDI � �Kddiag

�bip

p& ai

�qp� � �Kd�pI & A��B 'qp�

where Kd�� diagfkdig � and A

�� diagfaig � � The transfer matrixKd�pI&A

��Bis strictly positive real� hence output strictly passive with storage function

HF ��

��KdB

��

The overall closedloop consists of the passive map uDI �� 'qp� in feedback with anoutput strictly passive LTI system Kd�pI&A��B� hence it is still passive� The proofof GAS can be completed using passivity and detectability arguments� or taking thesum of the storage functions as a Lyapunov function candidate and invoking LaSalle�See �� for such a proof�

Let us prove now that this controller belongs to the EL class de�ned in Section �!� Towards this end it is convenient to de�ne the generalized coordinates as

qc�� Bqp� �

In terms of these coordinates we can rewrite the controller above as

'qc � �A�qc &Kdqp�

u � �Kp�qp� � ��qp��Kd�qc &Bqp�

which corresponds to an EL system with EL parameters

Tc�qc� 'qc � � Fc� 'qc ��

'q�c KdB

��A�� 'qc

Vc�qc� qp� � �

�qp� � ��qp��

�Kp�qp� � ��qp�� &�

�qc &Bqp�

�KdB��qc &Bqp�

Notice that� as in the TORA example� this corresponds to the �degenerate� case ofEL system with zero inertia and a standard quadratic damping� The �rst term of


Vc�qc� qp� corresponds again to the potential energy stored by a spring connectingthe motor axes to the hyperplane at angle ��qp�� The second term of Vc�qc� qp�corresponds to a �spring action� between 'qc and a zero reference�

The EL parameters of the closed loop are

T �q� 'q ��

'q�p D�qp 'qp� F� 'q �

�

'q�c KdB

��A�� 'qc� �!�

V�q ��

f�qc &Bqp�

�KdB��qc &Bqp� & �qp� � ��Kp�qp� � �g&

&�

qp�Kpqp & Vg�qp� �!� �

Now� we determine the values of Kp� Kd and � such that the conditions �i and�iii of Proposition !�� hold�

�i �Energy shaping

To verify this condition �rst notice that setting �V�q��q

�1q � yields�� K �K �K K &Kp &Kd KdB

��

Kd KdB��

�� 1qp�

1qp�1qc

��&

�� g�1qp�

��

�� Kp�

��

which has a �unique solution of the required form 1q � �q�p�� with

� � qp�� & �K�� &K��p gp��qp��

The second part of this condition is met if

��V�q�q�

�

�� K &


�qp��K

�K K &Kp &Kd KdB��

Kd KdB��

�� In� � � � q � IRn�

To satisfy this requirement we partition the two diagonal sub�blocks of the Hessianmatrix as

Q��

�K &


�qp��K

�K K & ��Kp

�� Q�

��

��Kp &Kd KdB

��

Kd KdB��

�

and look for conditions that ensure that both are bounded from below by some matrixInp�� for all q � IRn� The submatrix Q� is positive de�nite if ��Ka � kg wherewe have rede�ned

Ka��

�K �K�K K & �

�Kp

�then we can proceed as before observing that Ka satis�es the congruence transforma�tion �!� �

I I I

�Ka

�I I I

��

�K Kp

��

�� Examples �

on the other hand we can prove that there exists a permutation matrix P � IRnp�np

such that

P

�I II I

�P� � block�diagfEg

where

E��

��

��

Thus� the condition on K�Kp reduces to Kp� K � ��EkgImp� Following the same

procedure� it can be also shown that the submatrix Q� is positive de�nite for allKp� Kd � �

�iii �Dissipation propagationFinally� this condition is veri�ed as follows� we set the right hand side of

�Vc�qc� qp�

�qc� Kd�qc &Bqp�

to zero and consider qc � const� we then get that qp� � const� The proof is completedas for the previous EL controller exploiting the special triangular structure of therobot inertia matrix in order to conclude that also qp� is constant� �

Remark �� It is important to remark that the PBC discussed in Subsection B��modi�es the kinetic energy of the EL system� The dynamic extension is then of theorder of the EL system itself� On the other hand� the PBC in Subsection B� � leavesthe kinetic energy unchanged� and requires an n�th order dynamic extension only toinject the damping�

B�� Simulation results

In this section we illustrate through simulations the performance of both EL con�trollers presented above� We have used the two degrees of freedom simpli�ed modelof � �� whose EL parameters are �with zero payload

T �qp� 'qp��

�'qp�'qp�

�� cos�qp� & ��$$ �$� & �� cos�qp� �$� & �� cos�qp� ��

� �'qp�'qp�

��!�

Vp�qp �� $�� sin�qp� & ��! cos�qp� & qp� �!� !

Fp� 'qp � � �!� #

Moreover the bounds mentioned in properties P�� P�� for this particular caseare �assuming a payload of ml kg

dm � �#�� dM � �� kc � ��!� kg � �� kv � � �$� �!� �


and we have considered a joint sti�ness of K � !� I��

The constant reference to be followed is qp�� with zero initial conditions�

Figure !�# shows the result of the controller described in �� in this case we have setthe gains to K� � diag�� K� � diag��# � � �� ai � ! and bi � � �In Figure !�� we illustrate the response to the same reference using the controller of�!�� We have used the same model as before and the controller gains were set in sucha way to have a transient approximately similar in time to that of the �rst controller�These values are K� � K� � diag�� Rc � diag��!� � !� �� Gains canbe tuned to have a smoother but slower transient response�

0 1 2 3 40

0.2

0.4

0.6

0.8

1

time [s]

angl

e [r

ad]

First link position

0 1 2 3 40

0.2

0.4

0.6

0.8

1

time [s]

angl

e [r

ad]

Second link position

Figure !�#� EL Controller of �#�B� �

0 1 2 3 40

0.2

0.4

0.6

0.8

1

time [s]

angl

e [r

ad]

First link position

0 1 2 3 40

0.2

0.4

0.6

0.8

1

time [s]

angl

e [r

ad]

Second link position

Figure !�� EL Controller of �#�B��

In this brief section we have limited ourselves to illustrate the behaviour of thecontrollers contained in Subsection �#�B� further simulation results comparing theperformance of � �� and �� can be found in the latter reference�

� Bounded output feedback regulation ��

� Bounded output feedback regulation

In the previous section we characterized a class of Euler�Lagrange systems that canbe globally asymptotically stabilized via nonlinear dynamic output feedback� Thecontroller design� relies on the fact that the storage function for feedback intercon�nected systems is the sum of the corresponding storage functions� This basic propertyis expressed in terms of the physically appealing principles of shaping the potentialenergy and injection of the required damping� In particular� we considered the casewhere the controller is also an EL system� in this way the closed loop is still an ELsystem with total energy and dissipation function the sum of the corresponding plantand controller total energies and dissipation functions�

Motivated by practical problem of windup present in numerous applications� wewill consider in this section the setpoint control problem with amplitude constrainedinputs� More particularly� we focus our attention on saturated set�point control of aparticular class of fully�actuated EL systems�

�� Literature review

One of the �rst contributions in the robotics literature is due to �#�� who proposed asaturated PD plus gravity compensation like controller� to deal with stick�slip frictione�ects� Later� �� extended this result using precompensated gravity� however bothresults use velocity measurements� Some recent extensions of these works which onlyuse position measurements are given in �##� and �� Burkov �##� proposed a PD�likecontroller which uses exact gravity compensation� while we introduced a subclass ofEL controllers �which contains that of �##�� thus extending this methodology to thecase of fully�actuated EL systems under input constraints�

�� Problem formulation

In this section we assume that A�� holds� then under this condition we deal withthe

De�nition �� Set�point control under input constraints� For the system ��

D�qp-qp & C�qp� 'qp 'qp & g�qp � up� �!� �

assume that only generalized position measurements� qp� are available and that thesystem�s inputs are constrained to

jupij umaxpi

�i � n �!� $


then� �nd an output feedback controller which renders the closed loop system globallyasymptotically stable� that is� an output feedback controller such that

limt�

k�qp�tk � limt�

kqp�t� qp�k � � �!� �

where qp� is the desired constant position�

Based on the results of the previous section we de�ne in the sequel� a family ofEL controllers which yield bounded control inputs� We repeat for convenience thatthe aim of the EL controllers is to shape the closed loop energy V�q and to inject apartial damping� It is clear that the easiest way of shaping the potential energy ofthe closed loop� V�q� is to cancel the potential energy of the plant� Vp�qp� and thento impose a �new� potential energy shape� However� to enhance the robustness ofthe controller� instead of cancelling the potential energy we aim at dominating it�

According to the EL controllers methodology� the control input is de�ned by

up � ��Vc�qc� qp�qp

�

from this� we can deduce that the input constraint established by �!� $� shall entailsome growth restrictions on Vc�q� For instance� if we take Vc�q quadratic� as wehave done before� the control up will grow linearly with qp� and we cannot expect tosatisfy �!� $ for large kqpk� The basic idea is then to choose a Vc�q which growslinearly with respect to kqpk outside some ball� in this case we can expect to verify�!� $� As it will become more clear later� a suitable class of functions is of the form

Vc�qc� qp �nXi �

�Z fi�qp�qc�

�

sat�xdx

�& VcN�qc� qp

where fi � IR IR � IR� i n and VcN�qc� qp are linear in qp and sat�x is asaturation function de�ned below� Replacing this expression above we get

upi � ��fi�qc� qp

�qpisat�fi�qc� qp� �VcN �qc� qp

�qp

which is a bounded function of qp�

On the other hand� since Vc�q should be designed in a way that it dominatesVp�qp� it is also necessary to impose the same growth restrictions on Vp�qp� Thus�we consider in this section� a subclass of fully�actuated EL systems whose potentialenergy function satis�es � �� Loosely speaking� this condition restricts the growthrate of Vp�qp to be of order O�kqpk� for all qp in some ball B and to O�kqpk outsideB�

The following de�nition is in order�

�� Globally stabilizing saturated EL controllers ��

De�nition �� Saturation function�� A saturation function sat�x � IR � IRis a C� strictly increasing odd function that satis�es

� sat� � �

�� j sat�xj � ��

�� sat�x��x�

�� x �� IR�

Our motivation for considering saturation functions as de�ned above is that thesefunctions satisfy the following properties�

P��R �qpi�

sat�xdx � ��sat��qpi�qpi� �qpi � IR�

P�� There exists some � � such that

sat��qpi�qpi �sat��

��q�pi j�qpij � �� !� �

sat��qpi�qpi � sat��j�qpij j�qpij � �� !�!

For instance� we can take sat�x�� tanh� x� � � as proposed in �#��

�� Globally stabilizing saturated EL controllers

Since we are dealing with fully�actuated systems� the simplest way to �dominate�the plant�s potential energy is to cancel Vp�qp and to impose a desired shape to theclosed loop function� This however entails some potential robustness problems� hencewe favour a solution that does not rely on this cancelation� Interestingly enough� ifwe use a controller that does not cancel the vector of potential forces� the growth raterestriction on Vp�qp mentioned above is imposed only at the desired position� Theprice paid� however� is that in this case we need to use �high� gains in Vc�qc� qp todominate Vp�qp and this translates into sti�er requirements on the input saturationbound� umax

pi�

Proposition �� Control with cancelation of potential forces� Assume thatthe systems potential energy veri�es the strict inequality

supqpIRn

��Vp�qp�qp

�i

�� umaxpi

� i � n �!�!�

with ��i the i � th component of the vector� Under these conditions� there exists adynamic output feedback EL controller .c � fTc�qc� 'qc�Vc�qc� qp�Fc� 'qcg that insuresthe input constraint �� holds� and makes

� 'qp� qp� 'qc� qc � � � qp�� qcd �!�!

with qcd some constant� a GAS equilibrium point of the closed loop system� �


� A controller with cancelation of potential forces� Consider the EL controllercharacterized by


k 'qck�

Vc�qc� qp � Vc��qc & Vc��qc� qp� Vp�qpVc��qc �

�

q�c K�qc

Vc��qc� qp �nXi �

k�i

Z �qci��qpi�

�

sat�xidxi

where �qpi�� qpi � qp�i� k�i � k�i � � K�

�� diagfk�ig� Using Lagrange�s equations we

can derive the controller dynamics

'qci � �k�iqci � k�i sat�qci � �qpi �!�!!

upi � k�i sat�qci � �qpi &

��Vp�qp�qp

�i

�!�!#

which corresponds to that proposed by Burkov in �##��

Proposition �� Control without cancelation of potential forces� Assume that�at the desired reference� the gradient of the systems potential energy satis�es the in�equality ��

��Vp�qp

�qp��i

�� kmaxgi

� i � n �!�!�

with kmaxgi

� umaxpi

� and let its Hessian satisfy �� Under these conditions� thereexists a globally asymptotically stabilizing EL controller that does not cancel thepotential forces and insures the input constraints �� provided umax

piis su�ciently

large� �

� A controller without cancelation of potential forces� In this case the ELparameters of the controller can be chosen as


'q�c K�B

��A�� 'qc �!�!�

Vc�qc� qp � Vc��qc� qp� Vp�qp� & q�p�Vp�qp

�qp�

Vc��qc� qp �nXi �

�k�ibi

Z �qci�biqpi�

�

sat�xidxi & k�i

Z �qpi

�

sat�xidxi

�


where A�� diagfaig� B

�� diagfbig� K�

�� diagfk�ig � � and we select k�i �

su�ciently large� This choice yields the EL controller

'qci � �ai sat�qci & biqpi �!�!$

upi � �k�i sat�qci & biqpi� k�i sat��qpi &

��Vp�qp

�qp��i

�!�!�

A Some remarks on saturated EL controllers

� The propositions above characterize� in terms of the EL parameters Tc�qc� 'qc�Fc� 'qc� a class of output feedback GAS controllers for EL systems with sat�urated inputs� Thus� providing an extension� to the constrained input case� ofthe result presented in Section �!�

� A key feature of the controller given above is that� to enhance its robustness�we avoid explicit cancelations of the plant dynamics� As mentioned before� theprice paid for this is the requirement that the plants potential energy grow notfaster than linearly� also� higher gains have to be injected into the loop throughk�i �see Appendix D� As seen from our proposition� this imposes an additionalrequirement of su�ciently large input constraints for stability� The condition onk�i stems from the fact that� to impose a desired minimum point to the closedloop potential energy� now we have to dominate �and not to cancel the systemspotential energy� In this respect controller �!�!$� �!�!� supersedes the resultof �##� which relies on exact cancelation of Vp�qp�

� As a corollary of our proposition� we obtain an extension to the output feedbackcase of the result in �� where a full state feedback solution to the problem ofglobal regulation of rigid�joints robots with saturated inputs was presented� Itis also interesting to remark that if we write 'qci � �ai�qci&biqpi instead of 'qci ��ai sat�qci&biqpi� in �!�!$ we exactly recover the �approximate di�erentiationoutput feedback GAS controller of �� !�� Our proposition then shows thatby simply including the saturations we can preserve GAS even under inputconstraints�

B Proofs

The proofs of Propositions !�� and !��! are constructive� For this� we provide belowthe stability proofs of the above�proposed saturated EL controllers�

As pointed out before� the interconnection of two EL systems� yields an EL systemwith potential energy V�q � Vc�qc� qp & Vp�qp� hence the proofs of both results arecarried out by proving the conditions of Proposition !�� Notice that the di�cultylies in proving that V�q has a global and unique minimum including qp � qp�� Forthis� we will use Lemma C��


B�� Proof of Proposition ��

Notice �rst that the condition on the Rayleigh dissipation function of Proposition!�� is trivially satis�ed� hence we go on proving that the potential energy is adequatelyshaped and that the damping suitably propagates from the controller coordinates qcto the plant coordinates qp�

�i� �Energy shaping�

The potential energy of the closed loop system �!� �� !� $� �!�!# is given by

V�q � �

q�c K�qc &

nXi �

k�i

Z �qci��qpi�

�

sat�xidxi�

Now we use Lemma C�� to prove that V�q has a global and unique minimum atthe origin �qc � 1qc� �qp � � � � The positivity condition of Lemma C�� follows fromDe�nition !�� and Property P�� while the second condition follows from equalizing�V�q��q

� �

�K�qc &K� sat�qc � �qp�K� sat�qc � �qp

��

�

�� !�!�

Notice that� since sat�x is strictly increasing and vanishes only at x � � and K��K� are full rank� �!�!� is satis�ed if and only if �qc� �qp � � � �

�iii� �Dissipation propagation�

This condition is easily veri�ed by equalizing �Vc�qc�qp��qc

� �

K�qc &K� sat�qc � �qp � �

setting qc � const and observing that e�ectively� qp � const�

The proof is completed applying the triangle inequality to �!�!#� and using thefact that j sat�xj � �� to get the bound

j�upij � k�i & k�i &

��Vp�qp�qp

�i

��thus� under assumption �!�!�� we can always choose su�ciently small k�i� k�i � such that �!� $ holds� �

B�� Proof of Proposition ��


We provide in this section a stability proof for the controller �!�!$� �!�!�� Asin the previous proof� we verify the conditions of Proposition !��

We prove next that� if we take k�i su�ciently small and mini fk�ig � kmin�i

� withkmin�i

some suitably de�ned positive constant� then �!�! is a GAS equilibrium pointof the closed loop �!� �� !� $� �!�!� provided that the gradient of the systemspotential energy� evaluated at the desired reference� satisfy

umaxpi

�

��Vp�qp

�qp��i

�� & k�i� i � n� �!�#

We �nally show that� if in particular we take sat�x � tanh�x then

kmin�i

��

#kv

tanh��kvkg

� �!�#�

where kv and kg are given by � �� and � �� respectively�

�i� �Energy Shaping�

The closed loop potential energy is now

V�q �nXi �

�k�ibi

Z �qci�biqpi�

�

sat�xidxi & k�i

Z �qpi

�

sat�xidxi

�

&Vp�qp� Vp�qp�� q�p�Vp�qp

�qp�� !�#

Hereafter we show that� if there exists a kv as de�ned by � �� then there existskmin�i

� such that V�q has a global and unique minimum at the desired equilibriumfor all k�i � kmin

�i�

Notice that the �rst right hand term of �!�# is a nonnegative function of qc� qpwhich is zero at qc � �B��qp� Hence� to prove that �!�# has a global and uniqueminimum at �qp�� 1qc it su�ces to show that the last three terms have a global andunique minimum at qp � qp�� or equivalently� that the function f��qp � IR

n � IR

f��qp��

nXi �

�k�i

Z �qpi

�

sat�xidxi

�& Vp��qp & qp�� Vp�qp�� q�p

�Vp�qp

�qp��!�#!

has a global and unique minimum at zero� The proof of the latter statement requiressome lengthy but straightforward calculations which we include in Appendix D�

�iii� �Dissipation propagation�

The second condition is veri�ed by equalizing �V�q��qc

� �

K�B�� sat�qc &B�qp �


and observing that it holds true only if qc � �B��qp� since K� is full rank� hence�qc � const implies that qp � const�

The proof is completed applying the triangle inequality and using �!�!� to getthe bound

j�upij � k�i & k�i &

��Vp�qp

�qp��i

��thus under assumption �!�!�� we can always choose su�ciently small k�i � k�i � such that �!� $ be satis�ed� �

�� Examples

A The TORA system�

We will consider now the TORA system� whose model was given in Section #�� ofChapter �� and for which we already designed in Subsection �#�A an output feedbackEL PBC� The material in this section follows closely �$ � to which we refer the readersfor further details�

We will take the same approximate di�erentiation approach of Subsection �#�A�but to take into account the input saturation �!� $ we modify the EL parameters ofthe controller as proposed in �!�!�� That is� we choose

Vc�qp�� qc ��

b

Z �qc�bqp��

�

k� sat�sds&

Z qp�

�

k� sat�sds

Fc� 'qc ��

ab'q�c

where a� b are positive constants� and ki � � i � � !� The controller dynamics isthen given by

up � �k� sat�qc & bqp�� k� sat�qp�

'qc � �ak� sat�qc & bqp�

The conditions of Proposition !��! can be easily veri�ed� hence the trivial equilibriumis GAS if

k� & k� umaxp �

Computer simulations have been carried out to show the performance of the pro�posed controller� We use the parameters shown in table !�� with the physical con�straints jqp�j � � m and jupj �� Nm given in �#��

�� Examples �

Description Parameter Value Units

Cart mass M ��!� � KgArm mass m � �� Kg

Arm eccentricity l � �� mArm inertia I � �$� Kg%m�

Spring sti�ness k ��! N%m

Table !�� TORA parameters�

0 1 2 3 4 5 6 7 8 9 10−0.03

−0.02

−0.01

0

0.01

0.02

0.03

time [sec]

q_p1

Translational position

0 1 2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1

time [sec]

q_p2

Angular position

Figure !�� Transient behaviour for translational and angular positions�

All initial conditions are set to zero except the initial translational position which is

set at its extreme value qp�� We selected sat�s�� tanh�s for the control

law and after a few iterations in simulation to get the best transient behaviour wechose the following parameters� a � �� b � #�� k� � � !�� k� � � �� Noticethat k� & k� is much smaller than the allowable bound �� We observed� however�that performance was actually degraded for larger values of these gains because ofthe peaking phenomenon�


0 1 2 3 4 5 6 7 8 9 10−0.04

−0.02

0

0.02

0.04

time [sec]

u_p

Control torque

0 1 2 3 4 5 6 7 8 9 10−1.5

−1

−0.5

0

0.5

1

1.5

time [sec]

q_c

Controller state

Figure !�$� Applied control and controller state�

A typical response of the system in closed loop with the proposed controller isshown in Figs� !�� and !�$� As we see the system exhibits good settling behaviour� itstabilizes around ! sec�� with a control e�ort� jupj � # Nm� well below its admissibleupper bound�

It is interesting to compare our results with the ones obtained with the full statefeedback unsaturated controllers reported in �� First of all� notice that our resultspertain to the original system� while those given in �� are carried out for its scaledversion� Eq� �� in �� where the arbitrary value �� is given for the scale couplingfactor � � ml�

p�I &ml��M &m� This factor equals � for the benchmark prob�

lem of �#�� hence the plots are not directly comparable� Secondly� as pointed outin �� the best results were obtained with the controller P!� which is a passifyingcontroller consisting of a standard PD plus a nonlinear term that enforces the passiv�ity property� Thus is very similar� at least in spirit� to the controller presented here�although our controller is saturated and uses only output feedback� It is quite clearfrom the �gures of �� that the behaviour of the backsteppingbased controllersthat did not exploit passivity properties� that is P� and P � is signi�cantly inferior�Further comparisons of the two controllers may be found in �$ ��

In order to evaluate the robustness of the closed loop system with respect to theexternal disturbance f � we apply a pulse of amplitude ! N� and duration �� sec� tothe system once it has reached its equilibrium point� See Figs� !�� and !��


0 1 2 3 4 5 6 7 8 9 10−6

−4

−2

0

2

4

6x 10

−3

time[sec]

q_p1


0 1 2 3 4 5 6 7 8 9 10−0.1

−0.05

0

0.05

0.1

time[sec]

q_p2

Rotational position

Figure !�� Translational and rotational responses to an external disturbance�

0 1 2 3 4 5 6 7 8 9 10−0.01

−0.005

0

0.005

0.01

time[sec]

u_p

Control torque

0 1 2 3 4 5 6 7 8 9 10

−0.2

0

0.2

time[sec]

q_c

Controller state

Figure !�� Applied control and controller state for an external disturbance�


To illustrate the global nature of our controller we present a simulation where wewant to �unwind� the arm from an initial value of qp�� to the zero position�with all other initial conditions equal to zero� In Figs� !�� and !�� we show thetransient behaviour� Note from Fig� !�� that the controller actually saturates butglobal asymptotic stability is preserved as predicted by the theory�

0 10 20 30 40 50 60 70 80 90 100−40

−20

0

20

40

time [sec]

q_p2

Angular position

0 10 20 30 40 50 60 70 80 90 100−0.05

0

0.05

time [sec]

q_p1


Figure !�� Translational and rotational responses for qp��

�Notice that we are looking at the evolution of the system in Euclidean space� hence the pointsqp� � n�� n � � � � �� are di�erent� This is done just for the purposes of illustration�


0 10 20 30 40 50 60 70 80 90 100−10

−5

0

5

10

time [sec]

q_c

Controller state

0 10 20 30 40 50 60 70 80 90 100−0.06

−0.04

−0.02

0

0.02

0.04

0.06

time [sec]

u_p

Control torque

Figure !�� Applied control and controller state for qp��

B Robot manipulators

The rigid�joint robot manipulators � �!� is another system to which our results can beapplied� Note that for manipulators with purely rotational joints� the potential energyfunction has only linear and trigonometric functions of the generalized positions whilefor a purely translational�joints manipulators� the potential energy function has linearfunctions only� A more general case is the combination of the two previous ones thus�the restriction �!�!� holds and our results can be directly applied�

Using SimulinkTM of MatlabTM� we tested our algorithm in the two link robotarm of �� see also p� �� with a desired reference qp� � �� We haveimposed the input constraint umax

pi� Nm in �!� $� To meet the conditions

of Proposition !��! we chose A � diagf� � � g� B � diagf�! � �! g while thecontroller gains were set to K� � diagf�� g� K� � diagf� �� g according to�!�# �

Then� in order to evaluate the performance of our controller� we tested as wellthe one proposed in �� !� with exactly the same gain values and starting from initialconditions qp� � ��#� ��#�� and in accordance with the previous discussions we setqc� � ��! �� ! �� in order to make ��

In Fig� !�� we show the transient of the �rst link position using the algorithmof �� !�� i�e� the non saturated controller and the control input signal yielded by this


0 1 2 3 4 5 6 7 80.5

1

1.5

2

2.5

3

3.5

Time [s]

[rad

s]

First link response and reference

0 1 2 3 4 5 6 7 80

100

200

300

400

500

600

Time [s]

[Nm

]

NON saturated control input

<−− 503

Figure !�� EL Controller of Subsection �#�B�

controller� In Fig� !��! we show the response of the same link driven by the saturatedcontroller of Proposition !��! and its control input�

On one hand� notice that the transient produced by the non saturated controller ismuch faster than the response using saturated controls� On the other hand� it mustbe remarked that the control input yielded by the linear controller fails to satisfythe input constraint� in particular the maximum absolute value of up is � !Nm forthe �rst link� In contrast to this the saturated controller yields a control input withjupijmax � �� Nm�

Thus we verify what is not surprising� that there is a compromise between a fastand smooth transient and small control inputs�

Remark �� Stabilization of a marine vessel�� Another EL system to whichour saturated EL controllers can be applied is the Lagrangian ship model discussedin ��#�� The construction of the controller follows along the same lines as for therigid�joints robot� It is interesting to note however that since in this case g�q � bwhich is constant then there is no di�erence between cancelling and compensating forthe potential energy�

� Set�point regulation under parameter uncertainty ��

0 1 2 3 4 5 6 7 80.5

1

1.5

2

2.5

3

3.5First link response and reference

Time [s]

[rad

s]

0 1 2 3 4 5 6 7 8−100

0

100

200

300

Time [s]

[Nm

]

Saturated control input

<−− 216.65

Figure !��!� Saturated EL controller without cancelation of g�q�

Set�point regulation under parameter uncertainty

As it was previously discussed� in our EL controllers approach it is it assumed that thepotential forces are accurately known� which in practice rarely happens� In this sectionwe explore the set�point control problem with uncertain knowledge of the potentialenergy� We make particular emphasis on PID control� For the position feedbackcase we propose the so�called PI�D controller which� by means of a linear dynamicextension and a second integrator solves this problem without velocity measurements�

Notation In this section we adopt the compact notation km and kM to denotepositive constants for a positive de�nite matrix K � IRn�n� such that km kxk� x�Kx kM kxk�� for all x � IRn�

De�nition �� Set�point control problem under parameter uncertainty�For the system ��

D�q-q & C�q� 'q 'q & g�q � u� �!�##

assume that the potential energy function V�q is not known accurately but only thebounds kg and kv de�ned in �� are available� Then� �nd a �output feedback con�troller which renders the closed loop system asymptotically stable� that is� a �output


feedback controller such that

limt�

k�qp�tk � limt�

kqp�t� qp�k � �

where qp� is the desired constant position�


It is interesting to stress that the conception of the popular PID controller is appar�ently due to Nicholas Minorsky� and goes back to �� In the robotics literature�there is a huge body of research on the subject� among the �rst we can mention ��In � � � it is shown that the PID controller �!�#$� �!�#� is locally asymptoticallystable provided the gains Kp� Ki and Kd satisfy some complex relationships�

A tool commonly used in the proof of asymptotic stability are Lyapunov func�tions with cross terms of the form ��q�D�q 'q or ��q� 'q� � � � This type of functions�which are also used throughout this section� have been widely used in the literaturestarting probably with ��!�� see also �$� � � $� �#� and references therein� Themotivation being usually to construct a Lyapunov function with a derivative contain�ing more negative terms than � 'q�Kd 'q� for instance quadratic negative terms of theposition error� This allows to prove GAS of the origin by invoking standard Lyapunovtechniques�

Unfortunately the price paid for the extra terms in the Lyapunov function deriva�tive is the apparition of cubic terms of the form �kc k�qk k 'qk which can be dominatedonly locally� Interestingly enough� this technical problem can be overcome by nor�malizing the cross term� The use of normalized cross terms goes back at least to thework by Koditscheck ��!#��

In this section we will revise these techniques showing that the use of the crossterms leads in fact to the de�nition of passive maps with respect to outputs of theform ��q& 'q� This is important in the sense that the relative degree of the system withrespect to this output is one� and henceforth� properties as OSP can be claimed�

Unfortunately� all the above�mentioned approaches can be applied only to fully�actuated systems� As far as we know the only results where underactuation is con�sidered are � � and �#�� for the particular case of �exible�joint robots� In the �rstreference� the author makes use of the global contraction mapping theorem to provesemi�global ultimate boundedness of the solutions� In the second reference Burkovproposes a common PID controller� the novelty is the use of singular perturbationtechniques in order to prove that �there exists� a su�ciently small integral gainsuch that the closed loop system is globally asymptotically stable� Unfortunately� nospeci�c bounds for the integral gain are given� Both approaches are based on theassumption that velocity measurements are available�

�� Adaptive control ��

�� Adaptive control

We have shown above that an easy way of shaping the potential energy of the plantis by substituting it with another function of the generalized positions which hassuitable properties� most typically we choose a quadratic potential� The latter leads tothe PD controller �!�! which cancels the natural gravitational forces vector from thedynamics� In the case when this vector is not accurately known a natural remedy is totry to parameterize this function in terms of the unknown parameters usually thosedependent on the payload and use an adaptation law to estimate them� As shownin Eq� � �� the parameterization is linear� hence standard estimation techniques

may be used� That is� we can write g�q�� 0�q�� where 0�q contains some known

functions and � � IRq is the vector of unknown parameters� An adaptive version ofthe PD plus gravity cancelation controller is obtained as

u � �Kp�q �Kd 'q & 0�q(� �!�#�

where (� are the parameter estimates to be updated with some adaptation law� Thedynamics �!�## in closed loop with �!�#� are given by

D�q-q & C�q� 'q 'q &Kp�q &Kd 'q � 0�q��

where �� (� � � is the parameter error� It is easy to see that these dynamics de�nean OSP operator 0�q�� 'q with storage function

H�q� 'q � T �q� 'q &�

�q�Kp�q�

On the other hand� it is well known ��!� that the standard gradient estimator

'(� � ��0��q 'q �!�#�

with � � � de�nes a passive operator 'q �� 0�q�� with storage function

H ��

��

That is� the closedloop system consists of the feedback interconnection of a pas�sive and an OSP operators� Invoking Proposition A�� of Appendix A we have thefollowing�

Fact �� The system �!�## in closed loop with �!�#�� !�#� and external input� � Ln

�e� i�e��

D�q-q & C�q� 'q 'q &Kp�q &Kd 'q � 0�q��

'�� 0��q 'qde�nes an OSP �and in view of Proposition A�� L�stable operator � �� 'q withstorage function H &H�


Noting that the storage function is positive de�nite and invoking Proposition A�� of Appendix A we could conclude from here asymptotic stability if we could verifyzero�state detectability �with respect to the output 'q� Unfortunately� this is not thecase� To see this remark that 'q � does not imply that �� q� '�q� �

To go around this problem we need to establish a new passivity property for theEL dynamics� To this end we need to add to the total energy function cross�terms tode�ne the new storage function� This topic has been thoroughly studied �in the formof Lyapunov analysis in �$� � � $� �#� and will be explained in the next section�In particular we have the following proposition�

Proposition �� Passivity of PD plus gravity compensation control� Thedynamics

D�q-q & C�q� 'q 'q &Kp�q &Kd 'q � 0�q��

de�nes� an OSP operator .� � 0�q�� 'q & ��q� where

��

�

�� & k�qk�

with positive de�nite storage function

HN�q� 'q ��

'q�D�q 'q &

�

�q�Kp�q & ��q�D 'q

provided � � is su�ciently large� �

This property motivates us to consider the estimation law

'(� � �0��q�� 'q &

�q

� & k�qk��

�rst proposed by Tomei in � �� which we know de�nes a passive operator � 'q&��q ��0�q�� yielding the desired OSP for the closedloop� Unfortunately� we still do nothave the required detectability property� because �� has a manifold of equilibria� Weneed then Theorem �� from �#$� which states that for the system 'x � f�x� y � h�xthe following implication holds

x�t � L and y � L� � limt�

h�x�t �

We can apply this theorem to our system with the new �output� '�q & ��q� which issquare integrable in view of the OSP� Boundedness of the trajectories follows notingthat the function HN �q� 'q &

�� is nonincreasing�

�Notice the present of the cross product of q and �q in the third right hand term�

�� Linear PID control �

�� Linear PID control

For the sake of clarity� let us �rst review the passivity properties of the EL systemformed by the closed loop of �!�## �for simplicity let us assume M � I with a PDplus gravity compensation controller as depicted in Fig� !��#�

~qg(q

*)

. q

K d

1Σ

+

pK

-

-

Σ2

EL Plantv +

-+

v2

u

Figure !��#� PD plus gravity compensation�

From Proposition �� we know that the plant�s total energy function T �q� 'q&V�qquali�es as a storage function for the supply rate w�u� 'q � u� 'q� It is not di�cult tosee that this passivity property is conserved after the feedback u � �Kp�q& g�q� & vwith v an external input� By setting v � �Kd 'q & v� the passivity is strengthen tooutput strict passivity from the input v��

Interestingly enough� if the potential energy function is not known accurately thepassivity properties of the closed loop are preserved� Let v� � �g�q� & (g�q� & v�then the actual control input will be u � �Kp�q& (g�q�& v�� A simple analysis showsthat the closed loop system of �!�## with this new control input de�nes an outputstrictly passive map . � v� �� 'q for any positive de�nite Kd and storage function

H � T �q� 'q & Ug�q &�

�q � (��Kp�q � (�

where in analogy with �!�# (��q� � q� & K��p (g�q�� It is important to remark how�

ever that even though the condition for H to be positive de�nite is still that Kp besu�ciently large �speci�cally Kp � kgI the global minimum of H is not necessarilyq � q�� In other words� even though the input output properties are unchanged� thesystem is no longer GAS at the desired equilibrium� That is� with respect to Fig�!� this control law �pulls� the pendulum to a constant position di�erent from thedesired one q��

As it is well known� the steady state error resulting from the mismatch betweeng�q� and (g�q� can be compensated by means of a simple integrator� Unfortunately�the price paid in terms of passivity is that the property looses its global character�


that is� the resulting system is only locally output strictly passive � �� !$�� This issummarized in the following proposition�

~qg(q

d)

. q

1ε

K i0

t

K D

ε

-

+

+

++

z

g(q

PK

d)

-

+

Σ2

1

+

-

+

0

Σ

EL Plant

−ζ

ζ

z

-

u+

’

Figure !�� Passivity interpretation of PID Control�

Proposition �� PID is locally passive� Consider the feedback interconnectedsystem depicted in Fig� �� which corresponds to the closed loop of the EL plant�� and the PID controller

u � �Kp�q �Kd 'q & � �!�#$

'� � �Ki�q �!�#�

where �q�� q� q�� q� is the desired link position which is assumed to be constant� Kp�

Kd� Ki are positive de�nite diagonal matrices� Then for a su�ciently small constant� � the following is true�

� The map .� � z �� is locally output strictly passive�

�� The map .� � �� z is passive�

Proof� Let us choose any positive de�nite diagonal matrix K �p such that

Kp�� K �

p &�

�Ki

where � � is a �small constant to be determined� clearly Kp is also positive de�niteand diagonal for any � � � Then the error equations �!�##� �!�#$� �!�#� which

�� Linear PID control �

correspond to the internal dynamics of the maps .� and .� can be written as

.� � D�q-q & C�q� 'q 'q & g�q� g�q� &K �p�q &Kd 'q � z �!�#�

.� � 'z � ��q & 'q �!��

where we have de�ned �� g�q� and z � ��

�Ki�q� Consider then the storage

function

H��q� 'q ��

'q�D 'q & Ug � Ugd � �q�gd &

�

�q�K �

p�q & ��q�D 'q �!��

where we have dropped the arguments and de�ned Ugd

�� Ug�q�� gd

�� g�q� to

simplify the notation� We �nd it convenient to this point to split the kinetic� andpart of the potential energy terms as

�q�K �p�q � �� & �� & ��q

�K �p�q

'q�D�q 'q � �� & �� & �� 'q�D�q 'q

with � � �i � � i � �� !� Then one can show that if

k�pm � max

�kg��dM��

��

then function H��q� 'q satis�es the lowerbound�

H��q� 'q � �� q�K �

p�q &�� & ��

'q�D 'q

hence it is positive de�nite and radially unbounded� The motivation for this parti�tioning of the energy terms will become more evident in the sequel� Next� using thewell known bounds � � � and

we obtain that the time derivative of H��q� 'q along the trajectories of �!�#�� !�� is bounded by

'H� ��kdm �

�

kdM � �kc k�qk � �dM

�k 'qk� � �

�k�pm � kg � �

kdM

�k�qk� & z��

�!��

Notice that 'H� with z � is negative semide�nite for instance if

kdm � ��kdM & dM

k�pm � kg &�

kdM

k�qk kdm �kc

� �!��!


After completing some squares� one can show that there exist strictly positive con�stants �o and such that

'H� ��o k�k� & z��

thus completing the proof of �� Notice that the existence of �o � is conditioned to�!��! thus the local character of the passivity property�

To complete the proof notice that the operator .� is an integrator� which is thesimplest passive system one can think of� The property is established immediatelyby considering the storage function

H� ��

�z�K��

i z� �!��#

�

We have established above the passivity properties of a simple PID in closed loopwith a fully�actuated EL system� The convergence of the signals �q and 'q to the origincan be easily established using Lemma A�� and observing that � � Ln

� � � � ast � � and � � � Furthermore from �!�#�� !�� we conclude that 'z � andz�t� � From the de�nition of z we obtain the convergence of ��t� � Hence thesystem is zero�state detectable and GAS follows�

Alternatively one can take the Lyapunov function candidate V � H� &H� whichis positive de�nite under the conditions established above and whose time deriva�tive is negative semide�nite� Local asymptotic stability of the origin col��q� 'q� �� col� � � follows invoking Krasovskii�LaSalle�s invariance principle� Furthermoreone can de�ne a domain of attraction for the closed loop system �!�#�� !�� asfollows� De�ne the level set

B ��

�x � IR�n � V �x �

�where � is the largest positive constant such that 'V �x for all x � B �

�� Nonlinear PID control

As it is clear now from the proof of Proposition !�� what impedes claiming outputstrict passivity for a PID controller in the usual �global sense� is the presence of thecubic term �kc k�qk k 'qk� in the time derivative 'H�� This technical di�culty can beovercome by making some �smart� modi�cations to the PID control law� leading tothe design of nonlinear PID controllers�

To the best of our knowledge� the �rst non�linear PID controllers that appearedin the literature are �� and �$�� In this section we discuss these controllers from apassivation point of view�

�� Nonlinear PID control �

A The normalized PID

In order to cope with the cubic term �kc k�qk k 'qk� in �!�� and inspired upon theresults of Tomei revised in Section #� � Kelly �� proposed the �adaptive� PD con�troller

u � �K �p�q �Kd

'�q & 0�q�(� �!��

together with the update law

'�� '(� � ��

�0�q�

��'q &

��q

� & k�qk�

�!��

where �� is a small constant� In the original contribution �� Kelly proved thatthis �adaptive� controller in closed loop with a rigid�joint robot results in a globallyconvergent system� However� since the regressor vector 0�q� is constant the updatelaw �!�� together with the control input �!�� can be implemented as a nonlinearPID controller by integrating out the velocities vector from �!��

(��t � ��

�0�q��

��q�t &

Z t

�

��q�

� & k�q�kd�& (�� !��$

Notice that the choice Kp � K �p &Ki� with Ki �

��0�q�0�q�� yields the controller

implementation

u � �Kp�q �Kd 'q & � �!��

'� � ��Ki�q� �� IRn� �!��

where we have rede�ned

��

�� & k�qk � �!��

Since controllers �!�� !��$ and �!�� !�� are equivalent� following the steps ofKelly �� one can prove global asymptotic stability of the closed loop system �!�##��!��!��

Furthermore� following the same steps as in the proof of Propositions !�� and !��$one can show that the normalization introduced in the integrator helps in enlargingthe domain where the passivity property holds� hence rendering the system .� � z �� where we rede�ned

z�� Ki�q & �� !��

output strictly passive� This can be shown by evaluating the time derivative of thestorage function H� �using �!�� and �!�� along the trajectories of .� whosedynamics is de�ned by �!�#�� One obtains after some straightforward bounding

'H� ��kdm � � k�qk � ��dM � ��kdM

�k 'qk� � ��k�pm � kg � ��kdM

k�qk� & z��


which is similar to �!�� however� notice from �!�� that �kc k�qk k 'qk� kc k 'qk��hence the condition �!��! is no longer needed� The output strict passivity is thenestablished following similar arguments as in the proof of Proposition !��

It is worth remarking that even though the normalization of the cross term in thestorage function and in the integral gain� allows to claim OSP notice that the number��

��k�qk is multiplying the term �q in 'H�� While Lyapunov global asymptoticstability can be claimed using the Lyapunov function V � H�&H�� it can be expectedthat the rate of convergence for large position errors� be small since � 'V has then alinear growth in k�qk�

A second important remark is that the normalization just introduced can bethought of as the same e�ect of a saturation function as de�ned in Def� !�� Thisleads us to the second passive nonlinear PID controller�

B The saturated PID controller

An alternative trick to achieve output strict passivity for .� is to saturate the pro�portional feedback term instead� Even though with a Lyapunov design motivationsuch idea was �rstly presented as far as we know in �$� where the following nonlinearPID� was proposed�

u � �K �p sat��q� �

�Ki�q �Kd 'q & � �!��

'� � �Ki�q� �� IRn

where in this case � � is a small constant and sat shall be considered component�wise� Arimoto proved in �$� that if kpm � kg� and Ki is su�ciently small� the closedloop is passive and moreover it is globally asymptotically stable�

The key idea used in �$� is to dominate the cubic terms in the derivative of thestorage function by means of the saturated proportional feedback in �!�� Moreprecisely� one can prove that the map .� � z �� with input z � ��

�Ki�q� output

� � � sat��q & 'q and internal dynamics

.� � D�q-q & C�q� 'q 'q & g�q� g�q� &K �p sat��q &Kd 'q � z

is output strictly passive� This can be established by derivating the storage function

H�a��q� 'q ��

'q�D�q 'q & Ug�q� U�q�� q�g�q� &

�

�q�K �

p�q & � sat��q�D�q 'q

�which is positive de�nite for � � su�ciently small to obtain

'H�a��q� 'q ��kdm � �kc k sat��qk � �dM � �

kdM k 'qk�

��k�pm � kg � �

kdM �q� sat��q & z�� !��!

�It is worth mentioning that in �� Arimoto used a saturation function which is a particular caseof sat considered here� however this point is not fundamental for the validity of the result�

�� Output feedback regulation� The PI�D controller �

then using k sat��qk � and similar arguments as in the previous proofs�

Notice however� that as in the previous case one gets rid of the cubic terms butthe price paid for enhancing thereby the passivity property is a slower convergenceof the signal �q due to the linear growth of �q� sat��q for large error values�

�� Output feedback regulation� The PI�D controller

~qg(q

d)

. q

. q

K D p+ab

++

PK

ϑ

-EL Plant

Σ

+z

-

Figure !�� The EL controller of Kelly ��

In this section we solve the last practical problem raised at the end of Section ��that of position feedback setpoint control with uncertain gravity knowledge� Ourcontribution is the so called PI�D controller originally presented in � $�� In order toput the PI�D controller in perspective� let us brie�y summarize the results we havepresented so far concerning the set�point control problem for EL mechanical systems�

As we have seen� one can identify several solutions to both problems separatelythat is� on one hand the controllers which do not need measurement of generalizedvelocities need exact a priori knowledge of the potential energy� and on the otherhand� the di�erent approaches to regulation with uncertain potential energy knowl�edge needed the measurement of the generalized velocities�

To the best of our knowledge the �rst solutions to the problem of designing anasymptotically stable regulator that does not require the exact knowledge of g�qd northe measurement of speed appeared independently in � $� and �� The contributionof � $�� the PI�D controller� is a semiglobally stable control law that solves this prob�lem� The authors of �� obtained a similar result for robot manipulators consideringthe dynamics of DC actuators�

We proceed now to show how the PI�D controller can be constructed from theinterconnection of passive blocks we are already familiarized with� Our starting pointis the EL controller of Kelly �� revisited in Subsection �#�B� � As we have pointed


out� the closed loop system depicted in Fig� !�� constitutes a passive operator. � z �� 'q� On the other hand we know that the PID controller in closed loop withthe EL plant �!�## is locally output strictly passive�

To this point� we might wonder if there is a way to introduce the OSP block de�nedby the approximate di�erentiation �lter� into the PID controller �!�#$� �!�#� withoutdestroying the passivity properties of the closed loop� as we have done with the PDplus gravity compensation� Fortunately the answer is a�rmative as the followingproposition shows�

~qg(q

d)

. q

1ε

K i0

t

K D p+ab

ε

ε

-

+

-

+

+

z

g(q

PK

d)

ϑ

-

Σ2

1

+

-

+

0

Σ

EL Plantζ

−ζ

z

-

+

+

+

’

Figure !��$� PI�D Controller� block diagram�

Proposition �� Passivity of the PI�D controller�� For the system depictedin Fig� �� de�ned by

D�q-q & C�q� 'q 'q & g�q &K �p�q � � &Kd� � z �!��#

'� � �A� &B 'q �!��

'z � �Ki��q &�

�'q � � �!��

where the �rst two equations represent the dynamics of .� and �� is the integratorblock� there exist some control gains Kp� Kd� Ki� A and B such that

�i System .�� de�nes a Locally Output Strictly Passive �LOSP

operator z �� where �� q � �� and storage function

H��q� 'q� �� H��q� 'q &

�

��KdB

�� q � ��D�q 'q�

�� Output feedback regulation� The PI�D controller �

where H� is de�ned by ��

�ii System .�� de�nes a passive operator �� z with storage functionH��z� where H� is de�ned by ��

�

Proof� For clarity of exposition we present below the main steps of the proof� formore detail we refer the reader to the Appendix D�

�i� By de�ning

z��

�Ki�q & ��

we show in Appendix D that for all ��q� �� 'q satisfying some suitable conditions� thetime derivative ofH��q� 'q� � along the trajectories of �!��#� �!�� satis�es the bound

'H��q� 'q� � ��k�qk� � ��k 'qk� � ��k�k� & z��q � � & 'q� �!��$

where the strictly positive constants �� are de�ned in �D�$� Also notice that

k��q � �� & 'qk� ��k�qk� & k�qkk�k& k�k� & �k 'qk�k�qk& k�k & k 'qk�

from this� we obtain after some straightforward calculations that

��k 'qk� & ��k�qk� & ��k�k� � k��q � �� & 'qk�� !��

if �� & � and �� & �� This clearly imposes new conditions on thecontroller gains�

�

bmdm � �� !��

kpm �

#� � �!�$

kdmam#bM

� �� & �� !�$�

The condition imposed by inequality �!�� is met for su�ciently large bm while �!�$ holds for a su�ciently small � and �!�$� holds for su�ciently large am� Integrating�!��$ from to T � on both sides of the inequality and using �!�� we get

H��T �H�� k��q � �� & 'qk��T & hz j ��q � � & 'q�iTsince H��q� 'q� � is positive de�nite for all T � �

hz j ��q � � & 'q�iT � k��q � �� & 'qk��T � �

hence completing the proof�

�i� Straightforward� �.� is a simple integrator� �

Ch� !� Set�point regulation

Remark �� Notice that inequality �!��$ only holds locally� that is if �D�� issatis�ed� This is what leads us to local output strict passivity�

We have thus shown how to construct a PI�D controller by interconnecting passiveblocks� The following proposition which is the original contribution of � $�� de�nesthe PI�D controller�

Proposition �� SGAS of the PI�D controller�� Consider the dynamic model�� in closed loop with the PI�D control law

u � �Kp�q & � �Kd� �!�$

'� � �Ki��q � �� IRn �!�$!

'qc � �A�qc &Bq �!�$#

� � qc &Bq �!�$�

Let Kp� Ki� Kd� A��diagfaig� B �

�diagfbig be positive de�nite diagonal matrices

with Kp�� K �

p &��Ki and

B �#dMdm

I �!�$�

K �p � #kg

where kg is de�ned by ��

Under these conditions� we can always �nd a �su�ciently small integral gain

Ki such that the equilibrium x�� q�� 'q�� g�q�� is asymptotically

stable with a domain of attraction including�x � IR�n � kxk � c�

�where limbm� c� � �� In other words� given any �possibly arbitrarily large initialcondition kx� k� there exist controller gains that ensure limt� kx�tk � � �

Remark �� Notice that control law �!�$ can be alternatively written as

u�t� � �Kp�q�t��Ki

Z t

��q��d� �Kddiag

�bip

p� ai

�q�t��Ki

Z t

�diag

�bip

p� ai

�q��d��

The �rst three right hand side terms constitute the proportional� integral and �lteredderivative actions� while the presence of the fourth right hand term motivates thename PI�D� We have kept the notation D for the derivative term because in practicalapplications of PID regulators this is commonly implemented incorporating a �lter�

Proposition !� � establishes the semiglobal stability of the PI�D controller� in the sensethat the domain of attraction can be arbitrarily enlarged with a suitable choice of thegains� namely by increasing bm� However� as shown in the proof �cf� Subsection A �the stability conditions impose an order relationship between Ki and B such that wemust correspondingly decrease Ki�

�� Output feedback regulation� The PI�D controller

A Sketch of proof of Proposition ��

Once we have established the local output strict passivity of the map .� � z �� and accounting for the passivity of the integrator .�� the next step to prove internalstability would be to prove zero�state detectability for the system .� however� incontrast to Section #�! one cannot conclude much about the behaviour of the state��q� 'q� � from the fact that � � � Therefore we rely on Lyapunov theory�

Since we have established some passivity properties for the closed loop by usingstorage functions which are based on the total energy of the system� we can usenow the Lyapunov function candidate V � H� &H� which is positive de�nite underthe conditions established in the Appendix D� Further� the time derivative of Valong the closed loop trajectories �!��# � �!�� is negative semide�nite as it canbe appreciated from �!��$� Asymptotic stability of the equilibrium ��q� 'q� �� z � follows invoking Krasovskii�LaSalle�s invariance principle� observing that the latter isthe largest invariant set obtained in 'V � �

Furthermore as proven in Appendix D a domain of attraction is given by

kxk c��

�

kc

��

bmdm � dM

�r��

��

�

where �� and bm are positive constants such that bmI � B and

��kxk� V �x ��kxk��

A�� Semiglobal asymptotic stability

To establish semiglobal asymptotic stability �SGAS we must prove that� with a suit�able choice of the controller gains� we can arbitrarily enlarge the domain of attraction�To this end� we propose to increase bm and bM at the same rate� The key questionhere is whether this can be done without violating the order relationships betweenB and � imposed by the stability conditions �D�� D�� and �D�� The orderrelationship due to �D�� is ��bm � O��

pbm� while that of �D�� and �D�� is

��bm � O��bm� The latter being implied by the former for � su�ciently small�

On the other hand� for bm su�ciently large we can always �nd � � so that�� c��bm and �� c�� where c�� c� are constants independent of B� Replacing thisin �D��! we get

limbm�

c�� limbm�

cpbm

where c is also independent of B� This proves that there exists � � such that �the stability conditions� are satis�ed� i�e�� verifying � � O��

pbm� the domain of

attraction is arbitrarily enlarged� that is� limbm� c� ��

The proof is completed choosing� for the given �� 1��Ki � O��


Remark �� It is important to remark that when implementing the PI�D con�troller� one should be careful with setting the initial conditions of the dirty derivatives�lter �!�$#�!�$�� from the calculations above� the initial conditions �� should besmall enough to guarantee asymptotic stability� Also� in order to enlarge the domainof attraction� we proceed to increase B� Nevertheless� notice from �!�$� that theinitial condition �� qc� &Bq� � hence the larger B is� the larger �� may alsobe� A simple way to overcome this problem� is to set qc� � �� Bq� for some�xed �� q� and B�

B Simulation results

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4first link position error

[rad

s]

[sec]

Figure !�� PI�D control� First link position error�

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4first link position error

[rad

s]

[sec]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−10

−5

0

5

[sec]

[rad

s]

First link noisy velocity

Figure !�� PID control under noisy velocity measurement�

� Concluding remarks �

In this section we illustrate in simulation the performance of our algorithm in thetwo�rigid�link robot manipulator model of � �� see also p� �� For the sake of com�parison we tested the performance of a simple PID controller with velocity measure�ments� We assumed that the measurements are a�ected by a random noise of � )which obeys a uniform distribution� In both cases� the controller parameters were

set to KP�� diagf�# � # �g� KI

�� diagf� � #� � ��g� KD

�� diagf�! � ! �g�

A�� diagf�! � ! �g and B

�� diagf�� g� We started from zero initial

conditions to achieve the constant reference q��

In Fig� !�� we show the �rst link response using our PI�D controller� In Fig� !��we show the �rst link response using a common PID controller as well as the noisyvelocity measurements corresponding to the same link� As a criterion of comparison�we have evaluated the integral square position error� �ISE i�e� J �

R �

��qpdt� For

the PI�D controller� this calculation resulted in J � �� while for the ordinary PIDcontrol� J � !��$�

Concluding remarks

We have addressed in this chapter the set�point control problem of EL systems byoutput feedback� The controller� which we choose to be also an EL system� is designedusing the energy shaping plus damping injection ideas of the passivity�based approach�

Our contribution to this problem is the proof that asymptotic stabilization withoutgeneralized velocity measurement is possible via the inclusion of a passive dynamicextension provided the system satis�es a dissipation propagation condition�

An important extension of these results includes the EL controllers which yieldbounded inputs� We have identi�ed a class of EL plants which can be regulated withsaturated controllers depending on the growth rate of the potential energy function�In words� the key requirement is that this function does not grow faster than linearlyfor large position values�

We have illustrated this technique with the case studies of the TORA system andthe �exible joints robots� For the TORA system we proved the superiority of ourPBC with respect to the backstepping controllers of �� both in performance andsimplicity of implementation� For �exible joints robots we showed that several appar�ently unrelated controllers which appeared independently in the literature happen tobelong to the EL class� We illustrated in simulations the performance of the two ELcontrollers �!� � ��

Finally we revised the setpoint control problem with uncertain gravity knowledge�in particular� we revisited the popular PID control from a passivity point of view�For the case of unmeasurable velocities we have presented the PI�D controller� whichrelies on the passivity properties of the EL plant to ensure SGAS�

Chapter �

Trajectory tracking control

In this chapter we extend the passivity�based method� developed for regulation in theprevious chapter� to solve trajectory tracking problems� The �rst main modi�cationthat we have to make is that for tracking� besides reshaping the potential energy ofthe EL plant� we must also shape the �kinetic energy� function� Whereas modifyingthe potential energy function means to relocate the equilibria of the system� themodi�cation of the �kinetic energy� function can� roughly speaking� be rationalizedas imposing a specic pattern to the transformation of potential into kinetic energy�However� the quotes here are important because the storage function that we assignto the closed loop is not an energy function in the sense that it de�nes the equationsof motion� With an obvious abuse of notation we will still refer to this step as energyshaping� but it is better understood as passivation with a desired storage function �seeAppendix A� The damping injection step is added then to make the passivity strict�The passivation objective is achieved invoking the key passive error dynamics Lemma �$� which states that we can always factor the workless forces in such a way that� interms of the error signals s� the EL system behaves like a linear passive system�

In Section �� we have introduced already a PBC that modi�es the kinetic energy�namely the PBC discussed in Section �� #�B�� As pointed out in Remark !�� thismodi�cation requires an additional dynamic extension� besides the one used to injectdamping� It will become clear below that to address tracking problems the PBCprocedure gives �rst an implicit de�nition of this dynamic extension� which has tobe later explicitly realized� As we will see in Section �� the realization is immediatefor fully actuated systems� however� for underactuated systems like the �exible jointrobot discussed in Section !�! this implies an additional step� which involves somekind of �system inversion�� The importance of this step will be further underscoredin our applications to electromechanical systems in Part III�

A �nal di�culty for applying PBC in tracking is that� with respect to position�the EL dynamics is relative degree two� Hence� we cannot apply passivity principles�that require a unitary relative degree directly to this signal� This obstacle can

�!

� Ch� #� Trajectory tracking control

be overcomed via a suitable de�nition of the error signal s� for instance with theingenious inclusion of a linear �lter� as used in the robot control literature � ��

In Section � we address the problem of state feedback tracking control of fully�actuated mechanical systems� We consider the case of known parameters as well asadaptive schemes� PBC is particularly suited for the latter because� as we pointedout in Section ��#� � the standard gradient estimation algorithm de�nes also a passivemap� thus passivity �and consequently L� stability of the overall scheme are ensured�

Section ! is devoted to the extension of PBC to underactuated EL systems� alwaysassuming statefeedback� We take as case study the model of robots with �exiblejoints� This problem has a strong practical motivation for high performance robotswhere the elasticity phenomenon is no further negligible and has to be explicitly takeninto account in the design� typically adding a linear torsional spring as in regulationproblems� It is also a very challenging theoretical problem since the number of degreesof freedom of the system is twice the number of control actions and the matchingproperty between nonlinearities and inputs is lost�

We �nish the chapter with some results of PBC to the di�cult output feedbackcase�

� State feedback control of fully�actuated systems

The solution to the state feedback tracking control problem of fully actuated ELsystems �in particular� rigid�robot manipulators has been known from many yearsnow� for a literature review see e�g� � � � �!��

In this section we study the passivity based tracking control technique for fully�actuated systems� The approach is based on the passivity property of EL systemsestablished in Proposition �� More precisely consider the following problem�

De�nition �� State feedback tracking control of fully�actuated systems�Consider the system �� which we repeat here for convenience�

D�q-q & C�q� 'q 'q & g�q � u �#��

where u � IRn� Design a state feedback control law u � u�t� q� 'q that ensure

limt�

k�q�tk � limt�

kq�t� q��tk � �#�

for any desired time�varying trajectory q� � C��

Below we revise the well known controllers of Slotine and Li � �� and the PD&controller of Paden and Panja � �� from a passivity point of view�

� The PD� controller �

� The PD� controller

The PD& controller originally introduced in � �� was one of the �rst results guaran�teeing global tracking for rigid�joint robots� that is global uniform asymptotic stability�GUAS of the origin ��q� '�q � � � � It is the natural extension to tracking control�of the energy�shaping plus damping injection PD plus gravity cancelation used forregulation �see Section ��

Motivated by the passive error dynamics Lemma �$ it is reasonable to aim at anerror system of the form

D�q-�q & C�q� 'q '�q &Kp�q &Kd'�q � / �#�!

where Kd� Kp are positive de�nite matrices� This system de�nes an OSP map / �� '�qwith storage function

H� '�q� �q ��

'�q�D�q '�q &

�

�q�Kp�q� �#�#

The proof follows straightforward� di�erentiating the storage function �#�# with re�spect to time� along the trajectories of �#�! to obtain

'H� '�q� �q '�q�Kd

'�q &/� '�q�

Hence if we can set / � then '�q � Ln� � which is our �rst step towards the proof

of convergence of the errors to zero� A control law that achieves this objective isprecisely the PD& control law of � ��

u � D�q-q� & C�q� 'q 'q� & g�q�Kp�q �Kd'�q �#��

which borrows its name from the last three terms that correspond to the PD controllerused for set�point control purposes�

This controller was proved in � �� to globally asymptotically stabilize the closedloop system �#�! with / � � by relying on Lyapunov techniques� In particularthe authors of that reference used Matrosov�s theorem � �� which involves tediouscalculations due to the computation of -H and H��

It is important to stress� however� that the proof of this statement via OSP andzero�state detectability is yet to be established� The underlying obstacle to carry outthis proof is that we have to reduce the relative degree �with respect to �q of theerror dynamics� which in this case is two� The latter is achieved by the Slotine andLi controller presented below�

Remark �� In Section !�� we present a controller which borrows inspiration fromthe PD& algorithm� for tracking control of �exible�joint manipulators�

� Ch� #� Trajectory tracking control

� The Slotine and Li controller

To reduce the relative degree of the error system let us rede�ne the error signal as

s�� '�q & 2�q� �#��

with 2 � � Notice that �q � �pI & 2��s� hence it is the output of a strictly properasymptotically stable LTI system with input s� Invoking Lemma A�� of AppendixA we have that if s � L� then �q � � This motivates us to aim at an error dynamicsof the form

D�q 's& �C�q� 'q &Kd�s � / �#�$

which de�nes an OSP operator / �� s with storage function

Hd�q� s ��

s�D�qs� �#��

It is easy to see that the system �#�$ is equivalent to the system �#�� with

/ � u� �D�q-qr & C�q� 'q 'qr & g�q &Kds �#��

where we de�ned 'qr � 'q��2�q� The controller that sets / � is obtained then in anobvious manner� This is the celebrated Slotine and Li controller � ��

Remark �� In � !!� we study the disturbance attenuation properties of PBC forrobots manipulators� We show that adding a proportional gain around the positionerror� as proposed in � �� we can achieve arbitrarily good disturbance attenuation�without compromising the convergence rate� More precisely� we prove the followingproposition�

Proposition �� Disturbance attenuation� Consider the EL system

D�q-q & C�q� 'q 'q & g�q � u& �

where � � L�e represents an external disturbance� in closed�loop with the PBCcontroller

u � D�q-qr & C�q� 'q 'qr & g�q�Kds�Kp�q

with qr� s as de�ned above� and �� kd� kp positive scalars verifying

Kd ��

��

��& � & �

�Kp �

�

�

�� & � & �

�for a �xed � � � Under these conditions� the following inequality holds

k� '�q�� q��k�T �k�k�Tfor all T � � Consequently� arbitrarily good disturbance attenuation �in the L��sense is achievable by increasing the gain kd� �

� Adaptive trajectory tracking �

Remark �� In the original contribution of � �� and several further applications ofthis algorithm� only convergence of the signals '�q and �q is actually proved� However itshould be noted that the far stronger property of global exponential stability of theorigin �'�q� �q � � � follows considering the Lyapunov function proposed in � �#�� andrestricting �without loss of generality Kd� Kp and 2 to be diagonal� This Lyapunovfunction will be instrumental for the analysis presented in Section !�! and Chapters� and ��

Remark �� It is shown in � � � that via a suitable de�nition of the error signals above we can recover various global tracking controllers reported in the litera�ture� That is� in general we can set s � H��p�q� with H�p strictly proper andasymptotically stable� and the proof will go through� In the case of � �� we chooseH��p � p & �� while Sadegh and Horowitz considered in � �� a PID� Anotherinteresting controller� studied within this framework in � � �� is the one proposed in�� #��

Remark �� A procedure� similar to the one presented above� can be used for someunderactuated EL systems� e�g�� robots with �exible joints and diagonal inertia ma�trix� However� the extension is not straightforward and one must be careful withchoosing a suitable desired potential energy� See Section !�!�

� Adaptive trajectory tracking

It has been shown in the regulation problem of Section ��#� that one of the advantagesof ensuring OSP of the closedloop is that� in view of the the passivity of the gradientestimator� adaptive extensions are straightforward� In this section we present anadaptive implementation of the PBC of Slotine and Li� One important drawbackof this scheme is that� as shown in � �� the parameter estimates may drift in thepresence of measurement noise� To overcome this problem we present the robusti�edversion reported in � ��

� Adaptive controller of Slotine and Li

To derive an adaptive version of the PBC above we invoke �rst the Property P�� oflinearity with respect to the parameters of the EL dynamics of Chapter � to write

D�q-qr & C�q� 'q 'qr & (g�q � 0�q� 'q� 'qr� -qr�

where � � IRq contains the unknown parameters and the regressor matrix 0�q� 'q� 'qr� -qrconsists of known functions�

Then� we propose a certainty equivalent adaptive controller of the form

u � 0�q� 'q� 'qr� -qr(� &Kds

Ch� #� Trajectory tracking control

with (� the estimated parameters� This yields the error dynamics

D�q 's& �C�q� 'q &Kd�s � 0�q� 'q� 'qr� -qr�� / �#��

where� with some abuse of notation� we have used again /� Finally� we propose aparameter adaptation law

'(� � �3��0��q� 'q� 'qr� -qrs �#��

where 3 � 3� � � Analogously to the regulation case we have an OSP operator/ �� s de�ned by �#�� in negative feedback with a passive operator s �� / de�nedby the estimator �#�� Hence� the closedloop system is OSP� and s � L�� The proofof global convergence of �q is completed� as before� invoking Lemma A��

� A robust adaptive controller

It is well known ��!� that the equilibrium set of adaptive systems is unbounded�Therefore� in underexcited conditions and in the presence of noise in the adaptationlaw� the instability mechanism of parameter drift appears� This instability mechanismappears� in particular� in the adaptive controller given above� To exemplify thisphenomenon� consider a single link pendulum moving in the horizontal plane� that isd-q � u where the inertia d � is unknown� Excitation is lost in regulation� thereforeconsider the desired position q� � const� Assume further that velocity is corrupted bynoise with zero mean and variance �� that is� � � N� � �� In these circumstancesthe adaptation law �#�� looks like

'(d � �3�� 'q & �� 'q & � & ��q

whose expectation is

Ef '(dg � 3� 'q� 'q & ��q & 3��

Consequently� the integral of the right hand term introduces a parameter drift� thusthe controller is not robust to measurement noise� As it will become clear below� inthe adaptation law we propose here� there are no quadratic terms of 'q and the driftphenomenon disappears�

In this section we present a controller� originally introduced in � �� which hasthe following features� � enhanced robustness with respect to velocity measurementnoise� in particular� the aforementioned drift is avoided� does not require high gainloops� ! we provide a relationship between convergence rates and compensator gainsthat is independent of the desired trajectory� To achieve these objective we introducetwo key modi�cations to the Slotine and Li controller� the inclusion of an additional�Coriolisforcesdependent� term and the use of a normalization factor similar to theone presented in Section ��#� �

�� A robust adaptive controller

We present below the main result and refer the reader to � � for further detailsand some illustrative simulations� As standard in PBC� we start by remarking thepassivity property of a desired closed loop system�

Proposition �� OSP of the error equation�� The error equation

D�q-�q & C�q� 'q '�q & �C�q� �q 'q� &Kd'�q &Kp�q � / �#��

with Kp� Kd positive de�nite and satisfying

�� min

�kdm

!dM & kc�

#kpmkdM & kdm

�� #��!

de�nes a passive operator / �� s with storage function

Hd�s� �q ��

s�D�qs&

�

�q�Kp�q �#��#

where s � '�q & ��q� and�

��

�� & k�qk �#��

where ��

Proof� �sketch� Evaluating the time derivative ofHd along the trajectories of �#�� we obtain after some bounding and using the Properties P�� P��

'Hd�s�� q �� ks�k� � ��

��q � & s�/ �#��

where we have de�ned s�� '�q & �

��q and the constants

�� kdm � !��dM � ��kc

��

#kpm��

� kdM � ��dM � ��kc� �#��$

which are positive in view of �#��$� The proof is completed integrating on both sidesof �#��

Motivated by this passivity result we construct the stabilizing control law for theknown parameter case� as in previous sections� by setting / � and comparingequations �#�� and �#�� This yields the controller

u � D�q-q� & C�q� 'q � ��q 'q� & g�q�Kd'�q �Kp�q

�� 0�-q�� 'q�� q�� q� 'q��

�The normalization of the term � is due to ��

�� Ch� #� Trajectory tracking control

where the second equation is introduced to stress the fact that this control law is stilllinear in the parameters� Therefore� in the case of parameter uncertainty� an adaptiveversion is simply obtained as u � 0�-q�� 'q�� q�� q� 'q�(� with estimation law �#�� The

closed loop has exactly the form �#�� with /�� 0�-q�� 'q�� q�� q� 'q�� and we can

apply verbatim the analysis of the previous section to prove global convergence�

Remark �� In � � it is shown that� in the known parameter case� the closed looptrajectories � '�q�t�� q�t�� tend to the origin exponentially fast if

��

s#dMkpMd�M

�

It is important to remark that the bounds for the design parameter �� do not dependon any bounds on the desired trajectory q��t and its derivatives as is the case ofother results� for instance in ��!� �#�� Therefore tuning of the controller parametersis task independent�

� State feedback of underactuated systems

In this section we study the problem of global tracking of underactuated EL systemsvia state feedback� We concentrate our attention on the problem of robots with �ex�ible joints with block diagonal inertia matrix� First� following exactly the techniqueused in the previous section to derive the Slotine and Li controller for rigid robots�but with a suitably de�ned storage function� we design a global tracking static statefeedback PBC� Then� to provide some insight into this scheme� we compare it with thecontrollers obtained from the application of backstepping and decoupling principles�Some of the material in this section has been reported in �# ��

�� Model and problem formulation

The simplied model of an n link robot �which assumes the angular part of the kineticenergy of each rotor is due only to its own rotation was derived in Section ��#�!� Werecall that the EL parameters are given by

T �q�� 'q��

�

'q�D�q� 'q� V�q ��

q�Kq & Vg�q�

where

K ��

�K �K�K K

�� D�q�

��

�D�q� J

�q� � Rn and q� � Rn represent the link angles and motor angles� respectively� D�q� isthe nn inertia matrix for the rigid links� J is a diagonal matrix of actuator inertias

�� Literature review ��

re�ected to the link side of the gears� and K � is a diagonal matrix containing thejoint sti�ness coe�cients� For ease of reference� we repeat here the dynamical model� �!� in its compact form

D�q�-q & C�q�� 'q� 'q & G�q� &Kq �Mu �#��

and in the separate form�D�q�-q� & C�q�� 'q� 'q� & g�q� � K�q� � q�J -q� &K�q� � q� � u

�#��

As usual C�q�� 'q� 'q� represents the Coriolis and centrifugal forces and g�q� representsthe gravitational terms� As suggested throughout the book we de�ne C�q�� 'q� viathe Christo�el symbols� We will refer in the future to the �rst and second equationsabove as link dynamics and motor dynamics� respectively�

We are interested here in the following problem�

De�nition �� Global tracking problem� For the system �� de�ne an in�ternally stable control law that� for all q��t � C��Ln

and arbitrary initial conditions�ensures

limt�

k�q��tk � limt�

kq��t� q��tk � �


The �exible�joint robot model �#�� is globally feedback linearizable �by static statefeedback and therefore globally stable controllers can be derived with �classical�geometric techniques� see e�g� � �� Besides the intrinsic lack of robustness of schemesbased on nonlinearity cancelation� the proposed solutions su�er from the additionaldrawback that the control implementation relies on the availability of link accelerationand jerk� Even though these signals can be derived without di�erentiation from thesystems model� this is not a desirable procedure since the accuracy in their calculationwill be highly sensitive to uncertainty in the robot parameters� One way to overcomethis di�culty is to use parameter adaptation techniques� unfortunately it is not clearat this point how to make these schemes adaptive preserving the global stabilityproperty�

In �� a Lyapunov�based backstepping technique is applied to derive the �rstglobal tracking controller� In �� a scheme that is not based on feedback lineariza�tion is presented� Availability of link acceleration and jerk is still required� but thesensitivity problems mentioned above are claimed to be overcomed by the adaptiveimplementation� The controller is a complicated dynamic state feedback that requiresthe realization of several �ltering stages� Furthermore� some incorrect technical issuespointed out by Prof� Liu Hsu had to be addressed in �!�� leading to the modi�ca�tion of the controller� In �# � a globally stabilizing PBC that does not su�er from


the shortcomings of �� was derived� The new controller� which is a simple staticstate feedback� has a clear physical interpretation� is exponentially stable� and also ad�mits an adaptive implementation� In �# � the PBC is compared with the controllersobtained from the application of backstepping and decoupling principles using thefollowing performance indicators� continuity properties vis a vis the joint sti�ness�availability of adaptive implementations when the robot parameters are unknown�and robustness to �energy�preserving� �i�e�� passive unmodeled e�ects� Completestability proofs of all the resulting controllers are also given�

It should be remarked that� to the best of our knowledge� the global trackingproblem for the complete model of �� is as yet open� Some results concerningdynamic feedback linearization for certain particular robot structures are given in�� Other e�orts aimed at solving this problem may be found in ��#��

�� A passivity based controller

Following the PBC approach we propose to assign to the closedloop a desired storagefunction of the form

Hd ��

s�D�q�s &

�

�q�K�q �#�

where� as de�ned in the previous section� s � '�q & 2�q� with

�q�� q �

�q��q�d

��#� �

and �with an obvious partitioning 2 � blockdiagf2��2�g � � and 2� diagonal� Aswe will see below this particular choice of 2 is needed for the proof�

At this point we recall the reader an important notational convention that we usethroughout the remaining of the book� Notice that the de�nition of �q �#� � includesthe actual link reference q�� and a signal q�d to be de�ned below�� This is in contrastwith the de�nition of �q in our previous developments �e�g�� for regulation in Section�� !� and tracking with fully�actuated systems in Section � where it represents theerror between q and a given reference value q�� Remember that in our notation ��is used exclusively for external reference signals� while ��d denotes signals generatedby the controller� Another di�erence with the fullyactuated case is that� to de�neHd� we must take into account the presence of the potential energy term q�Kq whichcannot be removed� Similarly to the regulation problems of Chapter �� we proposeto shape it so as to have a global and unique minimum at �q � � It has been shownin that chapter that with di�erent choices of the desired �potential energy� term wecan obtain di�erent PBCs� two examples for tracking purpose are given below�

�This signal will be chosen so as to insure the energy shaping� i�e� such that the closed loop ispassive with storage function Hd�

�� A passivity�based controller ��

First� let the perturbed desired error dynamics be

D�q� 's& �C�q�� 'q� &Kd�s&K�q � / �#�

where Kd�� diagfKd�� Kd�g � is a diagonal matrix that injects the damping� and

the perturbation term is de�ned as

/��Mu�

�D�q� -qr & C�q�� 'q� 'qr &K

�q��q�d

�& G�q�

�&Kds

�#� !

where 'qr�� 'q�� 'q

��d�

� � 2�q� Comparing with the corresponding equations for thefullyactuated case �#�$� �#�� we remark that the error dynamics Lemma �$ is notdirectly applicable because of the presence of the term K�q� However� we notice thattaking the time derivative of Hd along the trajectories of �#� we get

'Hd � �s�Kds� s�K�q & '�q�K�q & s�/

� �s�Kds� �q�2K�q & s�/

�s�Kds& s�/

where the last inequality uses the fact that� for the particular choice of 2� we havethat 2K � � Hence� we have the desired OSP property�

The next step of the design procedure is to calculate� using �#� !� the controlsignals u and the functional relations for q�d� required to assign the desired storagefunction� that is� to ensure that / � �

The perturbation term is set equal to zero with the control law

q�d � q�� &K��uru � �Kd�s� & J�-q�d � 2�

'�q��K�q�� q�d�#� #

where ur is the control signal for the rigid�joint robot case derived in Section �� thatis

ur � D�q�-q�r & C�q�� 'q� 'q�r & g�q��Kd�s� �#� �

A very important remark at this point is that -q�d� required for the implementation of�#� #� is computable without di�erentiation� This fundamental property of �#�� islost in the complete model of ��

We are in position now to present the main result of the section�

Proposition �� The nonlinear static state feedback control �� solves the globaltracking problem� Furthermore� it ensures that the closed�loop system is globallyexponentially stable� �


Proof� The proof of global convergence follows exactly the same lines as in the fully�actuated case using the storage function Hd� To prove global exponential stability weconsider the Lyapunov function candidate proposed in � �#�

VPB ��

sTD�q�s& �q�2

T�Kd��q� & �qT� 2

T�Kd��q� &

�

�qTK�q

For which it can be shown that 'VPB ��VPB for some � � � �

�� Comparison with backstepping and cascaded designs

In �# �� besides the PBC given above� three di�erent global tracking controllers de�rived from considerations of cascaded systems or the backstepping technique ��# � arepresented� The former uses a cascade decomposition property of the robot model�and is motivated by the result on stability of cascaded connections of stable systemswith bounded orbits of � !�� The resulting closed loop is a cascade connection� Twobackstepping�based schemes� which use also the cascade decomposition property ofthe model but combined with the integrator augmentation stabilization of ��!$�� arepresented� Typical to backstepping designs� the closedloops are not triangular any�more but satisfy some �antisymmetric properties�� The �rst scheme results from adirect application of the technique and closely resembles the one reported in ��We also present a robusti�ed version� which incorporates some elements of PBC inthe construction of the Lyapunov function�

In this section we compare these controllers in relation to the following practicalquestions� What happens in the �almost rigid� case� that is� when the joint sti�nessK takes in�nitely large values"� Do they yield high gain designs" The latter questionis particularly important because of noise sensitivity considerations�

The four control laws derived in �# � are summarized in the equations below� Inall cases ur corresponds to the rigid robot control signal �#� � and s and �q are de�nedas above�

Decoupling�based Control

u � J -q�d �Kd��q� �Kd�'�q� &K�q� � q�

q�d � K��ur & q��#� �

Backstepping�based Control

u � J �-q�d � '�q� � �q� �K� 's� & s�� &K�q� � q�q�d � K��ur & q�

�#� $

Robustied Backstepping�based Control

u � J �-q�d � '�q� � �q� � � 's� & s�� &K�q� � q�q�d � K��ur & q�

�#� �

�� A controller without jerk measurements ��

Passivity�based Control

u � J -q�d &K�q�d � q��K�ds�q�d � K��ur & q��

�#� �

The following remarks are in order�

� The backstepping�based controller �#� $ becomes high gain design for increasingvalues of the joint sti�ness� due to the term K� 's� & s�� Notice that this e�ect doesnot appear as a consequence of a term K�q��q� because of the convergence of q��q�to zero as K �� This drawback is removed in �#� � and �#� �� and is conspicuousby its absence in the passivity�based designs� Notice in particular that the controlsignal u is independent of the gain K in �#� �� the dependence on K comes onlyfrom K�� and K�q� � q� which vanishes as K � &��

� As K grows unbounded the control �#� � converges to the controller of � �� forthe complete rigid robot � �!�� On the other hand� in the control �#� � a term J -q�is added to ur�

� For large K� the decoupling and backstepping based controllers feed directly intothe loop the signal -q� that is calculated using �#�� through -q�d� while �#� � usesinstead the noise�free reference -q�� Therefore� it is reasonable to expect better noisesensitivity properties for the latter�

It is very di�cult to derive a de�nite conclusion about the performance of thedi�erent classes of controllers out of the observations made above� This is particu�larly true since� as shown in �# �� modi�cations introduced at various stages of thebackstepping and decoupling designs yield signi�cant improvements� In this respectthe passivity�based technique yields robust �tuning knob free� designs in �one�shot��provided of course we can come out with the right desired storage function� One�nal remark is that it is not clear how to remove the noise sensitivity problem ofbackstepping and decoupling controllers�

�� A controller without jerk measurements

Motivated by the passivation property of the PD& controller� and the passivity of a�exible�joint manipulator one would like to extend this result to the underactuatedcase� Intuitively� a direct extension of control law �#�� could be

Mu � D�q-qd & C�q� 'qd 'qd & G�q� Kp�q � Kd'�q �#�!

where Kd�� blockdiagfKd�� Kd�g� Kp

�� blockdiagfKp�� Kp�g� are positive de�nite�

However� let us recall that in the underactuated case� the control u � � � u�f �� where

u � IR�n and uf � IRn� Thus in order for �#�! to be realized we should de�ne

uf � K�q�d � q�� & J -q�d �Kp��q� �Kd�'�q� �#�!�

q�d � K��D�q�-q�� & C�q�� 'q� 'q�� & g�q��Kp� �q� �Kd�'�q�� #�!


However� clearly the control input uf involves link jerk measurements due to theexplicit dependence of q�d on 'q��

The controller we present in this section� borrows inspiration from the PD& con�troller and includes a dynamic extension to avoid the explicit presence of 'q� in q�dand thereby� jerk measurements�

We start with establishing a passivity property for the system

D-�q & �C & Cd '�q &Kp�q &Kd� &K�q � / �#�!!

'qc � �A�qc & B�q �#�!#

� � qc & B�q �#�!�

where for simplicity we have omitted the arguments and A �� blockdiagfA�� A�g�

B �� blockdiagfB�� B�g� are positive de�nite and Cd �

� blockdiagfC�q�� 'q�� g�Notice that the equations �#�!# and �#�!� correspond to the dynamics of the

approximate di�erentiation �lter which was shown to belong to the EL class of con�trollers� We know from the previous chapter that this system de�nes an OSP operator� �� '�q� On the other hand� the error dynamics �#�!! with � � '�q and Cd � hasthe form of the closed loop system �#�!� From these facts� an immediate interestingpassivity property which can be established for the system �#�!!�#�!� is the fol�lowing� Using a storage function similar to �#�# with the additional term �

��q�K�q and

di�erentiating along the trajectories of �#�!!�#�!� we obtain that

'H�t� '�q� �q� � � '�q�Cd '�q � ��KdB��A�& '�q

�/� �#�!�

Assuming that the desired trajectory 'q��t is uniformly bounded and using the Prop�

erty P�� we obtain that � '�q�Cd '�q kc

'�q�� Integrating on both sides of the inequal�ity above from to T we obtain that

h/ j '�qiT � � kc '�q�

where we have de�ned �H� & ��KdB��A�� Roughly speaking the inequalityabove states that the system �#�!!�#�!� has a lack of passivity�

In the terminology of � !$� we say that this system with storage �#�! de�nes a map/ �� '�q which is Output Feedback Passive with negative index kc� in short OFP�kc�

The OFP property with a negative index is important in the sense that it estab�lishes that the map / �� '�q can be rendered passive if we are able to compensatefor that lack of passivity with an OSP map �that is OFP with positive index� Asa matter of fact� it can be proven that the OSP property of the linear approximatedi�erentiation �lter �#�!#��#�!� does the job� The proposition below formalizes thisclaim�

�� A controller without jerk measurements ��

Proposition �� Tracking without jerk measurements�� The system �� de�nes a locally output strictly passive operator / �� q��& '�q� Furthermoreif / � then system �� is semi�globally exponentially stable with a domain ofattraction including the set

fx � IRn � kxk � c�g �#�!$

where we de�ned x�� q�� '�q

��

Considering the passivity result established above the stabilizing passivity�based con�troller is constructed by setting / � and comparing equation �#�!! and �#��This yields

uf � J -q�d &K�q�d � q��Kp� �q� �Kd�� #�!�

and the desired motor�shaft trajectory is then chosen as

q�d�� K��D�q�-q�� & C�q�� 'q�� 'q�� & g�q��Kp� �q� �Kd�� & q��

�#�!�

which corresponds to the controller of ��!��

Notice that the calculation of uf in �#�!� requires -q�d however� in contrast with

other solutions to this problem� this controller does not require the calculation of q��

This stems from the use of 'q�d instead of 'q� in the second right hand term of q�d�and the use of the �lter� The second derivative of q�d still needs link acceleration andvelocity� Yet� only link velocity is considered to be available for measurement andacceleration can be computed using the �rst equation of �#��

Sketch of proof of Proposition �� We brie�y give below the main guidelines ofthe proof of the proposition above� For more detail we invite the reader to see ��

First� we notice that Property P�� holds true also for Cd� Now� consider thestorage function

H�t� x ��

'�q�D '�q &

�

�q��Kp &K�q & �

��KdB

��& ��q�D '�q � ��D '�q�#�#

which is positive de�nite for su�ciently small values of � � � In a compact form andafter a long but straight forward bounding it can be proven that for su�ciently largecontrol gains �and small � there exist some positive constants �� such thatthe time derivative of function H above satis�es

'H�t� x �� k�qk� � �� kc�k�k& k�qk� '�q� � �� &/��q � � & '�q��

�#�#�


Notice to this point that 'H above is locally negative de�nite� that is� if �k�k&k�qk ��kc which is satis�ed if �#�!$ holds with c� � �� kc� To this point� Lyapunovstability immediately follows from standard theorems by setting / � and noticing

that V �t� x�� H�t� x quali�es as a Lyapunov function�

Further� under the conditions above and after the completion of some squares onecan �nd a constant � such that

'H�t� x � ��q � � & '�q� &/��q � � & '�q�

from which the OSP property follows� �

It is important to remark however� that in the computation of the constant we have assumed that �#�!$ holds� from which we deduce the local character of thepassivity property� Interestingly enough� one can prove that by applying high controlgains� one can enlarge the constant �� so that the stability and passivity propertiesbecome semiglobal�

Output feedback of fully�actuated systems

In this �nal section we address the following problem�

De�nition �� Output feedback tracking control� For the system �� as�sume that only generalized positions is available for measurement� Under this con�dition� de�ne an internally stable �smooth control law �whose gains may depend onthe systems initial conditions that ensures �� for all qd � C�� kqd�tk� k 'qd�tk�k-qd�tk � Bd�

The output �i�e� position feedback tracking control problem stated above hasattracted a lot of attention in the robotics literature during the last decade� Froma practical point of view� the problem is clearly motivated due to the noisy velocitymeasurements� while from a purely theoretical perspective the problem of provingglobal uniform asymptotic stability is very challenging� This problem� as far as weknow continues open�

As in the regulation control problem� a natural approach is to design an observerthat makes use of position information to reconstruct the velocity signal� Then�the controller is implemented replacing the velocity measurement by its estimate�As far as we know� one of the �rst results in this direction was reported in ��#�where it was proved that a nonlinear observer asymptotically reproduces the wholerobot dynamics� in a PD plus gravity compensation scheme� The authors prove theequilibrium is locally asymptotically stable provided the observer gain satis�es somelower bound determined by the robot parameters and the trajectories error norms�Later� in � �� the authors presented a systematic procedure that exploits the passivity

�� Semiglobal tracking control of robot manipulators ��

properties of robot manipulators into the design of controller�observer systems� Localasymptotic stability was proved for su�ciently high gains�

In ��!�� based on a computed torque plus PD& controller �rst appeared in � �!��we added the n�th order �approximative di�erentiation �lter� studied in the previouschapter� to eliminate the necessity of velocity measurements� In that paper we provedsemiglobal asymptotic stability of the closed loop system hence showing that the do�main of attraction can be arbitrarily enlarged by increasing the �lter gain� Some morerecent and stronger results addressing the same problem are for instance� �� and�� Lim et al proposed the �rst adaptive controller for �exible joint robots by usingonly position measurements� In �� the authors proposed a globally asymptoticallystable observer�based controller needing only link position feedback�

In this section we brie�y present from a passivity point of view the position feed�back PD& controller of ��!� which is an extension to the output feedback case� ofthe controller of Proposition #�� and to tracking control of the EL controller byapproximate di�erentiation of Section �� #�B� �

�� Semiglobal tracking control of robot manipulators

We start by recalling the passivity property of system �#�!!��#�!�� Notice that theequivalent of this system for the fully�actuated case �i�e� u� q � IRn is

D-�q & �C & Cd '�q &Kp�q &Kd� &K�q � / �#�#

'qc � �A�qc &B�q �#�#!

� � qc &B�q �#�##

where Cd � C�q� 'q� 'q� and as before� all matrices A� B� Kp� Kd are diagonal andpositive de�nite� Moreover it is assumed that BD�q &D�qB � for all q � IRn�

Similar to the case when joint �exibilities cannot be neglected� a similar passivityproperty to that established in Proposition #�� holds for this system� We repeat thisbelow for convenience�

Proposition �� Tracking without velocity measurements�� The system�� de�nes a locally output strictly passive operator / �� q��& '�q� where/� q � IRn� Furthermore if / � then system �� is semi�globally exponentiallystable with a domain of attraction including the set

fx � IR�n � kxk � c�g

where we rede�ned x�� q�� '�q

��

Considering the passivity result established above the stabilizing passivity�based con�troller is constructed by setting / � and comparing equation �#�!! and �#��


This yields

u � D�q-q�� & C�q� 'q� 'q� & g�q�Kp�q �Kd� �#�#�

which corresponds to the controller of ��!�� Notice also that u above corresponds tothe terms in brackets in the de�nition of q�d in �#�!��

The proof of the proposition above follows along the same lines as the proof ofProposition #�� using the storage function

H�t� x ��

'�q�D '�q &

�

�q�Kp�q &

�

��KdB

�� & ��q�D '�q � ��D '�q�#�#�

which is positive de�nite for a su�ciently small � � � Then observing that its timederivative satis�es a bound like �#�#��

It is important to remark in the latter inequality the cubic terms �k�k�&k�qk '�q�

which are similar to those encountered in inequality �!�� Clearly� as in the reg�ulation problem of Section ��#�� case the presence of these cubic terms in the timederivative of H is a major technical obstacle to claim OSP in the usual �global sense�and therefore global asymptotic stability�

This is a common drawback of numerous articles in the literature of robot control�Even though it is beyond the scope to do an exhaustive review of the literature �seefor instance � �� we brie�y discuss below some of the latest attempts in solving theglobal output feedback tracking control problem�

�� Discussion on global tracking

As an attempt to bound the cubic terms in the time derivative of the storage functionwe presented in �� as far as we know� the �rst smooth controller which renders theone�degree�of�freedom �dof EL system globally uniformly asymptotically stable� Ourapproach relies on a computed torque plus PD structure and a nonlinear dynamicextension based on the linear approximate di�erentiation �lter� The main innovationin our controller� which allows us to give explicit lower bounds for the controllergains� in order to ensure GUAS� is the use of hyperbolic trigonometric functions in aLyapunov function with cross terms�

Global uniform asymptotic stability is ensured provided the controller and �ltergains satisfy some lower bound depending on the system parameters and the referencetrajectory norm� Unfortunately� the performance of our approach can be ensured onlyfor one dof systems and nothing can be claimed for the general multivariable case�

Independently� in �#!� Burkov showed by using singular perturbation techniques�that a computed torque like controller plus a linear observer is capable of making arigid joint robot track a trajectory starting from any initial conditions� The main

� Simulation results ��

drawback of this result is that no explicit bounds for the gains can be given� Thus�the author proves in an elegant way� the existence of an output feedback trackingcontroller that ensures GUAS�

Later� A�A�J� Lefeber proposed in �� an approach which consists on applyinga global output feedback set point control law �for instance an EL controller fromthe initial time t� till some �switching time� ts� at which it is supposed that thetrajectories are contained in some pre�speci�ed bounded set� At time ts one switchesto a local output feedback tracking control law �such as any among those mentionedabove� The obvious drawback of this idea is that the controller is no longer smooth�furthermore� the switching time may depend on bounds on the unmeasured variables�The results contained in �� concern the existence of the time instant ts such thatthe closed loop system is GUAS�

Most recently� the authors of � �#� proposed an apparent extension of the con�troller of �� to the multivariable case� The Lyapunov stability proof is carried outrelying on a nonlinear change of coordinates �See Eqs� !� and !� of that reference�Unfortunately this di�eomorphism is not invertible and therefore the controller theauthors propose in � �#� is not implementable without velocity measurements�

Last but not least in this brief review� we mention � !� where the author gives anelegant alternative result for one�degree�of�freedom systems� The controller proposedin � !� is based upon a global nonlinear change of coordinates which makes the systema�ne in the unmeasured velocities� This is crucial to de�ne a very simple controllerwhich has at most linear growth in the state variables� as a matter of fact the proposedcontroller is of a PD& type� This must be contrasted with the exponential growth ofthe control law proposed in �� due to the use of hyperbolic trigonometric functions�Hence� from a practical point of view� the controller of � !� supersedes by far that of�� Unfortunately� the nonlinear change of coordinates proposed in � !� does nothold true for n�degrees�of�freedom systems�

As far as we know the position tracking control problem stated at the beginningof this section for any initial conditions and for n�degrees�of�freedom systems remainsopen�

Simulation results

Using SimulinkTM of MatlabTM we tested the control algorithms of Propositions#�� and #��# in the two link robot arm model of � �� see also p� �� In Fig� #��we show the �rst link trajectory as well as the reference� In this case we �xed thecontroller gains to Kp �diag�� and Kd�diag��$ �� and the �lter pa�rameters to A �diag�� and B �diag�� The followed referencein both cases is q��

��Sin�� t � �

��Sin�� t�� Fig� #� shows the response

for the case when �exibility in the joints cannot be neglected� for which we have


set to K �diag�� In this case we have set Kpj �diag�� Kdj�diag��$ �� and we have considered actuator inertias to be J �diag��

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

-0.2

0

0.2

0.4ouput link trajectory q1(t)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

-0.2

0

0.2

0.4desired link trajectory q1d(t)

Figure #�� Output feedback of rigid�joint robots�

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

-0.2

0

0.2

0.4output link trajectory q1(t)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

-0.2

0

0.2

0.4desired link trajectory q1d(t)

Figure #� � State feedback of �exible�joint robots�


� Concluding remarks

We have shown in this chapter how to modify the passivity�based method� originallydeveloped for regulation� to solve trajectory tracking problems� While in the formercase one simply reshapes the potential energy of the EL plant� for tracking the desiredstorage function must also contain a suitably shaped �kinetic energy term�� Thekey technical result to achieve this end is the passive error dynamics Lemma �$�which essentially allows us to treat the EL system as a double integrator� A secondfundamental idea is the ingenious inclusion of the s variable� of � �� which allowsus to conclude convergence of the tracking errors from square integrability of s� Ofcourse� all this is possible because of the linearity with respect to s of the errordynamics in EL systems�

We have revisited from a passivity point of view some well known results instate feedback tracking control of fully�actuated mechanical systems such as the PD&scheme � �� and the Slotine and Li algorithm � �� of rigid robots� We have seen� inparticular� how to derive this popular scheme from the error dynamics Lemma �$�The idea behind this procedure is to render a closed loop map / �� s output strictlypassive� Setting / � de�nes the knownparameter controller� while / � 0�q� 'q��with 0�q� 'q a regressor and �� the parameter error� results in the adaptive case� Strictpassivity of the map / �� s ensures that a standard gradient algorithm� which in itsturn de�nes a passive map �s �� /� preserves the strict passivity �and consequentlythe L� stability of the overall scheme� The proof of convergence follows from LemmaA�� of Appendix A which shows that s � L� � �q � �

The extension of PBC to underactuated EL systems is illustrated with the exam�ple of robots with �exible joints� The results are restricted to the simpli�ed modelwith block diagonal inertia matrix� The main stumbling block to extend these resultsto the complete model is the unavailability �without di�erentiation of higher orderderivatives of the unactuated coordinates� which is related with the inability of ren�dering the system triangular and to decompose it into the feedback interconnectionof passive subsystems of Proposition �� This is a very interesting open problemthat apparently requires the development of new mathematical machinery�

An output feedback tracking controller which is the natural extension of the ap�proximate di�erentiation EL controller for regulation� was presented from a passiv�ity point of view� Finally it should be remarked that in spite of recent signi�cantadvances� the global tracking control problem by output feedback for n degrees�of�freedom systems� remains open�

�Notice that by taking s as the output of the operator we reduce its relative degree to one�

Chapter �

Adaptive disturbance attenuation

Friction compensation

�I don�t know if our scienti�c equations correspond with reality�because I don�t know what reality is� All that matters to me isthat theory predicts the results obtained by the measurements��

S� Hawking�

�In this chapter we illustrate one further advantage of PBC� the possibility of at�tenuating the e�ect of bounded external disturbance via highgain feedback� Roughlyspeaking� this feature stems from the fundamental property of in�nite gain margin ofpassive maps� hence stability is preserved when placed in closedloop with highgainoperators� The simplest application of this principle is in sliding mode control� wherea passi�able �e�g�� minimum phase and relative degree one system is controlled witha relay� which de�nes an operator with in�nite gain� albeit passive� Stability is thenpreserved because of the fundamental property of passivity being invariant underfeedback interconnection�

To illustrate this important feature of PBC we �rst present a novel adaptive fric�tion compensator based on a dynamic model recently proposed in the literature� Thecompensator ensures global position tracking when applied to an n degree of freedomrobot manipulator perturbed by friction forces with only measurements of positionand velocity� and all the system parameters �robot and friction model unknown�Instrumental for the solution of the problem is the observation that friction compen�sation can be recasted as a disturbance rejection problem as pointed out above� Thecontrol signal is then designed in two steps� �rst we �strictly passify the system withthe adaptive robot controller of Slotine and Li studied in Section �� and then adda relaybased outerloop that rejects the disturbance�

�The material reported in this chapter is based on work done in collaboration with Elena Panteleyand Magnus G�afvert�

��

�� Ch� �� Adaptive disturbance attenuation� Friction compensation

We �nish the chapter with a more general theorem applicable to a broader classof passi�able statespace systems�

� Adaptive friction compensation

It is well known that one of the major limitations to achieve good performance inmechanical systems is the presence of friction� which is a nonlinear phenomenondi�cult to describe analytically� Di�erent models have been proposed to capture thisphenomenon� starting with the classical descriptions via static maps between velocityand friction force� However� it appears that classical models are unable to capturesome of the behaviour experimentally observed in systems subject to strong frictione�ects� It is argued in �� that dynamic models are necessary to describe the frictionphenomena accurately� For a review of some of the existing dynamic friction modelssee �� and references cited therein�

Inspired by the works of �� and � �� a new dynamical model of the frictionforce �based on a bristle de�ection interpretation� is proposed in �� see also�� This model is brie�y presented in Section �� of this chapter� The frictioncompensation problem with this model is stymied by the fact that the parametersare uncertain and some of the state variables are inaccessible� Further� the structureof the equations is such that the existing techniques for adaptive output feedbackglobal tracking �based on transformations to special forms� see e�g�� $#�� # ��are �apparently inapplicable� In this chapter we present an adaptive global trackingcontroller for robot manipulators perturbed by friction forces represented by thisdynamical model� This scheme was �rst reported in � ��

The problem of adaptive friction compensation has a very long history that datesat least as far back as � �� It is presented in ��#�� as an application example ofmodel reference adaptive control� In �� the authors treated the problem of frictioncompensation of a DC motor by assuming that all the friction model parameters wereknown� A further extension of this result was reported in �� where the friction forceis linearly parameterized in terms of one unknown parameter� but all the remain�ing parameters assumed exactly known� See also �� where an impulsive model isconsidered and �� for some preliminary results on Coulomb friction adaptive com�pensation� The controller we present here extends these results in several directions�First� we treat the general case of position tracking of an n degree of freedom �dofrobot manipulator with only position and velocity measurable� Second� we considerall the parameters of the system �friction and robot to be unknown� Besides itstheoretical signi�cance� the proposed controller is of practical importance since thefriction compensation is achieved with a very simple adaptive law�

For the sake of clarity we present �rst the result for the DC motor in Section �� and then extend it in Section ��! to the general ndof robot case� To illustrate the

� The LuGre friction model ��

performance of the proposed controller we present some simulations corresponding tothe experimental setup studied in �� and �� in Section ��#� A simulationstudy that compares our controller with the one reported in �#� is also given� Inthe concluding remarks of Section ! we mention several immediate extensions of thepresent work and point out some of the open problems�

� The LuGre friction model

To capture the e�ect of friction in mechanical systems a bristlebased dynamicalmodel inspired by the works of �� and � �� was recently proposed in �� as

F � ��z & �� 'z & �� 'q ��

'z � ��a� 'qz & 'q ��

a� 'q��

j 'qj�� & ��e

�� q��

��!

where F is the friction force� z is the average de�ection of the bristles� 'q is the relativevelocity between the surfaces� �i� i � � �� and �i� i � � �� are some positivecoe�cients which are typically unknown� We should underscore the fact that� besidesthe parametric uncertainty� in this model neither F nor z are measurable� As pointedout in the concluding remarks this considerably complicates the task of adaptivefriction compensation�

Before proceeding with the adaptive controller design we recall an important prop�erty of the friction model that will be used in the sequel� As discussed in �� fromphysical considerations it is reasonable to assume that the initial bristles de�ection isbounded� that is� jz� j ��&�� In this way we ensure that it is uniformly bounded�namely

jz�tj 4��

�

�� & �� #

for all t � � This fundamental property will be used throughout the chapter�

It is di�cult to assess whether the equations above �or for that matter any othermathematical model constitute a bona de friction model� Particularly because ofthe distressing fact that it was not known whether the model above satis�es thefundamental property of de�ning a passive operator u �� 'q� This property capturesthe dissipation nature of friction and should be re�ected in any sensible model of it��

See � � for a detailed discussion on this topic�

We present below the complete answer to this question which was reported inthe recent paper ��#�� That is� we give necessary and su�cient conditions for the

�It is worth pointing out that the passivity established in �� pertains to the map u �� z� whichis not the one of physical interest�


passivity property to hold� The conditions are expressed in terms of a simple algebraicinequality involving the parameters of the model� If this inequality does not hold weconstruct an input signal that generates a periodic orbit along which the passivityinequality is violated� We present below the proof of su�ciency� The necessity partof the proof� being quite technical� is omitted and we refer the interested reader to��#��

Proposition �� Passivity of the friction dynamic system� The dynamical sys�tem �� de�nes a passive operator . � L�e � L�e � 'q �� F if and only if

��

��

� ��

�

Proof� �Su�ciency�

We will prove that if �� holds� then along the solutions of ��! with zeroinitial conditions we have

I� � T �

Z T

�

'qFdt � ��

for all 'q � L�e and all T � � First� it is clear from ��! that

a� 'q

j 'qj ��

�

�� & ��

�

��

��$

for all 'q �� Now� we will evaluate �� splitting it into two terms I � I� & I� with

I�� T ��

Z T

�

'q�� 'z & �� 'qdt

and

I�� T ��

Z T

�

z 'qdt

Replacing 'q from �� in I� we get

I�� T � ��

Z T

�

z� 'z & ��a� 'qz�dt

�� z��T & ��

Z T

�

z�a� 'qdt �

�� DC motor with friction ��

For the other term we replace 'z from �� and use the bounds ��#� ��$ to get

I�� T �

Z T

�

'q��a� 'qz & �� & �� 'q�dt

�Z t

�

'q��

�� & ��

��& �� & ��dt

�

Z T

�

'q��

��dt �

where we have used �� to establish the last inequality� This completes the proof ofsu�ciency� �

Remark �� In �� the author presents some results pertaining to the passivityproperty of the friction model discussed above� Motivated by the inability to conclu�sively establish passivity� it is proposed to make the damping coe�cient �� decreasewith increasing velocity� Physically this is motivated by the change of damping char�acteristics as velocity increases� due to more lubricant being forced into the interface�With this modi�cation it is possible to prove passivity of the operator�

� DC motor with friction

In this paragraph we consider a DC motor

J -q � u� F ��

perturbed by a friction torque modeled by ��!� We assume that position andvelocity are available for measurement and consider the case when the rotor inertia Jand all the parameters of the friction model are unknown� Our objective is to designan adaptive friction compensator that ensure global asymptotic position tracking ofa given reference signal q��

To derive the solution to the problem� we will �rst rearrange the friction model��! in such a way that the friction torque can be treated as a disturbance� Thecontrol signal is then designed in two steps� �rst we use in an innerloop a classicaladaptive controller for the DC motor that �strictly passi�es the system� and thenwe add a relaybased outerloop that rejects this disturbance� The amplitude of therelay is adjusted online via a suitably designed parameter estimator�

Towards this end� let us substitute �� in ��

F � �� & �� 'q & ��z � ��a� 'qz�� & �� 'q & Fz

where� for convenience� we have separated the viscous friction forces from the remain�ing terms which we denote Fz� We now design a SlotineLi adaptive controller �like


the one presented in Section �� for the motor model with viscous friction as

u � �Kps� (��il�il & uol ��

where Kp � � uol is an outerloop signal that will take care of the remaining frictionterms� and

s � '�q & ��q

�q � q � q� ��

with � � � The regressor is� as usual� de�ned as

�il��

� �-q� & � '�q� 'q

�

and the parameters (�il � IR� are updated as follows

'(�il � 3il�ils

with 3il � 3�il � � If we now de�ne the parameter error vector ��il�� (�il� �J� ��&��

�

and replace �� in �� we get

J 's&Kps � ��il �il & uol � Fz ��

Let us �rst look at the inputoutput properties of the inner loop� It is clear that thetransfer function �

Jp�Kpis SPR� hence is OSP in view of the KalmanYakubovich

Popov lemma� On the other hand� as we have seen before the operator s �� il�il�de�ned by the parameter update law� is passive� Consequently the operator �uol �Fz �� s is OSP�

We will see now how we can exploit this property to design the outerloop partof the scheme� To this end we notice that Fz can be bounded as

jFz� 'q� zj �� j�� a� 'q�zj 4&4

��j 'qj ��

where we have used the property ��# and the inequality a� 'q ��j 'qj� which follows

from ��!� Notice that the bound �� is independent of z�

The bound above suggests to close the outerloop with a relay with adjustableamplitude� which we parameterize as

uol � �(��ol�ol

where �ol�� sgn�s�� j 'qj�� De�ning the parameter error vector ��ol

�� (�ol�4��

��

we obtain the error equation

J 's&Kps � ��il�il � ��ol�ol � Fz �4sgn�s� ��4

��j 'qjsgn�s

�� DC motor with friction ��

A suitable choice for the estimator is then

'(�ol � 3ol�ols

with 3ol � 3�ol � � To see this consider the Lyapunov function candidate reportedin � �#�

V ��

�Js� & �Kp�q

� & ��3��

where �� il � ��ol�

�� il � ��ol�� 3

�� blockdiagf3il� 3olg� Taking the derivative

along the trajectories of the closed loop system we get

'V � �Kps� & �Kp�q '�q � Fzs�4jsj � ��4

��j 'qjjsj

�Kp'�q� � ��Kp�q

�

where we have used the bound �� to get the last inequality� From which weconclude that �q� '�q are square integrable� which implies� that �q � as t��

We have thus established the following result�

Proposition �� Consider the model of a DC motor �� with friction modeled by�� Let the adaptive control law be given by

u � �Kps� (��

with �� and the regressor de�ned as

��

��-q� & � '�q� 'q

sgn�sj 'qj sgn�s

��

where �� Kp are positive numbers� and the parameters (� � IR� are updated as follows

'(� � 3�s� 3 � 3� �

Then� the closed loop system ensures global asymptotic position tracking� that is�

limt�

j�q�tj �

�


�� Robot manipulator

In this section we extend the previous result to the case of an ndof rigid robotmanipulator described by

D�q-q & C�q� 'q 'q & g�q � u� F ��!

where q � IRn is the vector of joint angles� D�q is the inertia matrix� C�q� 'q is thematrix of Coriolis and centrifugal forces� g�q is the gravity� u � IRn are the controltorques� and F � �F�� Fn�

� are friction forces acting independently in each jointas

Fj � ��jzj & ��j 'zj & ��j 'qj ��#

'zj � ��jaj� 'qjzj & 'qj

aj� 'qj��

j 'qjj��j & ��je�� qj��j �

� ��

where j � 1n�� f�� ng� and all the remaining variables are as described above�

Again� proceeding from physical considerations� we assume that the initial bristlesde�ection are bounded� that is�

jzj� j 4j�� j & ��j� ��

In this way we can ensure the fundamental property that they are uniformly bounded�namely that jzj�tj �

��j��j & ��j� j � n for all t � �

We will use the wellknown properties mentioned in Section �� !� Namely thatD�q is bounded from above and below� that 'D�q� C�q� 'q is skew�symmetric andthat the robot model ��! can be parameterized as

D�q 'w & C�q� 'qw & g�q � 0��q� 'q� w� 'w�� $

where �� IRl contains the unknown parameters of the manipulator and the regressormatrix 0��q� 'q� w� 'w contains known functions�

The control problem can be now formulated as follows�

De�nition �� Assume that q and 'q are measurable� the initial bristles de�ectionsare bounded as �� and the parameters of the friction model �ij� �ij� j � 1n� i � � �� and the robot �� are unknown� Under these conditions� design an internallystable output feedback global tracking controller that ensures

limt�

jjq�t� q��tjj � ��

for any twice di�erentiable bounded reference q��t with known bounded �rst andsecond order derivatives�

�� Robot manipulator ��

To derive the solution to this problem we will proceed as in the case of the DCmotor� that is� �rst rearrange the friction model ��#�� in such a way that thefriction force can be treated as a disturbance� and then adopt a cascaded con�gu�ration based on the passivity and gain margin considerations explained above� Thedecomposition of F mimics the one above� that is�

F � Fz� 'q� z & 0�� 'q��

where we have de�ned a parameterization

diagf��j & ��jg 'q � 0�� 'q��

where �� & �� n & ��n�

� and 0�� 'q�� diagf 'qig� and the elements of

Fz� 'q� z satisfy

jFzj� 'q� zj �� j��j � ��j��ja� 'qj�zjj 4j &4j

��j��jj 'qjj

Hence� F will be treated as a �linearly bounded disturbance�

The design of the control law follows verbatim the DC motor case� and is summa�rized in the next proposition�

Proposition �� Consider the robot model �� with �� Let the adap�tive control law be de�ned as

u � �Kds & 0(�

with �� and 2� Kd diagonal positive de�nite matrices� The regressor matrix isgiven by

0�� 0��q� 'q� qr� 'qr� 0�� 'q� 0�� 'q� s� � IRn��l��n�

with qr�� 'q� � 2�q� �� and

0��s� 'q�� diagfsgn�sig� diagfj 'qijsgn�sig�

The parameters (� � IRl��n are updated as

'(� � 30�s� 3 � 3� � �

Then� the closed loop system ensures the global asymptotic position tracking objective�� with internal stability� �

Proof� The proof considers the Lyapunov function � �#�

V ��

hs�D�qs& �q�2Kd�q & ��3��

i


with �� (� � � and � de�ned in an obvious manner� Following the arguments abovewe can prove that

'V � '�q�Kd

'�q � �q�2�Kd2�q

From this inequality we conclude that all signals are bounded and �q� '�q are squareintegrable� The latter implies� that �q � as t��

Remark �� It is interesting to underscore that

0�� 'q� ss � � �In� diagfj 'qijg�� js�j� � � � � jsnj��

which reveals as in the DCmotor case the relaybased structure of the compensator�

�� Simulations

An experimental setup consisting of a DC motor connected to a gear box with sig�ni�cant friction is installed in the Laboratoire d�Automatique de Grenoble� see ��for a detailed description� The identi�ed parameters for the motor �� with frictionmodeled by ��! are

�J� �� $� � � ��$� ��

In all simulations we consider the sinusoidal position reference q� � � sin t�

To exhibit the e�ect of friction we �rst closed the loop with a ��xed parametercontroller which neglects friction

u � J�-q� � � '�q�Kps

with � � Kp � �� The behaviour of position and speed� starting from the initialvalues �q� � 'q� � � �� are shown in Fig�� and Fig�� Notice the presence of alarge tracking error� particularly near the zero speed region� This error can� of course�be reduced with a larger gain Kp but at the cost of larger input signals�

�� Simulations ��

0 5 10 15−1

−0.5

0

0.5

1Reference position and rotor angle position

t [s]

x, x

d [r

ad]

Figure �� Reference position q� and ro�tor angle position q without friction com�pensation�

0 5 10 15−2

−1.5

−1

−0.5

0

0.5

1Reference speed and rotor angle speed

t [s]

v, v

d [r

ad/s

]

Figure �� Reference speed 'q� and rotorangle speed 'q without friction compensa�tion�

The adaptive controller of Proposition ��! was then implemented with 3 � I� zeroinitial conditions� and the ideal relay function approximated everywhere by tanh��s�where � � � � The evolution of the relevant signals is shown in Fig� ��!Fig� ��$�A peak of �� Nm in the input torque due to the initial conditions is observed�As expected� eventhough parameter convergence is not achieved� the control signalasymptotically compensates the friction force� Actually� as depicted in Fig� ��$� theestimates converge to values very far from the real parameters� which are given by

� � � � �� #��#� ��

0 5 10 15−1

−0.5

0

0.5

1Reference position and rotor angle position

t [s]

x, x

d [r

ad]

Figure ��!� Reference position q� and ro�tor angle position q with adaptive frictioncompensation�

0 5 10 15−2

−1.5

−1

−0.5

0

0.5

1

1.5Reference speed and rotor angle speed

t [s]

v, v

d [r

ad/s

]

Figure ��#� Reference speed 'q� and rotorangle speed 'q with adaptive friction com�pensation�


0 5 10 15−0.2

0

0.2

0.4

0.6

0.8

1

1.2Position tracking error

t [s]

x−xd

[rad

]

Figure �� Position tracking error q� q��

0 5 10 15−2

−1.5

−1

−0.5

0

0.5Control signal

t [s]

u [N

m]

Figure �� Control signal u�

0 5 10 15−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25Estimated parameters

t [s]

thet

a

Figure ��$� Estimated parameters (��

0 5 10 15−0.6

−0.4

−0.2

0

0.2

0.4Friction force

t [s]

F [N

m]

Figure �� Friction force F �

For the sake of comparison we have repeated the same experiment for the controllerproposed in �#�� with measurable velocity� namely

u � J�-q� � � '�q�Kps& (F

with (F given by(F � J �z � kj 'qj� sgn� 'q

and

'z ��

Jk�j 'qj��u� (F sgn� 'q�

The behaviour of the tracking error for k � � and � � �� is shown in Fig� ��Comparing with Fig� �� we see that the performance of �#� is almost identical to theone of our controller� even though the former is designed for a simple Coulomb friction

� State�space passi�able systems with disturbances ��

model� Note however that� while for our controller all parameters are unknown� thecontroller of �#� requires knowledge of J � whose choice critically a�ects the transientbehaviour�

0 5 10 15−0.2

0

0.2

0.4

0.6

0.8

1

1.2Position tracking error

t [s]

x−xd

[rad

]

Figure �� Position tracking error q � q� for the controller of �#��

We have observed that this particular reference is quite benign� in the sense thata simple relay feedback u � �ksgn�s� with k � su�ciently large to overcomethe friction� will provide good performance� Of course� a more clever stabilizationmechanism is needed if other references are imposed� as would be the case in therobot example�

Remark �� It is interesting to try to check whether the passivity condition �� issatis�ed or not in the present application� Unfortunately� the key passivity inequalityis not satis�ed� Since the di�erence is of an order of magnitude the violation of theinequality can hardly be attributed to �unavoidable but small identi�cation errors�In spite of this discrepancy the model seems to provide a good �t for the experimentaldata� In this respect� and keeping in mind Stephen Hawking�s quote� we refer thereader to the experimental evidence reported in �� and ��

Also� we hope the analysis presented for the proof of Proposition �� providessome guidelines for the re�nement of the basic LuGre model� as proposed in � ��See also Remark ��

� State�space passi able systems with disturbances

� Background

Motivated by the results above and ��#�� we will present in this section a general theo�rem concerning the stabilization of nonlinear passi�able systems with time�dependentdisturbances� These results have been established in ��


To establish the connection with the adaptive friction compensation problem� letus recapitulate the main developments of the previous section� First� we use a nestedloop controller con�guration� an ubiquitous approach throughout the book� that issubstantiated by the fact that passivity is invariant under feedback interconnection�In the present problem of disturbance attenuation� the innerloop is a PBC whichstrictly passi�es the map �uol � Fz �� s de�ned by the closed loop system ��The outerloop control is a relaybased operator with input s that quenches� viahigh�gain� the e�ect of the disturbances� This outerloop operator is also passive�consequently passivity �and therefore stability of the overall system is ensured� Thisis� of course� the essence of the oftenquoted robustness of sliding mode control � ��

We consider nonlinear systems of the form

'x � f�t� x & g�t� x�u& d�t� x� ��

where x � IRn and we impose that the disturbance d � IR�� IRn � IRn satis�es thefollowing assumption�

A�� Each component of the vector function d is uniformly bounded in t by

jdi�t� xj ��i�i�kxk & ��i � � i n ��

where ��i and ��i are unknown non�negative constants and �i�kxk � IR�� IR��is a known continuous function�

An obvious implication of �� is the existence of a continuous function � � IR�� IR�� and positive constants �� and �� such that kd�t� xk ��kxk & �� or incompact form

kd�t� xk ��4�kxk ��

where we have de�ned �� and 4�kxk� �

� ��kxk� �� The disturbanced�t� x may be the result of noisy measurements or parameter uncertainty a�ectingthe plant�

Mimicking the derivations of the previous section�� we make the following assump�tion

A�� A statefeedback controller u � u��t� x and an output function y � Lm� �

y � h�t� x� where h � C�� are known such that the system

'x � f�t� x & g�t� x�u��t� x & v�t� �� !

de�nes an OSP operator . � v �� y with a positive de�nite and proper storagefunction V ��t� x�

�The same approach will be taken for interconnected systems in Chapter ��

�� A theorem for passi�able a�ne nonlinear systems ��

We are seeking for an outerloop statefeedback v � v�t� x guaranteeing that thetrivial solution x�t � of �� with arbitrary initial conditions t�� x� � x�t� beglobally uniformly convergent� that is�

limt�

kx�tk �

for any initial conditions �t�� x��

� A theorem for passi�able a�ne nonlinear systems

In order to formulate our main result in terms of input�output properties� we need tointroduce the following concept�

De�nition �� Strong zero�state detectability�� The nonlinear system ��

with input v�� u & d and output y � h�t� x where h is a continuous function of its

arguments� is said to be strongly zero�state detectable if

fv�t � Ln and y�t � g � lim

t� kx�tk � � �� #

The above de�nition di�ers from the one given in Appendix A �see also �#�� in thefact that we need the property to hold for all uniformly bounded inputs�

It is well known� e�g�� $�� that a detectability assumption is always needed totransfer input�output properties to statespace properties� Thus it will become clearlater� that the condition �� # will allow us to prove our main result using passivityarguments�

Theorem �� Robust adaptive stabilization �� Consider the system �� under the Assumptions �� and

A�� System �� ! is strongly zero�state detectable�

Under these conditions� the controller

u � u��t� x� (��4�kxksgn�y ��

'(� � 34�kxksgn�y�y ��

where (� is the estimate of � and 3 � 3� � � in closed loop with �� guarantees that the state x�t is globally uniformly convergent� �

Proof� We start by noticing that since �� ! de�nes an OSP operator with storagefunction V ��t� x then there exists a positive constant such that the time derivativeof V � along the trajectories of �� ! satis�es

'V ��t� x v�y � kyk� � �� $


De�ne

v�� d�t� x� (�

�4�kxksgn�y� ��

then the bound �� $ on the derivative of the storage function V �� coincides withthe following bound on time derivative of the Lyapunov function V � along the closedloop solutions ��

'V ��t� x� �� kyk� � d�y � (��4�kxksgn�y�y� ��

Consider on the other hand the Lyapunov function candidate

V �t� x� �� V ��t� x &

�

��3��

where �� (� � �� which is clearly positive de�nite and proper� Using the inequalities

�� and �� the time derivative of V along the closed loop solutions �x�t� ��tof system �� is bounded by

'V �t� x� �� kyk� � ��4�kxk�jyj � kyk ��!

where jyj stands for the Frobenius norm �c�f� Notations� Notice that the subtractionof the last two terms on the right hand side of ��! is non�positive hence

'V �t� x� �� kyk� � ��!�

Inequality ��!� implies that V �t is decreasing and hence bounded so the completestate �x� �� Ln��

� Furthermore� integrating ��!� from t� to � one concludesthat y � Lm

� � Next� notice from �� that since � is constant and 4�kxk iscontinuous� v�t is also uniformly bounded� From the continuity of h�t� x it followsthat y � Lm

� Then� from the closed loop dynamics �� !� �� we obtain that'x � Ln

and since h � C� we conclude using Barbalat�s lemma that y�t � ast � �� The result is �nally obtained using the assumption of strong zero�statedetectability� �

Interestingly enough the theorem above applies to non�autonomous non�linearsystems� In other words� if we think of the state x as a tracking error Theorem�� can be used to design robust tracking controllers such that the tracking error isguaranteed to converge to zero uniformly as t � �� It may clarify the reason forintroducing condition �� # that a proof in the spirit of the proof of Theorem !� of�#�� requires that v � �see also Lemma !� �! of � $ � which cannot be guaranteed inour case� It may be also clear that our motivation for calling condition �� # �strong�zero�state detectability stems from the fact that this condition is more restrictive thanthe usual de�nition�

It is also worth remarking that in Theorem �� we have implicitly assumed thatall parameters of our system are known� however� notice that the perturbation d�t� xmay also contain terms which come from uncertainties in the model�


More precisely� consider a nonlinear non�autonomous system de�ned by

'x � f�t� x� � & g�t� xu ��!

y � h�t� x

where x � IRn� and � � IRl of constant unknown parameters� Using Theorem �� acontroller u � u�t� x guaranteeing that x�t � as t � � can be designed if thedynamics ��! admit the following parameterization

'x � f ��t� x & g�t� x�u& d�t� x� ��

where the nonlinearity d�t� x� � satis�es

kd�t� x� �k ��4�kxk & �� !!

with �� and �� unknown constants� and 4�kxk a continuous known function�

Notice furthermore� that the parameters vector � does not need to be constant�as long as it de�nes continuous uniformly bounded signals� that is� if ��t � Ll

anda bound like ��!! can be determined�


We have illustrated in this chapter how� via an adaptive high�gain design� it is possibleto attenuate the e�ect of disturbances for passi�able systems� The control task isdivided into two steps� � A passifying control law is designed for the system withoutdisturbance� A discontinuous control loop� with adaptive gain� is added around theoutput of the resulting passive map of the �rst step� A theorem for general passi�ablestatespace systems is given� The principle is applied to solve the tracking controlproblem of robot manipulators a�ected by frictional forces�

The key observation that allows us to design an adaptive friction compensator isthat� in view of property �� the e�ect of the friction force can be �dominated��with measurable functions� and hence treated as a �linearly bounded disturbance�This� together with the fact that passivitybased control of robot manipulators en�sures that the closedloop is a passive map� allows us to implement a �highgain�design that rejects this disturbance�

We should underscore that� even though the resulting controller is non�smooth�the stabilization mechanism does not rely on the generation of sliding regimes� Fur�thermore� it is clear that for all practical purposes the signum function can be replacedby a smooth function with minor �quanti�able performance degradation�

It is interesting to note that even in the simple DC motor case� the problem ofglobal tracking is �apparently not solvable with the techniques reported in ��# ��


��$#�� Leaving aside the presence of a nonsmooth nonlinearity� this stems from thefact that the system� which can be written in state space form as

'� �

��

� �J�� & ��

J�� a� 'q

� ��a� 'q

�� &

��

�J

�� u

where �� q� 'q� z�� is not transformable into any of the special forms considered in

the literature for robust �or adaptive output feedback tracking� see e�g�� $#�� # ��

Among the possible practically �and theoretically interesting extensions to thisresult we have the problems of PD regulation with adaptive friction compensation�the case of robots with �exible joints and actuator dynamics� and the removal ofvelocity measurement� While the solution to the �rst three problems follow mutatismutandis from the present analysis� the latter seems to require the development of aradically new approach�

Part II

Electrical systems

�!!

Chapter

Modeling of switched DC�to�DC

power converters

We start with this chapter the second part of the book which is devoted to electricalsystems� and in particular to DCtoDC power converters� The study of these devicesconstitutes an active area of research and development in both power electronicsand control theory� Switched DCtoDC converters have an ubiquitous variety ofindustrial and laboratory applications thanks to their reduced cost� simplicity ando��the�shelf availability� This part of the book consists of two chapters� In view ofthe presence of the switches some new considerations with respect to those madein Chapter � and Appendix B� are needed to formalize the mathematical modeling�This is done in Chapter �� which is fully devoted to modeling and the exploration ofthe structural properties useful for PBC� which as we will see later� applies verbatimfor this class of systems� This material is presented in Chapter ��

� Introduction

Modeling of switched regulated DCtoDC power converters was initiated by thepioneering work of Middlebrook and �Cuk in �� and ��!� in the mid seventies� Thearea has undergone a wealth of practical and theoretical development as evidencedby the growing list of research monographs and textbooks devoted to the subject� seee�g� � !�� $� where interesting relations between power electronics and control arediscussed� The reader is also invited to see the recent text �!#� where an extensivebibliography� and a rather complete historical perspective of power electronics� ispresented through seminal and cornerstone articles�

In this chapter� both a Lagrangian and a Hamiltonian dynamics approach are usedfor deriving physically motivated models of switch regulated DC�to�DC power con�verters� The Lagrangian approach consists in establishing the EL parameters of the

�!�

�� Ch� �� Modeling of switched DC�to�DC power converters

circuits associated with each of the topologies corresponding to the two possible posi�tions of the regulating switch� This consideration immediately leads us to realize thatsome EL parameters remain invariant under the switching action while some othersare de�nitely modi�ed by either� the addition or annihilation of certain quantities� Aswitched model of the non�invariant parts of the EL parameters can then be proposedby their suitable inclusion through the switch position parameter� This inclusion iscarried out in a consistent manner so that� under a particular switch position param�eter value� the original EL parameters corresponding to the two intervening circuittopologies� are exactly recovered�

The switched EL parameter considerations immediately lead� through the useof the classical Lagrangian dynamics equations� to systems of di�erential equationswith discontinuous right hand sides� describing the actual behavior of the treatedconverters� The obtained switch�regulated models entirely coincide with the statemodels of DCtoDC power converters introduced in ��!� ��

The Hamiltonian modeling approach complements and generalizes the Lagrangianapproach in various respects� In modeling switched power converters the use of idealswitches is sometimes insu�cient due to several practical realization constraints� Also�actual DCtoDC power converters are synthesized using isolation transformers andsometimes� magnetic couplings are usually purposely introduced in the circuit toachieve desirable smoothing e�ects� Ideal switches are often replaced by suitable com�binations of transistors and diodes� While transistors can e�ectively behave as idealswitches� diodes have physical characteristics which are not suitable for Lagrangianmodeling� Using some recent results contained in �$#�� we show in this chapter thatthe Hamiltonian viewpoint leads to a natural modeling methodology in which notonly ideal switches have a suitable representation but also diodes� transformers� iso�lation transformers etc� The theoretical basis for these developments may be foundin ��$�� $!��

Section begins with some general issues about the modeling of switch�regulatedEL systems� In that section� the switched models of the traditional DCtoDC powerconverters� such as the Boost� the Buck�boost and the �Cuk converters� are derivedfrom the Lagrangian viewpoint� The modeling approaches are shown to be also validfor DCtoDC power converters with non�ideal switches including parasitics� We alsopresent an extension to the afore�mentioned modeling approaches� which includes themultivariable version of the Boost converter constituted by the cascaded connection oftwo such devices� The Hamiltonian modeling viewpoint of switched power convertersis presented in Section !� In Section # the Lagrangian approach is applied to obtainthe average state space models of pulse�width�modulation �PWM regulated DCtoDC power converters with ideal switching devices�

� Lagrangian modeling ��

� Lagrangian modeling

In this section� a Lagrangian approach is used for deriving mathematical modelsof the most commonly found switched DCtoDC power converters� The approachis suitable to be applied to a large class of physically existing DCtoDC powerconverters�

The Lagrangian modeling technique is based on a suitable parametrization� interms of the switch position parameter� of the EL functions describing each interven�ing system and subsequent application of the Lagrangian formalism� The resultingsystem is also shown to be a EL system� hence� passivity�based regulation may beproposed as a natural controller design technique�

� Modeling of switched networks

A large class of technological systems are characterized by the presence of one orseveral regulating switches� i�e�� devices that can only adopt one of two possible pos�tions� each of these giving rise to a determined dynamical behaviour of the system�Switchregulated systems are quite common in everyday life specially eversince thecommercial development of modern electronics�

We are particularly interested in dynamical systems containing a single switch�regarded as the only control function of the system� The switch position� denotedby the scalar u� is assumed to take values on a discrete set of the form f � �g� Weassume that for each one of the switch position values� the resulting system is an ELsystem characterized by its corresponding EL parameters� In other words� we assumethat when the switch position parameter takes the value� say� u � �� the system�denoted by .�� is characterized by a known set of EL parameters� fT��V��F��Q�g�Similarly� when the switch position parameter takes the value u � � we assume thatthe resulting system� denoted by .� is characterized by fT��V��F��Q�g�

De�nition �� A function �u� 'q� q � �� 'q� q� u� parameterized by u� is said to beconsistent with the functions �� 'q� q and �� 'q� q whenever

�uju � � �� uju � � ��

We introduce the set of switched EL parameters fTu�Vu�Fu�Qug as a set of func�tions parameterized by u which are consistent� in the sense described above� withrespect to the EL parameters of the systems .� and .� for each corresponding valueof u�

�In the class of systems studied in this part of the book the input enters bilinearly� hence we donot use the EL parameterM� On the other hand� there is always interaction with the environment�thus the EL parameter Q� will be nonzero�


A switched system .u� arising from the EL systems .� and .� is a switched ELsystem whenever it is completely characterized by its set of switched EL parametersfTu�Vu�Fu�Qug�

The basic problem in an EL approach to the modeling of switched systems� arisingfrom individual EL systems� is the following�

De�nition �� Modeling problem for switched EL systems� Given two ELsystems .� and .� characterized by EL parameters� fT��V��F��Q�g and fT��V��F��Q�g� respectively� determine a consistent parameterization of the EL parameters�fTu�Vu� Fu�Qug in terms of the switch position u� such that the model obtained bydirect application of the EL equations� results in a parameterized model .u� which isconsistent with .� and .��

Consistent parameterizations of the EL parameters� by means of the switch po�sition parameter u� may be� generally speaking� carried out in an in�nite number ofways� However� the general rule is to carefully consider those parameterizations thatnot only account for the e�ect of the change of the switch position in a particular ELparameter but� also� those for which the obtained switched EL parameter respects theessential nature of the fundamental physical laws that intervene in the constitutionof such an EL parameter� For instance� if a change in a switch position inserts� orremoves� a current source of value I into a node� to which a single resistive elementR is attached� along with capacitive branches that are una�ected by the switchingaction� then the fundamental law to be respected� in obtaining a suitable parameteri�zation of the Rayleigh dissipation function� is Kircho��s law of addition of currents ata node� Note that the parameterization Fu � �� R��

Pij

� & �� uI�� where theij�s are the currents in capacitive branches that do not change with the switch posi�tion� is a consistent parameterization� However� the correct parameterization wouldbe Fu � �� R�

Pij & �� uI�

� A variational argument

Suppose a consistent� phisically meaningful� parameterization has been carried outon the set of EL parameters in terms of the switch position u� Assume that the non�conservative switched Lagrangian function of the system has been derived as Lu� 'q� q�

To derive the dynamical model associated with the nonconservative swtiched La�grangian function we depart from Hamilton�s principle� Recall that Hamilton�s prin�ciple establishes that the trajectory of the system minimizes the action integral� whichis de�ned as the integral of the Lagrangian function� The variational condition for astationary value of the action integral is given by

�

Z t�

t�

L� 'q� q� udt � ��

�� A variational argument ��

for any arbitrary but �xed endpoints t� and t� of the considered time interval� Thevariational argument speci�cally establishes that for any virtual � i�e�� arbitrary variation of the system trajectory� � 'q�t� q�t� the corresponding variation of theaction integral should be zero� The virtual variations in the system trajectories mustbe in�nitely di�erentiable� of in�nitesimal nature� and moreover� with zero values atthe end points of the considered time interval�

The explicit dependance of the nonconservative Lagrangian function Lu� 'q�t� q�ton the switch position parameter u� plus the causality relation existing between u�considered as a function of time� and the system trajectories � 'q�t� q�t� makes onewonder if the admissible variations of the trajectories should be synthesized fromvariations of the control input u or� if due to their arbitrariness one should exercisethem directly on the state trajectory without regard for the intrinsic causality relation�An examination of the issue in the class of switch regulated systems establishes thatthe �rst road must be discarded as argued in the next paragraph�

Suppose an arbitrary time�realization of the switch position u� viewed now as afunction of time u�t� is given� The class of actual admissible variations �u�t of theswitch position function u�t is then represented by �pulses� of in�nitesimal duration�t� taking values now in the set f�� g� These pulses satisfy the restriction that anegative pulse may only take place while u � � and a positive pulse can take placewhen u � � The problem with this class of discontinuous� and restricted control inputvariations� is that each of them produces corresponding variations ��q�t� � 'q�t in thegeneralized coordinates �q� 'q which are not in�nitely di�erentiable and furthermore�the obtained perturbed trajectory may not be itself� an in�nitesimal variation of theoriginal trajectory in the course of time�

A third obstacle is also represented by the fact that the corresponding trajectoryvariations may not be zero at the end points of the time interval� The �rst fact possesa technical problem in the application of the calculus of variation� The second factis just inadmissible while the third would further restrict the control input variationsto those that produce zero e�ects at the end points� Moreover� each control inputvariation of the described class results in an actual system trajectory which in itselfminimizes the action integral� This argument clearly leads to the bizarre situation ofhaving in�nitely many minima of the action itegral which are in�nitesimaly close toeach other�

Hence the involved action integral minimization must be understood in the sensethat comparisons of the values of this functional are to be performed between itsevaluation on an actual system trajectory� produced say by a �xed u�t� and itsevaluation on in�nitesimal and in�nitely di�erentiable system trajectory variations�q�t & �q�t� 'q�t & � 'q�t� that cannot and should not be synthesizable from discon�tinuous control input variations �u�t taking values in the set f�� g� One is thenconfronted with the following choice� either one carries the variational arguments�without any regard for the presence of the parameter u� in terms of trajectory vari�


ations alone� or else� one introduces arbitrary in�tesimal and in�nitely di�erentiablevariations �i�e�� virtual variations �u�t on the action integral� even if they are notphisically synthesizable as a switch position function� We choose the �rst avenue� notonly because it is conceptually simpler but also because it allows one to regard u asa constant parameter simply indicating one of two possible switch positions� Thisconforms to the idea of consistency in a more natural manner�

Following standard� and well known arguments� in the calculus of variations weobtain the following development� Virtual variations of the system trajectory areallowed which result in the following corresponding variation of the action integral

�

Z t�

t�

Lu�q�t� 'q�tdt �

Z t�

t�

�Lu�q�t� 'q�tdt

�

Z t�

t�

�Lu�q�t & �q�t� 'q�t & � 'q�t� Lu�q�t� 'q�t� dt

�

Z t�

t�

��Lu

�q�q�t &

�Lu

� 'q� 'q

�dt

�

Z t�

t�

��Lu

�q� d

dt

�Lu

� 'q

��q�tdt&

��Lu

� 'q�q

��t�t�

� �

From the fact that admissible trajectory variations should have no contribution fromthe end points� the second term in the last expression above is identically zero� Thewell known Fundamental Lemma of the Calculus of Variations �see� for instance ��#��can now be applied to obtain the following result�

Proposition �� The EL equations are valid for the nonconservative switched La�grangian function parameterized by the switch position u� treated as a constant� Theresulting dynamical model of the switched system is a dynamical model parameter�ized also by u which is consistent with the intervening dynamical models for eachswitch position parameter value� that is

d

dt

�Lu�q� 'q

� 'q� �Lu�q� 'q

�q� � ��

�

�� General Lagrangian model� Passivity property

In this section we present a general procedure to derive the EL models of a class ofswitched DCtoDC converters� We also underscore some properties of the model�in particular its passivity� which will be exploited for the design of PBCs in the nextchapter�

�� General Lagrangian model� Passivity property ��

A Structural considerations

DC�to�DC power conversion is accomplished in switched circuits through ingeniouscyclic energy transfers� regulated by the position�s of the commanding switch�es�Generally speaking� switching actions allow that energy� obtained from the externalsources� be stored in �input� inductive branches by placing those inductors in the samemesh as the sources� The stored kinetic energy is then transfered to either �internal�or to �output� potential energy reservoirs �i�e�� capacitors� The stored potentialenergy in �internal� capacitors is transfered again� in the form of kinetic energy to adi�erent �internal� inductive branch� usually connected to output capacitors� In thecase of �output� capacitors� the energy may be directly drawn by the resistive loads�

Generalized forcing parameters� or external voltage source terms� will be asso�ciated with �input� inductor current variables in the EL equations� while� generallyspeaking� the external voltage sources will be absent from the equations correspondingto �output� capacitor charges�

Potential energy delivery to the output loads is accomplished thanks to the factthat �output� storing capacitors will be connected in parallel to the resistive loads�This RC �output sturcture� is always �xed and the switches do not a�ect it directly�Thus� nodes containing output storing capacitor branches will always have the outputresisitive load branch attached to them� We assume without loss of generality� thatthe switch�invariant capacitors are all �output� capacitors�

In summary� the switching action in DC�to�DC power converters� accomplishesone or two of the following three possibilities during the transfer phase�

�� Switchings insert �respectively remove a constant external voltage source into�resp� from a mesh containing an �input� inductor branch�

� Switchings insert �respectively remove a charged inductor branch �whether�input� or �internal� branch into �resp� from a node containing either an�output� RC branch or an �internal� storing capacitor branch�

!� Switchings insert �resp� remove an �internal� storing capacitor into �resp froma mesh containing an �internal� inductor branch�

The following developments include all of the previously treated ideal DC�to�DCpower converters �i�e�� non perturbed by parasitics� In the formulation we take intoaccount all previous remarks� The switch position parameter� u� is regarded to be avector so as to include the multivariable versions of the DC�to�DC power converters�


B Energy and dissipation functions

We hypothesize a general kinetic energy EL parameter of the form

Tu ��

'q�LL 'qL � 'qL � RnL

where 'qL denotes the vector of inductor currents and L is a diagonal positive de�nitematrix�

The generalized potential energy EL parameter� V� contains two terms� Onedescribing the stored potential energy in capacitors which are not directly a�ected bythe switchings and a second term including the potential energy of capacitors that areremoved from an �inductive mesh and inserted into another mesh by means of theswitching action� We denote the �rst vector of �switch�invariant� capacitor charges asan nCI �dimensional vector� qCI � The vector of charges associated with the capacitorsthat migrate from one mesh to another is deemed as a vector of �non�switch�invariant�charges� These will be denoted by an nCV �dimensional vector qCV � However� thesecharges are not generalized coordinates since their values depend on the particular�owing charge associated with the inductive mesh where these capacitors happen tobe for a particular switching position� In other words� qCV � V �uqL� where V �u isan nCV nL matrix� parameterized in terms of the switch position u� We call thematrix V �u� the �Capacitor mesh insertion matrix��

Vu ��

q�CIC

��I qCI &

�

q�CV C

��V qCV

��

q�CIC

��I qCI &

�

q�LV

��uC��V V �uqL

The generalized Rayleigh dissipation EL parameter is obtained by considering thecurrents �owing through the output load resistors� These currents are given by thedi�erence between the currents �owing from the inductive branches which are insertedinto the output nodes by the switching action and the currents �owing through the�output capacitors� �which� as assumed before� constitute the set of switch�invariantcapacitors� The currents through the output capacitors are described by the vectorterm� 'qCI � The vector of inductive currents inserted into� or removed from� the outputnodes is� evidently� a subvector of the vector qL� This subvector is expressed in theform Z�u 'qL where the nonqL matrix� Z�u� is the �inductive branch node insertionmatrix�� The Rayleigh dissipation EL parameter is then given by

Fu ��

�Z�u 'qL � 'qCI �

�R �Z�u 'qL �N 'qCI �

The matrix R is a diagonal matrix containing all the output resistance values�

Finally� the vector of external sources� Qu� usually contains a single nonzero ele�ment that may� or may not� be inserted into the �input� mesh by the switching action�

�� General Lagrangian model� Passivity property ��

However� we consider that the subvector E of nonzero entries� representing the nEexternal voltage sources� is of dimension nE � ni� where ni nL is the number of�input inductors� which may form �input� meshes �i�e�� inductor�charging meshes with the external sources by means of the switching action� The vector of generalizedforcing functions Q may then be expressed as

Qu ��Q�uE�� nL�ni�

�where Q�u is a �square �input mesh external source insertion matrix�� of dimensionni nE � ni ni�

C Properties of the EL model

Applying the EL equations to the above set of EL parameters we obtain a rathergeneral model of a large class of ideal DC�to�DC power converters� including the classof multivariable DC�to�DC converters�

L-qL & V ��uC��V V �uqL � �Z��uR �Z�U 'qL � qCI � &Qu

C��I qCI � R �Z�u 'qL � 'qCI � �

A state space representation of the generalized switched circuit may be obtained byde�ning a composite state vector as� xL � 'qL � xCV � C��

V qCV � C��V V �uqL and

xCI � qCI � After some algebraic manipulations� and di�erentiations where the vectoru is treated as a vector of constant parameters� one obtains the following general statespace model for a large class of switched DC�to�DC power converters

!" L

CV CI

#A

�� 'xL

'xCV'xCI

��&

!" �Z��u �V ��u

Z�u V �u

#A

�� xLxCVxCI

��

&

!"

R��

#A

�� xLxCVxCI

��

��

�F �uE

�

��

The set of equations above can be rewritten in the following general form�

D 'x& G�ux&Rx � E

where x�� x�L � x

�CV� x�CI �

�� D is diagonal and positive de�nite� G�u is a skew�symmetric matrix for any allowable values of the switching parameter vector compo�nents and R is a diagonal positive semide�nite matrix�


To reveal some structural properties of this model notice �rst that the total energyof the circuit is given by

Hu � Tu & Vu ��

x�Dx�

Di�erentiating Hu� taking into account the skewsymmetry of G�u� and integratingback� we obtain the energybalance equation

Hu�T �Hu� � �z stored energy

&

Z T

�

x�Rx�tdt� �z dissipated energy

�

Z T

�

x��tEdt� �z supplied energy

��!

The following observations are in order�

� As expected� the circuit dynamics de�nes a passive operator from the suppliedvoltages E to the inductance currents xL�

� The model� though an EL system� di�ers from the ones considered previously�since the control signal enters bilinearly� However� action of the control signalis �workless�� in the sense that it does not a�ect the energy balance equation�

� The system is in general only partially damped� Thus� additional damping willneed to be injected to achieve strict passivity�

Let us elaborate further this points with the example of the Boost converter� Inthis case

D �

�L C

�� G�u � �uJ � u

� ��

�� R �

� ��R

�� E �

�E

�

where x � �x�� x�� and the energy balance equation reduces to

Hu�T �Hu� �

Z T

�

Ex��tdt� �

R

Z T

�

x��tdt�

Eventhough� as pointed out above� the inductance current x�� being a passive output�is �easy� to control� in these devices the output to be controlled is the capacitorvoltage x�� The problem is further complicated by the fact that� as we will showlater� the system is non�minimum phase with respect to x�� We will� therefore�control x� indirectly via the regulation of x��

In this circuit the workless role of the control is quite clear since the switch simplytransfers the magnetic energy stored in the inductance to the RC circuit� with partof it being stored as electric energy in the capacitor and the rest dissipated by theresistance�


�� Examples

A The Boost converter

Consider the switchregulated Boost converter circuit of Fig� ��

u � �

x�

R

�

E

x�

L

u � � C

Figure �� The Boost converter circuit�

The di�erential equations describing the circuit were derived in �� using classicKircho� laws� Such set of equations are given by

'x� � �� u�

Lx� &

E

L��#

'x� � �� u�

Cx� � �

RCx� ��

where x� and x� represent� respectively� the input inductor current and the outputcapacitor voltage variables� The positive quantity E represents the constant voltagevalue of the external voltage source� The parameter u denotes the switch position�The switch position parameter takes values in the discrete set f � �g�

We consider separately the Lagrange dynamics formulation of the two circuitsassociated with each of the two possible positions of the regulating switch� Our aimin carrying out this formulation is to gain some insight on the physical e�ects of theswitching action in terms of the EL parameters of the two circuit topologies� In orderto use standard notation we refer to the input current x� in terms of the derivativeof the circulating charge qL� as 'qL� Also the capacitor voltage x� will be written asqC�C where qC is the electrical charge stored in the output capacitor�

Consider then u � �� In this case two separate� or decoupled circuits are clearlyobtained and the corresponding Lagrange dynamics formulation can be carried out asfollows� De�ne T�� 'qL and V��qC as the kinetic and potential energies of the circuitrespectively� We denote by F�� 'qC the Rayleigh dissipation function of the circuit�These quantities are readily found to be$ T�� 'qL � �

�L � 'qL

� � V��qC � ��Cq�C

F�� 'qC ��R � 'qC

� � Q�qL

� E � Q�qC

� ��


where F�qL

and F�qC

are the generalized forcing functions associated with the coordi�nates qL and qC � respectively�

Evidently� the EL equations used on these EL parameters immediately rederiveequations ��#� �� with u � �� as it can be esasily veri�ed�

Consider now the case u � � The corresponding Lagrange dynamics formulationis carried out in the next paragraphs�

De�ne T�� 'qL and V��qC as the kinetic and potential energies of the circuit� re�spectively� We denote by F�� 'qL� 'qC the Rayleigh dissipation function of the circuit�These quantities are readily found to be�$ T�� 'qL � �

�L � 'qL

� � V��qC � ��Cq�C

F�� 'qL� 'qC ��R � 'qL � 'qC

� � Q�qL

� E � Q�qC

� ��$

where� Q�qL

and Q�qC

are the generalized forcing functions associated with the coor�dinates qL and qC � respectively�

The EL parameters of the two situations generated by the di�erent switch positionvalues result in identical kinetic and potential energies� The switching action merelychanges the Rayleigh dissipation function between the values F�� 'qC and F�� 'qL� 'qC�Therefore� the dissipation structure of the system is the only one a�ected by theswitch position� One may then regard the switching action as a �damping injection��performed through the inductor current�

The following set of switched EL parameters are proposed for the description ofthe switched system�

$ Tu� 'qL � ��L � 'qL

� � Vu�qC � ��Cq�C

Fu� 'qL� 'qC ��R �� u 'qL � 'qC �

� � QuqL

� E � QuqC

� ��

Note that in the cases where u takes the values u � � and u � � one recovers�respectively� the dissipation functions F�� 'qC in �� and F�� 'qL� 'qC in ��$ fromthe proposed dissipation function� Fu� 'qL� 'qC� of equations �� The proposed ELparameters are therefore consistent�

The switched Lagrangian function associated with the above de�ned EL parame�ters is given by�

Lu � Tu� 'qL� Vu�qC � �

L � 'qL

� � �

Cq�C

One then proceeds� using the EL equations �� to formally obtain the switch�position parameterized di�erential equations de�ning the switch regulated system


which corresponds to the proposed switched EL parameters �� This results in thefollowing set of di�erential equations

L-qL � �� uR �� u 'qL � 'qC � & E

qC�C � R �� u 'qL � 'qC �

which can be rewritten� after substitution of the second equation into the �rst� as

-qL � �� uqCLC

&E

L

'qC � � �

RCqC & �� u 'qL

or using the state�space coordinates x� � 'qL and x� � qC�C one obtains

'x� � �� u�

Lx� &

E

L��

'x� � �� u�

Cx� � �

RCx��

The proposed switched dynamics �� coincides with the classical statespace model developed in �� and ��!��

B The Buck�boost converter

The circuit of the Buck�boost converter is shown in Fig� �� We summarize all therelevant formulae and equations� leading to the switched model of the Buckboostconverter circuit� through an EL formulation� in the Table ��

x�

R

�u � �

L

u � �

E x�

C

Figure �� The Buckboost converter circuit�

Remark �� The Lagrangian approach to modeling of the Buck�boost converterreveals that only the �dissipation structure� and the �external forcing functions� arenon�invariant with respect to the switching action�


BUCK�BOOST CONVERTER

EL parameters for possible switch positions

u u � u � �

Kinetic energy T�� 'qL � ��L � 'qL

� T�� 'qL � ��L � 'qL

�

Potential energy V��qC � ��Cq�C V��qC � �

�Cq�C

Rayleigh dissipation F�� 'qL� 'qC ��R � 'qL & 'qC

� F�� 'qC ��R � 'qC

�

External forces Q�qL

� � Q�qC

� Q�qL

� E � Q�qC

�

Switched EL Parameters

Kinetic energy Tu� 'qL � ��L � 'qL

�

Potential energy Vu�qC � ��Cq�C

Dissipation function Fu� 'qL� 'qC ��R �� u 'qL & 'qC �

�

External Forces QuqL

� u E � QuqC

�

Lagrangian for the Buck� boost converter model

Lu � Tu� 'qL� Vu�qC � ��L � 'qL

� � ��Cq�C

Switched model in generalized coordinates

L-qL � �� uR �� u 'qL & 'qC � & uE

qCC

� �R �� u 'qL & 'qC �

De�nition of state variables

x� � 'qL � x� � qC�C

Switched model for the Buck� boost converter

'x� � �� u �Lx� & u E

L

'x� � �� u �Cx� � �

RCx�

Table �� An EL approach for the modeling of the Buck�boost converter�


C The �Cuk converter

The �Cuk converter model is shown in Fig� ��!� In Tables �� and ��! we summarizeall the relevant formulae and equations leading to the switched model of the �Cukconverter circuit through our proposed EL formulation�

E

u � � u � �

�

R�

C�

C�

L�x�

L��x�

x�x�

Figure ��!� The �Cuk converter circuit�

Remark �� The Lagrangian approach to modeling of the �Cuk converter revealsthat only the �potential energy� structure of the system is non�invariant with respectto the switching action�

D Multivariable DC�to�DC power converters

A series of cascaded DC�to�DC power converters constitutes an interesting and newclass of multivariable switch�regulated converters with potential practical value andwhich nonetheless� seem to have been overlooked in the literature�

Consider the boostboost converter circuit of Fig� ��#�

u� � � �

R

E

L�

x� x�

L�

�x�

C�C�

x�

u� � �u� � �

u� � �

Figure ��#� The Boostboost converter circuit�

In Table ��# we present the EL parameters for the four possible combinations ofthe switch position parameters u� and u�� The Table �� contains the switched ELparameters� the system equations in generalized coordinates as well as the switchregulated state equations�


�CUKCONVERTER

ELParametersforpossibleswitchpositions

u

u�

u��

Kineticenergy

T� �'qL� �'qL� �� L

��'qL�

�&�� L�'qL�

�

T� �'qL� �'qL� �� L

��'qL�

�&�� L�'qL�

�

Potentialenergy

V� �qL� �qC� �

��C� q

�L�

&

��C� q

�C�

V� �qL� �qC� �

��C� q

�L�&

��C� q

�C�

Rayleighdissipation

F� �'qL� �'qC� �� R�'qL� �'qC� �

F� �'qL� �'qC� �� R�'qL� �'qC� �

Externalforces

Q�q

L�

�E

�

Q�q

L�

�

�

Q�q

C�

�

Q�q

L�

�E

�

Q�q

L�

�

�

F�q

C�

�

SwitchedELparameters

Kineticenergy

Tu �'qL� �'qL� �� L

��'qL� �&�� L�'qL� �

Potentialenergy

Vu �qL� �qL� �qC� �

��C�

��uqL�&uqL� � �&

��C� q

�C�

Dissipationfunction

Fu �'qL� �'qC� �� R�'qL� �'qC�

�

Externalforces

Quq

L�

�E

�

FuqL

�

�

�

FuqC

�

�

Table�� AnELapproachforthemodelingofthe�Cukconverter�ELparameters


�CUK CONVERTER

Lagrangian for the ��Cuk� converter model

Lu� 'qL� � 'qL�� qL� � qL� � qC� � Tu� 'qL� � 'qL�� Vu�qL� � qL� � qC��

�L� � 'qL�

� & ��L � 'qL�

� � ��C�

�� uqL� & uqL��

�C�q�C�

Switched model in generalized coordinates

L�-qL� � �� u �C�

�� uqL� & uqL�� & E

L�-qL� � �u �C�

�� uqL� & uqL� �� R � 'qL� � 'qC�qC�C�

� R � 'qL� � 'qC�

De�nition of state variables

x� � 'qL� � x� ��C�

�� uqL� & uqL� � � x� � 'qL� � x� �q�C�

Switched model for the �Cuk converter

'x� � �� u �L�x� &

EL�

'x� � �� u �C�x� & u �

C�x�

'x� � �u �L�x� � �

L�x�

'x� ��C�x� � �

RC�x�

Table ��!� An EL approach for the modeling of the �Cuk converter� Dynamic equa�tions�


BOOST�BOOST

CONVERTER

ELParametersforpossibleswitchpositions

�u� �u�

� �

��

� ��

��

Kineticenergy

T�� L

�'q �L�&�� L

�'q �L�

T�� L

�'q �L�&�� L

�'q �L�

T��T��

T��T��

Potentialenergy

V��

��C��qL� �qL�

�

&

��C� q

�C�

V��

��C� q

�L�&

��C� q

�C�

V��V��

V��V��

Dissipationfunction

F�� R�'qL� �'qC� �

F��F��

F�� R'q �C�

F��F��

Generalizedforce

Q��

qL��qL��qC�

� ��E ��

Q��


� ��E ��

Q��


� ��E ��

Q��


� ��E ��

Table��#�ALagrangianapproachtomodelingofthemultivariableBoost�boostconverter�ELparameters�


BOOST�BOOST CONVERTER

Switched EL parameters

Kinetic energy Tu�u� � ��L� 'q

�L�

& ��L� 'q

�L�

Potential energy Vu�u� � ��C�

�� u�qL� � qL� �� & �

�C�q�C�

Dissipation function Fu�u� ��R �� u� 'qL� � 'qC� �

�

External forces Qu�u�qL� �qL� �qC�

� �E �T

System equations in generalized coordinates

L�-qL� &�C�� u� �� u�qL� � qL� � � E

L�-qL� � �C�

�� u�qL� � qL� � � �� u�R �� u� 'qL� � 'qC� �qC�C�

� R �� u� 'qL� � 'qC� �

State variables de�nition

x� � 'qL� � x� ��C�

�� u�qL� � qL� � � x� � 'qL� � x� �qC�C�

Switched system state equations

'x� � �� u��L�x� &

EL�

'x� � �� u��C�x� � �

C�x�

'x� ��L�x� � �� u�

�L�x�

'x� � �� u��C�x� � �

RC�x�

Table �� A Lagrangian approach to modeling of the multivariable Boost�boostconverter� Dynamic equations�


E Modeling of non�ideal switches

In this section we extend the modeling approach presented in the previous section toinclude more realistic models of the switching device and of the circuit elements� Idealswitches do not exist in practise and these must be replaced by suitable arrangementsof a transistor and a diode �see � !�� $�� Such physical components exhibit nonidealities generally addressed as �parasitics�� For instance� these are represented bya small �ON resistance� in the transistor and a small �forward resistance� of thediode� An important parasitic element is constituted by an �o�set� voltage sourcefor the diode� Also� real life inductors and capacitors exhibit small resistances� Anadditional parasitic e�ect which� in the interest of simplicity� will not be included inthis study� is the so called �storage time modulation� e�ect� This topic is treated atlength in �� and also in � ��

Consider the more realisitic Boost converter model shown in Fig� ��

rDS

�VF

rC

C x�

R

u � �

E

�

�

rL

�

x�

RF

u � �

L

Figure �� Boost converter model with parasitic components�

This model� which includes an ideal switch� has been proposed in �� Theparasitic resistances rDS and RF represent� respectively� the ON resistance of thetransistor and the forward resistance of the diode� VF represents the o�set voltage inthe diode model� The resistances rL and rC are the resistances associated with theinductor and the capacitor of the circuit� The model� however� neglects the outputcapacitances of the transistor and of the diode�

Consider �rst the switch position parameter value u � �� As in the ideal case� twoseparate� or decoupled� circuits are clearly obtained and the corresponding Lagrangedynamics formulation can be carried out as follows� The EL parameters of the circuitare readily obtained as�%%�

%% T�� 'qL � �

�L � 'qL

� � V��qC � ��Cq�C

F�� 'qC ��rC &R � 'qC

� & ��rL & rDS 'q

�L

Q�qL

� E � Q�qC

�

��


Then� using EL equations and letting� x� � 'qL� x� � qC�C� one obtains after someelementary algebraic manipulations� the following set of di�erential equations�

'x� � � �

L�rL & rDS x� &

E

L

'x� � � �

�rC &RCx�

and the output load voltage� denoted now by xo� is readily obtained as

xo �R

rC &Rx��

Consider now the when case u � � The corresponding EL parameters of theresulting circuit are given by�%%�

%% T�� 'qL � �

�L � 'qL

� � V��qC � ��Cq�C

F�� 'qL� 'qC ��rL &RF 'q

�L & �

�rC� 'qC

� & ��R � 'qL � 'qC

�

Q�qL

� E � VF � Q�qC

� �

��!

Note that the appearance of the diode o�set voltage source VF modi�es the generalizedforce QqL � Alternatively� this voltage source could have also been taken into accountas part of the potential energy term� as V��qC � �

�Cq�C � VF qL �see ��$�� The

resulting dynamical model equations are identical in any case�

Using the EL equations and the previous de�nitions for x� and x�� one obtains�after some rearrangement� the following set of di�erential equations

'x� � � �

L�rL &RF & �rC jjR�x� � R

L�rC &Rx� &

E

L� VF

L

'x� �R

�rC &RCx� � �

�rC &RCx�

where the symbol �rC jjR denotes the resistance of the parallel arrangement of rCand R and the output load voltage� xo� is now obtained as�

xo �R

rC &Rx� &

rCR

rC &Rx�� #

Following the procedure proposed in the previous sections one obtains the follow�ing set of switched EL parameters�%%%%�

%%%%

Tu� 'qL � ��L � 'qL

� � Vu�qC � ��Cq�C

Fu� 'qL� 'qC ��rL� 'qL

� & �� uRF & urDS� � 'qL

�&��rC� 'qC

� & ��R �� u 'qL � 'qC �

�

QuqL

� E � �� uVF � QuqC

�

��


Note that in the cases where u takes the values u � � and u � � one recovers�respectively� the Rayleigh dissipation functions F�� 'qC in �� and F�� 'qL� 'qC in��! from the proposed dissipation function� Fu� 'qL� 'qC� in equations ��

We de�ne the Lagrangian function associated with the above de�ned switched ELparameters as�

Lu � Tu� 'qL� Vu�qC � �

L � 'qL

� � �

Cq�C ��

Using the EL equations� on �� one obtains the di�erential equationsdescribing the switched system which corresponds to the proposed EL parameters��

L-qL � �R�� u �� u 'qL � 'qC � & E � �� uVF � �rL & �� uRF & urDS� 'qL

qC�C � R �� u 'qL � 'qC �� rC 'qC

which can be rewritten� after substitution of the second equation into the �rst� as

-qL � � �

Lr 'qL � �� u

R

rC &R

qCLC

&E

L� �� u

VFL

'qC � �� uR

rC &R'qL � �

rC &RqC

where r denotes an �equivalent switched resistance� de�ned as

r ��rL & urDS &&�� uRF & �� u��rC jjR

Using x� � 'qL and x� � qC�C� as the state variables representing the inductor

current and the capacitor voltage� one obtains

'x� � � �

Lr x� � �� u

R

L�rC &Rx� &

E

L� �� u

VFL

��$

'x� � �� uR

�rC &RCx� � �

�rC &RCx� ��

It may be easily veri�ed that if the values of the parasitic elements� rC � rL� rDS�RF and VF are all set to zero in ��$� �� one obtains the ideal switched modelfor the Boost converter presented in ��

On the basis of this switched circuit model� the output voltage accross the loadresistance� denoted by xo� is readily obtained as

xo �R

rC &Rx� & �� u

rCR

rC &Rx�

which is in complete accordance with expressions �� and ��#� describing theload voltages found for each one of the circuit topologies�

� Hamiltonian modeling ��

� Hamiltonian modeling

Broadly speaking� a generalized Hamiltonian system is a lossless nonlinear system�provided with external control inputs� which is de�ned in a traditional state spaceframework� However� a generalized Hamiltonian system is identi�ed by an explicitinclusion of the contribution of the gradient of the total energy of the system in thedi�erential equations describing the controlled evolution of the state vector� In gen�eral� electric circuits are not considered to be Hamiltonian due to the presence ofresistive elements and some non dissipative elements which� nevertheless� disrupt thefundamental �canonical structure� of a lossless system� Thus� circuits with trans�formers� magnetic couplings between inductors� diodes� etc� are not per se consideredto be truly generalized Hamiltonian systems�

Yet the Hamiltonian formalism and much of its rich geometric structure and ap�pealing physical properties� can still be suitably preserved in the case of electric cir�cuits by allowing a basic lossless structure� constituted by the underlying LC circuit�while considering the rest of the elements to be modeled as external ports or inputsalong with constitutive relations describing their particular input�output properties�In this manner� resistive circuits� switched circuits and other interesting circuits canbe systematically modeled with an extended Hamiltonian viewpoint in a relativelystraightforward manner �see ��$�� $!��

The Hamiltonian approach allows for the systematic modeling of networks includ�ing resistors� transformers� diodes and switches� something that is not easy� and insome cases not even possible� to achieve within the Lagrangian formalism� Thesenon�energetic terms are �rst extracted from the circuit leaving an energy� conservingLC circuit with ports corresponding to the various extracted elements� The LC cir�cuit with ports can be represented as a generalized linear Hamiltonian system withexternal input variables� By terminating the ports of the Hamiltonian system withthe constitutive relations characterizing each of the extracted terms� one obtains amathematical description of the original switched circuit in forced generalized Hamil�tonian form with external inputs� One of the advantages of the proposed approach isthat for all operating modes of the switched circuit� the description of the system usesthe same state variables� the same Hamiltonian and the same dissipation functions�The obtained mathematical model is readily suitable for the application of the PBCfeedback regulation design methodology�

Generalized Hamiltonian systems have been extensively treated in the literature�Fundamental references� with plenty of physical examples� are constituted in thebooks �� $� and the recent monograph � $ �� The background results for thesystematic treatment of this topic� within the realm of general circuit theory� is foundin ��$�� $!�� The results in this section follow the work in �$#��


�� Constitutive elements

Within the generalized Hamiltonian modeling approach� all circuit elements are rep�resented as electrical ports� in which currents are identi�ed as �ow variables� denotedby f � and voltages are identi�ed as e�ort variables� denoted by e�

We consider �rst the continuous constitutive elements� such as resistors� capaci�tors� inductors and transformers� Next� we discuss the discontinuous elements� suchas switches and diodes� Denote by qC and �L the circulating charge in the capacitorand the linkage �ux in the inductor� respectively�

A Continuous constitutive elements

The constitutive laws for �linear constant resistive� capacitive and inductive elementsare then given by�

� Resistance�

eR � RfR

where fR is the current circulating through the resistance�

� Capacitor�

fc � 'qC � ec �qCC

� Inductor�

fL ��LL

� eL � '�L

In terms of the natural coordinates qC and �L the electric energy in the capacitor�playing the role of a potential energy V and the magnetic energy in the inductance�playing the role of a kinetic energy T are given� respectively� by�

V ��

Cq�C � T �

�

L��L

The Hamiltonian� which represents the total energy in an LC circuit� is simplygiven by

H � T & V ��

Cq�C &

�

L��L

� The ideal transformer�

�� Constitutive elements ��

fT�fT�

eT� eT�

Figure �� An ideal transformer circuit�

The case of an ideal transformer� shown in Fig� �� requires the consideration of theprimary and secondary terminals as two separate ports� each with its correspondingde�nition of �ow and e�ort variables� The ports� however� interact with each other�in an external fashion� according to one of the following constitutive relations

�a

�eT�fT�

�� J �n

�fT�eT�

��b

�fT�eT�

�� JT �n

�eT�fT�

��

where

JT �n �

� n�n

�is a skew symmetric matrix parameterized by n� thus re�ecting the power�conservationproperty of the transformer circuit�

B Discontinuous constitutive elements

The only discontinuous devices that we consider here are the ideal switch� and thediode� although the approach is suitable for the modeling of some other discontinuousdevices �such as transistors� for instance�

� The ideal switch�

An ideal switch can be considered as a lossless element� due to the fact that itcan conduct current at zero voltage �while being closed and it is capable of holdinga voltage with zero current �when it is open� The variable describing the switch isu which takes values from the discrete set f � �g� The following relations among the�ow and e�ort variables describe the behaviour of a switching element�

u �

�� fSW � IR � eSW � � � fSW � � eSW � IR�


Thus the ideal switch can� thus� conduct current in both directions�

� The ideal diode�

An ideal diode is a particular case of a unidirectional controlled switch� Its input�output behaviour is represented by the following relations�

Mode � eD � fD � Mode fD � � eD � � fDeD � �

�� LC circuits

As pointed out above in the Hamiltonian modeling approach the non�energetic el�ements such as resistors� transformers� diodes and switches are �rst extracted fromthe circuit� thereby leaving an energy�conserving LC circuit with ports correspondingto the various extracted elements� This LC circuit with ports can be represented inan intrinsic way as a Hamiltonian system with port variables� The representationof the original circuit is then obtained by terminating the ports of this Hamiltoniansystem by the extracted non�energetic elements� We brie�y review in this subsectionthe modeling of LC circuits�

It has been shown in ��$�� that an n�element LC circuit with m external ports�and total energy given by

H�x ��

x�Qx

where x � IRn is the state vector of the circuit� consisting of inductance �uxes �Liand capacitor charges qCi and Q being a diagonal symmetric matrix containing thecircuit parameters ��Ci� ��Li can always be written in the form�

'x � JQx &Gu ��

where u � Rm is the vector of external inputs to the system� The matrix G iscalled the input matrix and J is an nn skew�symmetric matrix� which is called thestructure matrix� The matrices G and J are determined from Kircho��s laws�

It can actually be shown that the natural outputs of the generalized Hamiltoniansystem �� can always be written in the form

y � G�Qx &Du � y � Rm

where D is a skew�symmetric matrix� called the throughput matrix� that appearswhenever there are static relations between port variables� The skew�symmetry ofthese matrices stems from the fact that the interconnections are all energy conserving�

Furthermore� it immediately follows that along the trajectories of the system� wehave 'H � u�y� which expresses energy conservation� then integrating from to T weverify the passivity property of the circuit�


Remark �� It is important to mention that the linear Hamiltonian systems we aredealing with are a special class of a more general class of systems called the generalizedHamiltonian systems which include systems which are described in local coordinatesby the following equations�

'x � J �x�H

�x&G�xu

y � G��x�H

�x&D�xu

where the matrices� J �x� G�x and D�x are not constant� but functions of the state�In such a general case the Hamiltonian need not be a quadratic function of x�

�� Examples

A Boost converter

Let us consider the ideal Boost converter circuit shown in Fig� �� The circuitcoordinates are x � ��L� qC �

�� and the electric and magnetic energies are given by

V ��

Cq�C

T ��

Lq�L�

So the Hamiltonian is then given by

H ��

Cq�C &

�

Lq�L �

�

��L� qC �

��

�L

�

C

� ��LqC

��

�

x�Qx

and the gradient vector of the Hamiltonian is simply

�H�x

� Qx �

��L�LqC�C

��

We replace the single switching element in the Boost circuit by two ideal dependent�or conjugated� switches as shown in Fig� ��$


L

R

E

�� u

qC

�L&

eSW�u

C

Figure ��$� Boost converter model with conjugate switches�

Using the concept of extracting the switches and the resistors from the circuit wehave� �

'�L'qC

��

��

��eSW� &

� �

��fSW� &

� �

��fR &

��

�E

and the corresponding outputs for the source� resistance and switches are given by��

fsfSW�

eSW�

eR

��

��

� � � �

��Qx &

��

��

��

E�eSW�

fSW�

�fR

��

From the constitutive relations for resistances and equation �� we have

fR ��

ReR �

�

RCqC

thus for each switch position we have

u �

�%%�%%

��

eSW� � qC�C eSW� � fSW� � fSW� � �L�L

� ��

eSW� � eSW� � qC�CfSW� � �L�L fSW� � �

Notice that even though the number of switches is two we have only two modes ofoperation instead of four due to the fact that the ideal switches are dependent �whenone is closed the other is open and vice�versa� We can then write

eSW� � �� uqC�C � fSW� � �� u�L�L

Summarizing the derivations presented above� we obtain the dynamical equations forthe Boost converter model�

'�L'qC

��

� �� u

�� u

� ��L�LqC�C

�&

� � �

R


�&

��

�E�


Letting x� � �L�L and x� � qC�C we recover the traditional switched model�� for the Boost converter derived using the Lagrangian approach�

B �Cuk converter

Consider the �Cuk converter� shown in Fig� ��! provided with an ideal switch� Thesystem coordinates are given by x � ��L�� qC�� L�� qC��

� and the electric and magneticenergies are given by

V ��

C�

q�C� &�

C�

q�C�

T ��

L��L� &

�

L��L��

As before� we introduce two single ideal conjugate switches that replace the singleswitch in order to obtain a graphical model from which the switch extraction proce�dure is more clearly accomplished �see Fig� ��

E

R

�

�

�

qC�

C�L�

�L�

L�

�L�

qC�

�� uu C�

eSW�eSW�

Figure �� Cuk converter model with conjugate switches�

Then we have��

'�L�'qC�'�L�'qC�

��

��

��

��L��L�

qC��C�

�L��L�

qC��C�

�� &

��

�

��E

&

��

�

�� fSW� &

��

� ��

�� eSW� &

��

�

�� fR


with the outputs for the source� switches and resistance given by

��

fseSW�

fSW�

eR

��

��

� � � ��

��Qx &

��

��

��

E�fSW�

�eSW�

�fR

��

where fR � ��ReR � ��RC�qC�� The e�ort and �ow variables for the two possibleswitch positions are given by

u �

�%%�%%

� ��

fSW� � eSW� �

�eSW� � �qC��C�

fSW� � �L��L� � �L��L�

��

eSW� � fSW� �

�fSW� � �L��L� � �L��L�

eSW� � qC��C�

then we can write

eSW� � �uqC��C�

fSW� � u ��L��L� � �L��L� �

In summary we obtain the following dynamical equations

��

'�L�'qC�'�L�'qC�

��

��

�� u �� u u

�u ��

��L��L�

qC��C�

�L��L�

qC��C�

��&

��E

�� &

��

� qC�RC�

��

fs � �L��L��

C A Boost converter model with switches and diodes

The ideal switches considered above allows the �ow of current in both directions� Inactual converters� this is not the case due to the fact that ideal switches are usuallyrealized by means of transistors and diodes�

We illustrate the main di�erences with the above developments by considering theinclusion of a clamping diode in the Boost converter circuit as illustrated in Fig� ��


E

L

R

&

&

x�

eSW

eD x�

C

Figure �� Boost converter model with a diode�

As before� one obtains the model by extracting both� diodes and switches� fromthe circuit and treating them as external ports� This procedure leads to�

'�L'qC

��

� ��


�&

��

�E &

� �

��fR &

��

� � �eD�fSW

�eR � � ��Qx � qC�C

with outputs for the port variables given by��fDeSW

��

��

� �qL�LqC�C

�&

� ��

� �eDfSW

��

and the output current� �owing through the resistor given by

fR ��

ReR �

�

RCqC

Analyzing now each one of the two positions of the switch u � f � �g� we have

u �

�%%�%%

� fSW � ��

eD � eSW � qC�CfD � �L�L

� � eSW � ��

eD � �qC�CfSW � �L�L� fD�

The values of fSW and eD can then be parameterized in terms of u as� �eD�fSW

��

�uqC�C � �� ueDu��L�L& fD

�which yields the following switched model placed in terms of the e�ort and �owvariables of the diode�

�'�L'qC

��

� �� u

�� u


�&

�E

�&

�

� qCRC

�&

� �� ueDufD

��


For each switch position� there are two modes of operation depending on whetheror not the diode is in conduction mode� Thus we can distinguish four modes ofoperation�

Mode � � u � � eD � � fD � �L�L ��

'�L � �qC�C & E'qC � �L�L� �

RCqC

Mode � u � �� fD � � eD � �qC�C ��

'�L � E'qC � �qC�C

Mode ! � u � � fD � �L�L � � eD � E � qC�C ��

'�L � 'qC � ��RCqC

Mode # � u � �� eD � qC�C � � fD � ��

'�L � E'qC � �

In order to know the mode which the circuit is operating in� one must �rstly observethe switch position value and then look at the states of the system� For u � thereare two modes� if the signal �L�L � then the system is in mode �� and if �L�L � then it is in mode !� For u � �� if qC�C � then the system is in mode � but ifqC�C � then it is operating in mode #�

D Converter circuits with switches� diodes and transformers

To conclude this section� we consider the circuit depicted in Fig� �� commonlyknown as the Flyback�

R

&

E

eSW

eD

C

Figure �� The Flyback converter circuit�

It is in fact a Buck�boost converter with an ideal transformer isolating the outputcircuit from the input circuit�

Contrary to previous cases we proceed to establish the generalized Hamiltonianmodel of the switched circuit for each switch position� This demonstrates the �exi�bility of the method�


Case u � �� The equations for this case are given by�'�L'qC

��

��

�es &

� �

��fR &

� fD

�with the corresponding outputs

eR � qC�C

eD � �qC�C � �

nes � �qC�C � �

nE

fs � �L�L &�

nfD

where fs is the current �owing through the source and

fR � eR�R ��

RCqC �

If the converter is in the normal mode of operation� i�e�� qC�C � � then eD � �This in turn� implies that fD � � and we have only one mode of operation� For thiscase then� the model results in

Mode � �

�'�L'qC

��

��

�es &

� �

��

RCqC�

Case u � � The equations are�'�L'qC

��

� �

��fR &

� fD

�&

�eT�

�with corresponding outputs

eR � qC�C

eD � �qC�C & eT�

fT� � �L�L�

and from the constitutive relations of the ideal transformer� we have

eT� � �neT� � fT� � nfT�

From this and the fact that fD � fT� we obtain that

eT� � �neD � nqC�C � fD � n�L�L

whereas for the resistance�

fR ��

RCqC �


Substituting these expressions in the model� we obtain�'�L'qC

��

� �nn


�&

� �

��

RCqC &

� �neD

��

As we can see for the case u � there are two modes depending on whether thediode is conducting or not�

Mode � eD � � fD � n�L�L

hence �'�L'qC

��

� �nn

� �qL�LqC�C

�&

� �

��

RCqC

Mode ! � fD � � eD � �qC�C ��

'�L'qC

��

� �

��

RCqC

The mode in which the system is operating is found out by �rst observing in whichposition is located the switch� If u � then the circuit is in mode � of operation� Ifu � � then we should observe the state of the circuit� If the quantity �L�L � thenwe are in mode � if� on the other hand� it is exactly zero then we are in mode !�

Average models of PWM regulated converters

In this section a Lagrangian dynamics approach is used for deriving a physically ori�ented models of the average behavior of pulse width modulation �PWM regulatedDC�to�DC power converters� As in the switched case� the approach consists in �rstestablishing the EL parameters of the circuits associated with each one of the topolo�gies corresponding to the two possible positions of the regulating switch� Secondly�an average PWM model of the non�invariant EL parameters can then be proposed bytheir suitable modulation through the duty ratio function� This modulation is donein a consistent fashion so that� under extreme duty ratio saturation conditions� theoriginal EL parameters� corresponding to the two intervening circuit topologies� areexactly recovered� Secondly� an intermediary condition which is consistent with thephysically plausible interpretation of the average value of the EL parameter guidesthe choice of the weighting of each intervening EL parameter by means of a simplelinear function of the duty ratio function�

The average EL parameter considerations lead� through use of the classical La�grangian dynamics equations� to systems of continuous di�erential equations� describ�ing the average PWM converter behavior� These equations are interpretable in termsof ideal equivalent circuit realizations obtained by replacing the switching device bya suitable ideal transformer� This particular result� �rst reported in � #�� is in ac�cordance with well�known circuit equivalents of PWM switches already derived in

�� General issues about pulse�width�modulation ��

�� and � �� The obtained average PWM models entirely coincide with the stateaverage models of DC�to�DC power converters introduced in �� and also with theinnite switching frequency model reported in � # ��

We begin this section by considering some general remarks about PWM controllednonlinear systems� We then present some considerations about the average modelingof discontinuously controlled EL systems� The general modeling procedure is appliedto obtaining the average models� from the Lagrangian formalism viewpoint� of DC�to�DC power converters of the �Boost� � the �Buck�boost� and the �Buck� type�equipped with ideal switching devices�

�� General issues about pulse�width�modulation

PWM is a widely used regulation technique for switch commanded systems� Typi�cally� a switch regulated system exhibits as a control input called the switch positionfunction� whose values can always be constrained to be found in the discrete set f � �g�In such a context� a �pulse� refers to a scalar time signal corresponding to the switchposition function which is zero everywhere except on a �nite interval of time whereit takes the value of one� The PWM regulation technique is used in conjunction witha sampling process carried out at a �xed frequency� At each sampling instant� thestate of the controlled system is determined and� on its basis� the width of a pulse isspeci�ed for the current sampling interval� The width T of the sampling interval isusually referred to as the duty cycle and the fraction of this time interval occupied bythe pulse is usually referred to as the duty ratio and denoted by �� Although manyparticular forms of PWM schemes do exist� we assume here that the pulse beginsprecisely at the sampling instant tk and that it ends before the sampling interval isover� i�e� before tk&T � If the pulse occupies all of the sampling interval� or no part ofit� the duty ratio is said to be operating under saturation conditions� Hence� the dutyratio function is a feedback function of the state� taking values in the open interval� � ��

Basic work related to PWM control schemes� in linear and nonlinear dynamicalsystems� can be found in � � � �� Seminal work is due to the e�orts of Tsypkin� �� For a more recent account with a rather complete survey of early work see��##�� Also� some generalizations and applications of PWM control in the area ofDC�to�DC power conversion can be found in � !�� # ��

Consider a nonlinear single�input single output system of the form�

'x � f�x & g�xu ��

where u is a switch position function taking values in the discrete set f � �g� Generallyspeaking� a PWM regulation policy is speci�ed as follows�

u �

$� for tk � t tk & ��x�tkT

for tk & ��x�tkT t � tk & T � tk��

�� !


The exact determination of the state x�t of the controlled system �� !�at the sampling instants tk � k � � �� for a given duty ratio function is anextremely di�cult problem� except for the simplest drift vector �elds f�x and controlinput �elds g�x� For the linear systems case� the problem admits only approximatesolutions even for low dimensional systems�

Continuous time average models of PWM controlled systems have been used assimple and useful substitutes of exactly discretized switch regulated systems� Herewe explore one such average model usually known as average state model� It consistsin the obtained continuous time description of the PWM regulated system when thesampling interval is reduced to zero by a limiting argument�

Consider an arbitrary sampling time interval �tk� tk�� tk� tk&��x�tkT �� tk&��x�tkT� tk�� of length T � We shall determine� on the basis of �� and �� !�the value of the state� at the end of the sampling interval� by rewriting the di�erentialequation as an integral equation�

x�tk & ��x�tkT � x�tk &

Z tk��x�tk��T

tk

�f�x�� & g�x�� d�� #

On the other hand� at time tk & T � tk�� the state is given by

x�tk & T � x�tk & ��x�tkT &

Z tk�T

tk��x�tk��T

f�x��d� ��

Substituting �� # into �� one obtains�

x�tk & T � x�tk &


tk

f�x�� & g�x��d� &

Z tk�T

tk��x�tk��T

f�x��d�

� x�tk &

Z tk�T

tk

f�x��d� &


tk

g�x��d�

The di�erence of the states at the beginning and at the end of the sampling interval�tk� tk & T �� divided by the sampling period T is given by

�

T�x�tk & T � x�tk� �

�

T

Z tk�T

tk

f�x��d� &�

T


tk

g�x��d�

Taking limits as T � and letting the time instant tk take the generic value t oneobtains�

limT��

�

T�x�t & T � x�t� � lim

T��

��

T

Z t�T

t

f�x��d� &�

T

Z t��x�t��T

t

g�x��d�

�� 'x�t � f�x�t & g�x�t��x�t� ��


The continuous average model of a PWM regulated system is thus given by thesame nonlinear model except that the discrete control input u is substituted by thecontinuous� but limited� duty ratio function �� In order to establish a di�erencebetween the actual nonlinear model and its continuous time average �� we de�note the average state vector by z�t rather than by x�t� In summary we have thefollowing�

Proposition �� The continuous state average model of a PWM regulated system�

'x � f�x & g�xu �� $

with�

u �

$� for tk � t tk & ��x�tkT

for tk & ��x�tkT t � tk & T � tk��

��

is given by

'z � f�z & g�z� ��

�

The continuous time approximation �� of PWM regulated systems �� $�� has been extensively used in the literature devoted to the control of nonlinearPWM regulated systems to transform an essentially discrete�time nonlinear controlsynthesis problem into an equivalent continuous time nonlinear control problem� Oncethe continuous time control�constrained regulation problem has been solved for theaverage approximation� in the sense that a speci�cation of the duty ratio has beenfound in the form of a sate feedback function� then the obtained average closed loopduty ratio synthesis law is used in the actual PWM regulated model using the actualsampled state variables instead of their averaged values� The nature of the approxi�mation depends on how close is the actual �xed sampling interval to the ideal valueof zero� Generally speaking� the higher the sampling frequency the more accurate isthe average continuous time approximation�

�� Examples

In this subsection we derive EL models to describe the average behaviour of PWMcontrolled DCtoDC converters� We will see that the average models exactly coincidewith the exact switched models derived in the two previous sections� simply replacingthe switch position u by the duty ratio �� For the sake of brevity we present only theexample of the boost and the Buckboost converters�


A Boost converter

Consider the switchregulated Boost converter circuit of Fig� �� The di�erentialequations describing the circuit are given by ��#� �� and are repeated here forconvenience as

'x� � �� u�

Lx� &

E

L��!

'x� � �� u�

Cx� � �

RCx�� !�

A PWM policy regulating the switch position function u� may be speci�ed asfollows�

u�t �

$� for tk t � tk & ��tkT

for tk & ��tkT t � tk & T

tk�� tk & T � k � � �� !

where tk represents a sampling instant� the parameter T is the �xed sampling period�also called the duty cycle� the sampled values of the state vector x�t of the converterare denoted by x�tk� The function� �� is the duty ratio function� truly acting as anexternal control input to the average PWM model of the converter �see � # �� Thevalue of the duty ratio function� ��tk� determines� at every sampling instant� tk� thewidth of the upcoming �ON� pulse as ��tkT �during this period the switch is �xed atthe position represented by u � �� The actual duty ratio function� �� is evidentlya function limited to take values on the closed interval � � �� of the real line�

We recall� from previous sections� the EL parameters associated with each one ofthe two electric circuits obtained as a result of �xing the switch at each one of thetwo possible regulating positions�

For u � � we have�$ T�� 'qL � ��L 'q�L � V��qC � �

�Cq�C

F�� 'qC ��R 'q�C � QqL � E � QqC �

��!!

For u � � we have�$ T�� 'qL � ��L 'q�L � V��qC � �

�Cq�C

F�� 'qL� 'qC ��R � 'qC � 'qL

� � QqL � E � QqC � ��!#

Note that� according to the PWM switching policy ��! � on every samplinginterval of period T � the Rayleigh dissipation function F�� 'qC is valid only a fraction ofthe sampling period given by ��tk while the Rayleigh dissipation function F�� 'qL� 'qCis valid a fraction of the sampling period equal to �� tk�


There are� of course� a variety of ways in which one could reasonably propose anaverage value of the Rayleigh dissipation function for a circuit of the form ��! ��!� undergoing a switching policy of the form ��! � One possible and perhapsnatural way is to propose the following set of average EL parameters�

$ T�� 'qL � ��L 'q�L � V��qC � �

�Cq�C

F�� 'qL� 'qC ��R � 'qC � �� 'qL�

� � Q�qL

� E � Q�qC

� ��!�

Note that in the cases where � takes the extreme saturation values � � �� or � � �one recovers� respectively� the dissipation functions F�� 'qC in ��!! and F�� 'qL� 'qC in��!# from the proposed average dissipation function� F�� 'qL� 'qC� of equation ��!��Indeed� such a �consistency� condition is veri�ed by noting that�

F�� 'qL� 'qC��

� F�� 'qL� 'qC � F�� 'qL� 'qC��

� F�� 'qC�

Also� it is easy to see that the proposed average Rayleigh dissipation functionsatis�es an important �intermediary� condition of the form�

min fF�� 'qL� 'qC�F�� 'qCg � F�� 'qL� 'qC � max fF�� 'qL� 'qC�F�� 'qCgfor any � lying in the open interval � � ��

We note that the Lagrangian function associated with the above de�ned averageEL parameters is actually invariant with respect to the switch position function�Nevertheless� to keep the notation consistent� we denote it by�

L� � T�� 'qL� V��qC � �

L 'q�L �

�

Cq�C �

One then proceeds� using the EL equations to obtain the di�erential equationsde�ning the average PWM model as

L-qL � �� R � 'qC � �� 'qL� & E ��!�qCC

� �R � 'qC � �� 'qL� ��!$

which can be rewritten� after substitution of ��!$ into ��!�� as

-qL � �� qCLC

&E

L

'qC � � �

RCqC & �� 'qL�

Using z� � 'qL and z� � qC�C one �nally obtains

'z� � ��

Lz� &

E

L��!�

'z� � ��

Cz� � �

RCz� ��!�


where we denote by z� and z� the average input current and the average outputcapacitor voltage� respectively� of the PWM regulated Boost converter� We establishthis distinction with the nonaveraged variables x� and x� so that the state variablesassociated with the average PWM model are not mistakingly confused with the actualPWM regulated circuit variables�

Note that the proposed average dynamics ��!�� !� coincides with the stateaverage model developed in �� and with the in�nite switching frequency model� orFilippov average model� found in �� # �� To obtain the average model ��!�� !��one simply replaces the switch position function� u� in ��! � ��!� by the duty ratiofunction � and the actual state variables x�� x� by their averaged values� z�� z�� Thedevelopments above are formalized in the following proposition�

Proposition �� The state average model of the Boost converter �see �� givenby �� is an EL system corresponding to the set of average EL parametersgiven by �� These parameters are� in turn� obtained by suitable modulation�through the duty ratio function �� of the EL parameters� given by �� and �� which are associated to each one of the intervening circuit topologies arising from aparticular value of the switch position function� �

Similarly to the developments of Section � for ease of reference we will be usingthe following� more compact� matrix representation of ��!�� !�

DB 'z � �� J z &RBz � EB ��#

where

DB �

�L C

�� J �

� ��

�� RB �

� ��R

�� EB �

�E

��#�

Remark �� It is easy to realize that the average model ��!�� !� has a circuittheoretic interpretation by letting the quantity �� z�� in the �rst equation� rep�resent a controlled voltage source while also letting the quantity �� z�� in thesecond equation� represent a controlled input current source� Fig� �� depicts theideal equivalent circuit describing the average PWM model�

L

z�

z�

�� z�� z� R

�

C

E

��

�

Figure �� Equivalent circuit of the average PWM model of the Boost convertercircuit�


In such a circuit� a quadrapole �shown in Fig� �� is connecting the �input� and�output� circuits which e�ectively replaces� in an average sense� the actual switchingdevice�

z�

�� z� �� z�

z�&

&�

Figure �� Ideal transformer representing the average PWM switch position func�tion�

With some elementary power considerations we can establish that the quadrapoleis a lossless� ideal �average power transferring device� where the average input voltageto the quadrapole� �� z�� is ampli�ed to the value z� at the output� while theinput current to the quadrapole� z�� is attenuated to the value ��z� at the output�The switching element has been thus e�ectively replaced by an ideal transformer withturn ratio parameter given by ��

B Buck�boost converter

Following exactly the same considerations as for the Boost converter we have theproposition below for the switchregulated Buckboost converter�

Proposition �� The state average model of the Buck�boost converter �see �� given by

'z� � ��

Lz� & �

E

L��#

'z� � ��

Cz� � �

RCz� ��#!

is an EL system corresponding to the following set of average EL parameters

T�� 'qL ��

L 'q�L � V��qC � �

Cq�C

F�� 'qL� 'qC ��

R � 'qC & �� 'qL�

� � Q�qL

� � E � Q�qC

�

obtained by suitable modulation� through the duty ratio function ��


Fig� ��! depicts the equivalent circuit of the average PWM regulated dynamicsfor the Buckboost converter circuit�

L

�� z� �� z� R

z�

�z�

�E�

� C�

�

Figure ��!� Equivalent circuit of the average PWM model of the Buckboost con�verter circuit�

We will be using the following matrix representation of ��# � ��#!

DBB 'z & �� J z &RBBz � �EBBwhere

DBB �

�L C

�� J �

� ��

�� RBB �

� ��R

�� EBB �

�E

��##

�� Some structural properties

In Section we have shown that the switched model of the DCtoDC converterssatis�es the energy balance equation ��!� Obviously this equation also holds true forthe averaged model� thus establishing the passivity of the operator mapping externalvoltage to the current� Besides this fundamental property� DCtoDC convertershave some other structural properties that will be considered for the design of thePBC� We present in this section those pertaining to the Boost and the Buckboostconverter�

A Boost converter

We will �rst prove below that the system ��!�� !� is non�minimum phase withrespect to the output z�� This poses a serious di�culty for the PBC designs� since aswe have pointed out before� the implementation of the controller relies on some kindof the system inversion� Fortunately� as we will see later� the equilibria of z� and z�are in a onetoone correspondence� and furthermore the zero dynamics with respectto z� are stable� This suggests to control z� indirectly via the regulation of z��

�� Some structural properties ��

A�� Zero dynamics

A straightforward elimination of z� from the set of di�erential equations ��!�� !�leads to the following nonlinear inputoutput di�erential representation�

-z� &

��

RC&

'�

��

�'z� &

�

LC

�� &

L

R

'�

��

�z� � ��

E

LC

The �zero dynamics� at a desired equilibrium point z� � z�� associated with thisinputoutput representation� is obtained by letting 'z� � and -z� � �see �� Theresulting di�erential equation describing the �remaining dynamics� of the duty ratiofunction � is simply obtained as

'� �R��

Lz��E � �� z�� #�

The equilibrium points of ��#� are given by

� � � � � � �� E

z��

among which the equilibrium value � � �� E�z�� has physical signi�cance�provided z�� E� This fact con�rms the �amplifying� features of the Boost converter�However� the phase�plane diagram of equation ��#�� shown in Fig� ��# � readilyreveals that this equilibrium point is unstable� We conclude that the average PWMmodel of the Boost converter with ouput� the average capacitor voltage z�� is actuallya non�minimum phase system�

Figure ��#� Zero�dynamics of Boost converter corresponding to constant averageoutput voltage�

Consider now the output of the circuit to be represented by the average inputcurrent� z�� One obtains the following di�erential inputoutput representation forthe average system

-z� &

��

RC&

'�

��

�'z� &

��

�

LC

�z� �

E

L

��

RC&

'�

��

�� #�


The �zero dynamics� at an equilibrium point z� � z�� associated with the inputoutput representation ��#�� is obtained as�

'� ��

RCE

�� Rz�� E

� ��#$

and the equilibrium points of ��#� are given by

� � � � � � ��r

E

Rz�� &

rE

Rz��

The equilibrium value� � � �� pE�Rz�� has physical signi�cance provided

Rz�� the average steady state voltage across the load resistor� satis�es Rz�� E�This fact con�rms� once more� the �amplifying� character of the Boost converter�The phase�plane diagram of equation ��#$� shown in Fig� �� reveals that thisequilibrium point is now locally stable� We conclude that the average PWM model ofthe Boost converter� with output represented by the average input inductor currenty � z�� is a minimum phase system�

Figure �� Zero�dynamics of Boost converter corresponding to constant averageinput current�

A�� Equilibria

We proceed now to establish the relationship between the equilibria of the averageoutput voltage and the average input current� To this end assume a constant dutyratio function � � �� It easily follows from the average PWM model equations��!�� !� that the corresponding stable equilibrium values for the average inputcurrent� denoted by z�� and the average output voltage� denoted by z�� are given by

z�� E

�� R� z��

E

��

�� Some structural properties ��

Henceforth� given a desired equilibrium value z�� for the output voltage� which corre�sponds to a constant value of the duty ratio function � � �� E�z�� the uniquecorresponding equilibrium value for the average input current is given by

z�� z��

R��

�

REz�� #�

This means that if we desire to regulate z� towards an equilibrium value z�� which isknown to correspond to a steady state value �� of the duty ratio function �� then� sucha regulation can be indirectly accomplished by stabilizing the average input currentz� towards the corresponding equilibrium value z�� computed from ��#��

B Buck�boost converter

Similarly to the Boost converter case� one can easily establish the nonminimumphase character of the average model of the PWM regulated Buckboost convertersystem when the output of the system is taken as the average capacitor voltage z��When the output of the system is taken to be the average input inductor current� z��the resulting inputoutput system is seen to be locally minimum phase �see � ##��

Given a constant duty ratio function � � �� it easily follows from the averagePWM model equations ��# � ��#! that the corresponding stable equilibrium valuesfor the average input current� denoted by z�� and the average output voltage� denotedby z�� are given by

z��

��

�E

R� z��

��

��

�E ��#�

This means that� depending on the particular value of the steady state duty ratiofunction� �� the Buckboost converter can accomplish� in steady state� either sourcevoltage �ampli�cation� or �attenuation�� modulo a polarity inversion� at the load�

It follows from ��## that� given a desired equilibrium value� z�� for the outputvoltage� which corresponds to a constant value �� of the duty ratio function �� thenthe unique corresponding equilibrium value for the average input current� z�� is givenby

z�� z��R��

�

�z��RE

� �

E

�z��

Hence� if we desire to regulate z� towards an equilibrium value z�� which correspondsto a steady state value �� z��z�� E of the duty ratio function �� then� such aregulation can be indirectly accomplished by stabilizing the average input current z�towards the corresponding equilibrium value z�� computed from ��#��


Conclusions

We have shown that well�known models of DCtoDC power converters constitute aspecial class of EL systems with switch�dependent EL parameters� Ideal switchingdevices were �rst considered and the corresponding switched models of the traditionalconverters structures were derived by appropriately combining the EL parametersassociated with the intervening circuit topologies� The Lagrangian formalism wasalso extended to handle multivariable versions of switched�regulated power convertersand realistic models of traditional switchregulated power supplies including parasiticresistances and parasitic voltage sources�

It has also been shown that using the generalized Hamiltonian formalism of ��$��and � $!� there exist possibilities for the systematic derivation of mathematical modelsthat describe a wider range of DC�to�DC power converters� In particular� the Hamil�tonian viewpoint naturally allows for the introduction of some continuous� as well asdiscontinuous� devices which are not easy to model from the Lagrangian viewpoint�Speci�cally� external sources� resistors� diodes� isolating transformers and parasiticsources may be systematically included in lossless LC circuits as port elements alongwith their characterizing constitutive relations� This procedure results in a morerealistic description of switched power converters in general�

Classical state average models� or in�nite switching frequency models� of DC�to�DC power converters were shown to be EL systems for a suitable set of average ELparameters� The derived average PWM models were also shown to be interpretable interms of ideal circuit realizations including internal controlled sources and modulatedexternal inputs�

As it has been discussed throughout the book the Lagrangian formulation is con�sistent with our PBC approach� which has emerged as an advantageous physicallymotivated controller design technique which exploits the energy structure of EL sys�tems� Thus� this chapter shall be regarded as a �rst step towards the formalizationand development of PBC for a variety of switched regulated models of DCtoDCpower converters� which is addressed in the next Chapter $�

Chapter �

Passivity�based control of

DC�to�DC power converters

� Introduction

The feedback regulation of DCtoDC power supplies is� broadly speaking� accom�plished through either PWM feedback strategies� or by inducing appropriate stabi�lizing sliding regimes� PWM control of these devices is treated in several books�among which we cite ��$� !�� The topic has been also extensively treated� amongmany others� by the third author an collaborators in � ##� #�� where emphasis hasbeen placed in using advanced nonlinear feedback control design techniques for theregulation of average PWM models of the various converters�

Sliding mode �SM control of switched power supplies was �rst treated in � $��from a linearized model viewpoint� The subject bene�ted from the advances in thegeometric understanding of sliding modes in nonlinear systems� Some referencesadopting this last viewpoint are constituted in � # � #!� and � #$�� More recently� inthe context of motion control systems� the problem has been successfully cast as partof a larger regulation problem in � ��

The feedback controller design approaches mentioned above entirely overlook the�energy related� physical properties of either the original converter circuit or of itsclosed loop structure� The controller design philosophy� primordially insists on amathematically motivated� average closedloop linearization� geared to solve the sta�bilization or tracking task�

The main advantage of underscoring the often overlooked physical properties ofDCtoDC power converters in an EL or Hamiltonian modeling context is that� in thisway� we can advantageously exploit these properties at the feedback controller designstage� In particular� we explore the relevance and implications of a �passivity�based�approach in the feedback duty ratio synthesis problem� In this chapter we describe two

��

�� Ch� $� Passivity�based control of DC�to�DC power converters

approaches for the passivity based regulation of DC to DC power converters withinthe context of a stabilization task� First� we prescribe feedback duty ratio designsbased on the average PWM model of the circuit� In the second part of the section acombination of passivity based regulation and sliding mode control is explored�

In order to e�ectively deal with a sometimes realistic issue of constant but un�known resistive loads for the converters under study� an adaptive feedback regulationscheme has also been explored within the various design methodologies available fornonlinear systems and in particular within the PBC methodology� Adaptive feedbackregulation of power supplies has been our subject of study in � #�� and � #�� Morerecently the passivity based regulation schemes for switched power converters havealso been extended to the adaptive case in � #��

In Section we study the passivity based stabilization of average models of PWMregulated DCtoDC power converters� An extension of the PBC design which suit�ably combines the energy shaping plus damping injection methodology with slidingmode control is introduced in Section !� Section # is devoted to extend the PBCdesign for the adaptive regulation of DCtoDC power converters� The last sectionof this chapter� Section �� reports an experimental comparison of several linear andnonlinear regulators for the stabilization of DCtoDC power converters� includinglinear control� PBC and feedback linearization� The chapter is closed with someconclusions and suggestions for further study in this area�

� PBC of stabilizing duty ratio

In this section we develop PBCs for DCtoDC power converters described by theaverage models derived in Section ��#� Therefore� for the sake of validation of themodel� we implicitly assume that the sampling frequency is su�ciently high� Eventhough we work out the details only for the Boost and the Buckboost converter theapplication of the technique to other converters follows mutatis�mutandis�

Due to the non�minimum phase nature of the average output voltage variable� adirect application of the passivity based design method� aimed primarily at outputvoltage regulation� leads to an unstable dynamical feedback controller� This is dueto an underlying partial inversion of the average system model� carried out at thecontroller design stage� For this reason� an indirect approach� consisting of outputvoltage regulation through inductor current stabilization is undertaken� Indirect con�troller design for non�minimum phase systems has been justi�ed in the context ofDCtoDC power converters� in � ##�� The indirect control technique also naturallyarises� from module�theoretic results� in sliding mode control of linear multivariablenon�minimum phase systems� as inferred from ��#��

The performance of the derived� indirect� dynamical state feedback controllersis successfully tested� via computer simulations� for the Boost converter example�

�� The Boost converter ��

The model used for the switched Boost converter included an unmodeled stochasticperturbation input� directly a�ecting the external voltage source� as well as unmodeledparasitic resistances attached to each one of the circuit elements�

� The Boost converter

We consider the average model of the Boost converter ��# ��#�� which we repeat herefor ease of reference

DB 'z � �� J z &RBz � EB �$��

DB �

�L C

�� J �

� ��

�� RB �

� ��R

�� EB �

�E

��$�

The control objective is to regulate the output capacitor voltage z� to a constantvalue� z�� E�

A Strict passivation

Following the PBC methodology we will achieve the control objective by making theclosedloop passive with respect to a desired storage function� Motivated by the formof the total energy function of the average system model� which� as it was shownbefore� is given by H � �

�z�DBz� we propose as desired storage function

Hd ��

�z�DB�z �$�!

where �z�� z � zd� and zd is a �desired� value for z� yet to be de�ned� It is important

to observe at this point that the system is underactuated� therefore as for the �exiblejoint robot problem of Section ��!�!� we cannot select arbitrary functions for the�desired� signals� they will result instead from the de�nition of the error dynamics�

The desired error dynamics associated to the storage function �$�! is then

DB'�z & �� JB�z &RBd�z � / �$�#

where we have added the required damping by choosing a desired Rayleigh errordissipation term�

Fd ��

�z�RBd�z �

�

�z� �RB &R�B �z

where

R�B �

�R�

�� R� � �


It is clear that the error dynamics �$�# de�nes an OSP map / �� z with storagefunction �$�!� Notice that in this particular example the OSP is obtained with respectto the full state vector� therefore we do not need to invoke detectability arguments toprove asymptotic stability as in previous examples� Actually� in this simple exampleit is easy to show that the unperturbed error dynamics

DB'�z & �� JB�z &RBd�z � �$��

is exponentially convergent� by taking the time derivative of Hd along the solutionsof �$�� to get

'Hd � ��z�RBd�z ��

Hd � � �z ��

where � may be taken to be � � minfR�� Rg and � maxfL�Cg�The next step is then to derive the controller dynamics by setting / � � It is

easy to see from �$�� and �$�# that the perturbation term

/�� EB � �DB 'zd & �� JBzd &RBzd �R�B�z

hence� setting / � we get

DB 'zd & �� JBzd &RBzd �R�B �z � EBUsing �$� the equations above are explicitly written as

L 'z�d & �� z�d � �z� � z�dR� � E �$��

C 'z�d � �� z�d &�

Rz�d � �$�$

The equations above give an implicit de�nition of our controller� To obtain anexplicit expression we use the degree of freedom that we have because there are threefree variables �� z�d� z�d and only two equations to be satis�ed�

B Direct output voltage regulation

At this stage� one is tempted to �x z�d � z�� in this case �z � would automaticallyensure the control objective� Thus� the control problem is the following� given adesired constant output voltage value z�d � z�� nd a bounded function z�d�t and asuitable duty ratio function �� such that �$�� and �$�$ be satis�ed� We proceed toeliminate the variable z��t from these equations as follows� From �$�$ we obtain

z�d�t �z��

R�� t�

�� The Boost converter ��

Substituting this expression into �$�� we obtain after some algebraic manipulations�an expression for the dynamical feedback duty ratio synthesizer of the form�

'� �R��

Lz��

�E � �� z�� &R�

�z� � z��

R��

�� $��

This controller stabilizes z� and z� towards their desired values z�d and z�d � z��respectively� Unfortunately� the controller �$�� is not feasible due to its lack of sta�bility� Indeed the �remaining�� or zero�dynamics associated with the above controllerresults in

'� �R��

Lz��E � �� z��

which coincides with the zero�dynamics already found in ��#� and shown to beunstable around its only physically meaningful equilibrium point�

C Indirect output voltage regulation

In the previous paragraph we have shown that a direct output voltage control scheme isnot feasible �with internal stability� In this paragraph we provide a feasible regulationalternative based on an indirect output capacitor voltage control� achievable throughthe regulation of the input current��

Suppose it is desired to regulate z� towards a constant value z�d � z�� In orderto �nd a suitable feedback controller for this task� one proceeds now to eliminate thevariable z�d from the set of equations �$�� $�$� Using �$�� z�d�t is given by

z�d�t �E & �z� � z��R�

�� t�$��

Substituting �$�� into �$�$� we obtain after some algebraic manipulations�

'� ��

C�E & �z� � z��R��

n�� z�� E & �z� � z��R�

R� R�C

L�E � �� z��

o�$��

The zero�dynamics associated with the controller �$�� is obtained by letting z� andz� coincide with their respective desired values thet is�

'� ��

RCE

�� Rz�� E

�

�$��

�We remark that some other possible alternatives include proposing a di�erent error energyfunction for the system� In this instance� we have just chosen to explore the implications of usingthe most natural energy function for the system�


The zero�dynamics �$�� coincides with the zero�dynamics derived in equation ��#$�which was shown to be locally stable around the only physically meaningful equilib�rium point� Therefore� the indirect controller �$�� is feasible�

We will now complete the proof that the equilibrium point �z�� z�� z�� z�� of the overall system �$�� $�� is locally asymptotically stable� To this end� weintroduce the following auxiliary variable

� ��

�E & �z� � z��R�

��

��

� z��

�$��

which is well de�ned for � in a neighborhood of the equilibrium point� � � �� It is easy to show that � satis�es the following linear di�erential equation�

'� � �

RC� &

z��R�

RCE�z� � z�� $��!

Recalling that �z� � z� � z�� exponentially fast� we conclude that � � as well�It follows that z�d � z�� locally which in turn implies that ��

The derivations above are summarized in the following claim�

Proposition �� Given a desired constant value� z�� E� for the output capacitorvoltage of a Boost converter� The dynamically generated duty ratio function �� with z�� given by �� locally asymptotically stabilizes the state trajectories of theaverage PWM model �� towards the desired equilibrium point �z�� z�� with �converging to a constant value given by � � �� E�z��

D Discussion

The passivitybased dynamical duty ratio synthesizer design is carried out under theassumption that the average PWM model ��!�� !� of the converter captures theessential behavior of the actual switch�regulated circuit� described by ��! � ��!��This assumption has shown to be only approximately valid due to the fact thatin practice� in�nite sampling frequency and corresponding in�nitely fast switchingsare impossible to achieve� Yet� for su�ciently high sampling frequencies� feedbackcontrollers designed on the basis of average models can indeed be used to regulate theactual switched converter� with rather satisfactory results �see ��$�� The scheme�shown in Fig� $�� is based on this philosophy� The underlying approach has beenextensively used for similar nonlinear dynamical feedback controllers and its validityhas been justi�ed both� from a theoretical viewpoint as well as through extensivecomputer simulation results �see for instance � ##� and the references therein�

�� The Buck�boost converter ��

Id

x1

x2

1

1

(actual)

DC-to-DCpower

converterPWM

Nonlinearpassivity-baseddynamic duty

ratio synthesizer

u

(calculated)

Limiter

μ

μ

Figure $�� PWM feedback control scheme for indirect PBC output voltage regulationof DC�to�DC power converters�

Two additional remarks are in order� regarding the use of a feedback PWM schemesuch as that of Fig� $��

� The averagebased duty ratio synthesizer produces a computed duty ratio func�tion� As such� it is entirely possible that these computed values exceed thephysical bounds of the required actual duty ratio function� which is necessarilylimited to the closed interval � � �� For this reason� a hard limiter must beused in conjunction with the derived dynamical feedback regulator� as shownin Fig� $�� As a consequence of this limitation� only local asymptotic stabilityof the closed loop system may actually be guaranteed� Large initial state devi�ations may induce destabilizing saturation e�ects which are not accounted forin the previous developments�

� The duty ratio synthesizer �$�� requires the online values of the averagePWM circuit states z� and z�� These average states can be approximatelyobtained by low pass ltering the actual circuit states� x� and x�� However�note that in the presented scheme depicted in Fig� $�� the actual circuit statesx�� x�� are being used for feedback� rather than their averaged or �ltered versionsz� and z�� In this respect� it is worth pointing out again� that for large samplingfrequencies the di�erences between using one or the other set of states is entirelynegligible� due to the underlying low pass �ltering e�ects of the system itself�

� The Buck boost converter

Following exactly the same procedure as in the previous case one concludes that� forthe Buckboost converter� a direct regulation policy of the output voltage is unfea�

� Ch� $� Passivity�based control of DC�to�DC power converters

sible due to nonminimum phase phenomena� We� thus� summarize in a propositionthe dynamical feedback regulation scheme achieving indirect output capacitor voltageregulation� towards a given desired equilibrium value z�� through input current sta�bilization towards a desired constant value z�� computable in terms of z�� as givenby equation ��

Proposition �� Given a desired constant value z�� for the output capacitor voltageof a Buck�boost converter� the dynamically generated duty ratio function given by

'� ��

C�E & �z� � z��R��

n�� z�� E & �z� � z��R�

R� R�C

L��E & z��

o�$��#

locally asymptotically stabilizes the state trajectories of the average PWM model�� towards the desired equilibrium point �z�� z�� with � converging to aconstant value given by� � � �� p

E�Rz�� with z�� obtained from z�� fromequation ��

Note that the zero�dynamics associated with the controller �$��# is given by

'� ��

RCE

�� Rz�� E

�which has three equilibrium points given by�

� � � � � � � &E

Rz��

s�E

Rz��

��

&E

Rz��

Two of the equilibrium points �� and the one corresponding to the plus sign ofthe square root are unstable while the remaining one� which is the only physicallysigni�cant� is locally asymptotically stable�

�� Simulation results

Simulations were performed for the closed loop behaviour of a Boost circuit regulatedby means of the passivity�based indirect PWM controller �$�� In order to testthe e�ectiveness and robustness of the proposed feedback controller with respectto unmodeled parasitic resistances and unmodeled realistic switching devices� thefollowing stochastically perturbed version of a Boost converter circuit� taken from��#�� was used for the simulations�

'x� � � �

Lr�u� �� u

R

L�R & rCx� &

E & �

L� �� u

VFL

'x� � �� uR

�R & rCCx� � �

�R & rCCx�

�� Simulation results �

where r�u � rL&urDS&��u�RF & rC jjR with rL being the resistance associatedwith the inductor� rDS is the resistance associated with the �ON� state of the transis�tor used in the realization of the switching element constituted by a transistor�diodearrangement� RF is the forward resistance of the diode� rC is the resistance associatedwith the output capacitor while rC jjR denotes the resistance of a parallel arrangementof rC and R� The voltage VF represents a small constant voltage drop associated withthe conducting phase of the diode� The signal �� added to the external source voltage�represents an external stochastic perturbation input a�ecting the system behaviour�

Figure $� � Closed loop performance of PBC in a stochastically perturbed Boostconverter model including parasitics�

Note that the perturbation input � is of the �unmatched� type� i�e�� it enters int

the system equations through an input channel vector �eld� given by ��L �� which

is not in the range space of the control input channel� given by the vector �eld��

rDS�RF�rC jjRL

x� �R

L�R�rC�x� �

VFL

R�R�rC�C

x�

��


The peak�to�peak magnitude of the noise was chosen to be� approximately� ) ofthe value of E�

The circuit parameter values were taken to be the following �typical� values�

C � �F � R � ! * � L � mH � E � �� V � VF � �$ V

rL � � � * � rC � � * � rDS � �� * � RF � � � *

The sampling frequency for the PWM policy was set to � KHz� The duty ratiofunction is obtained from a sampling process carried out on the output ��t� of thesmooth dynamical duty ratio synthesizer �$�� To avoid the use of low pass �lters�instead of using the averaged state variables� z�� z�� for feedback on the duty ratiosynthesizer� we used as it is customarily done� the actual PWM controlled states x��x�� on the controller expressions� The desired ideal average input inductor currentwas set to be z�� !�� Amp�� with a steady state duty ratio of �� Thiscorresponds to an ideal average output voltage� z� � z�� !$�� V� Fig� $� shows theclosed loop state trajectories as well as the duty ratio function and a realization ofthe computer generated stochastic perturbation signal ��

As it can be seen from Fig� $� � the proposed dynamical feedback controller �$�� achieves the desired indirect stabilization of the output voltage for the non�idealstochastically perturbed model around the desired equilibrium value� The averagesteady state errors� with respect to the desired equilibrium values� approximatelyrange from �� ) in the average inductor current variable to a �� ) in the averagecapacitor voltage variable� The ideal duty ratio is achieved within less that �� )error� The controller performance also exhibits a high degree of robustness withrespect to the external stochastic perturbation inputs�

Figure $�!� Robustness test of controller performance to sudden load variations�

� Passivity based sliding mode stabilization ��

Unknown load resistance variations generally a�ect the behaviour of the closedloop performance of the controlled converter� Simulations� shown in Fig� $�!� wereperformed to depict the sensitivity of the regulated input current� the output capacitorvoltage and the duty ratio with respect to abrupt� but temporary unmodeled changesin the load resistance R� An unmodeled sudden change of the load resistance� was setto an � ) of its nominal value� As seen from the �gures� the controller manages torapidly restore the desired steady state conditions� right after the load perturbationdisappears� As expected� the state variable most a�ected by such a perturbation isthe output voltage� The duty ratio function� on the contrary� is seen to be littleuna�ected by such sudden load changes�

In this section we have derived some physically�motivated dynamical feedbackduty ratio synthesizers for the indirect average output voltage stabilization of DCtoDC power converters of the Boost and Buckboost types� In accordance withthe passivity�based methodology undertaken in this book� the dynamical feedbackcontrollers are based on the modi�cation of the total energy function of the averageconverter circuit model� In the following section we explore the possibility of com�bining classical techniques of sliding mode control together with the passivity�baseddesign developed here�

� Passivity based sliding mode stabilization

�� Introduction

The main objective of this section is to extend the PBC methodology to includeswitch�regulated models of DCtoDC power converters� without appealing to ap�proximate ��average� PWM models� For this kind of models� where the controlvariable is either or �� sliding mode control policies are naturally de�ned� Thereforewe propose a new mixed passivity�based sliding mode controller in short� SM&PBC�We only treat in detail the regulator design for the Boost case� however� our resultscan be extended to other DCtoDC power converters such as the Buck� Buck�boostand the �Cuk converter�

The advantages of the passivity�based sliding mode controller over the traditionalsliding feedback regulator are established on the basis of an evaluation of the totalerror energy� which is a weighted version of the integral square state stabilizationerror of the system� Depending on the choice of the controller�s initial state� thepassivity�based sliding mode controller is shown to possess a smoother and� in fact��nite weighted integral square state stabilization error� The corresponding measureof performance of the traditional sliding controller is shown to be in�nite� Further�it is shown that �in any given �nite interval the total energy absorbed by the circuitfrom the external source is strictly smaller for the proposed mixed controller than forthe classical sliding mode scheme�


In this section we revisit the traditional �current mode� sliding mode controllerfor the ideal version of the Boost converter circuit� Under this mode of operation� thecontroller adopts as a sliding surface a desired constant inductor current correspond�ing to a desired equilibrium value of the capacitor voltage� It should be remarked� asalready found in � #!�� that a sliding surface based on a desired constant equilibriumcapacitor voltage leads to an unstable closed loop dynamics for both converters� Thisphenomenon is due to the underlying non�minimum phase zero�dynamics associatedwith the capacitor voltage output discussed in the previous section� The error energybehavior is analytically derived and shown to have an unbounded time integral� Thetime derivative of this error energy is shown to also exhibit an in�nitely large discon�tinuity� Such a discontinuity occurs� precisely� at the moment of reaching the slidingmode phase from a typical zero state �start�up� condition�

We present the derivation of the passivity based sliding mode controller for theBoost converter� As in the traditional sliding mode controller� our proposed regulatoralso results in an output feedback scheme but� this time� of dynamic rather than staticnature� However� as a counterbalance to the controller complexity� it is shown thatfor suitable set of controller initial conditions �a possibility denied in the traditionalstatic sliding mode controllers� the performance index is not only smooth but it alsoexhibits a nite value as long as the controller initial state does not coincide with theplant initial state� Some simulations are presented depicting the satisfactory closedloop behaviour of the derived PBC with respect to unmodeled parasitic resistancesand parasitic voltage sources� Such parasitic elements are typical of the switchingdevices usually constituted by a suitable transistor�diode arrangement�

�� Sliding mode control of the Boost converter

In the previous chapter we have derived the model of the switch�regulated Boostconverter circuit as ��#� ��#� which we repeat here for ease of reference given by

'x� � �� u�

Lx� &

E

L�$��

'x� � �� u�

Cx� � �

RCx� �$��

Notice that� in contrast to the average model� �$�� here x� and x� represent� respec�tively� the actual input inductor current and the output capacitor voltage variables�and u denotes the switch position function� which takes values in the discrete setf � �g�

A Direct sliding mode control

In this section we show the unfeasibility� due to instability� of a �voltage mode�sliding mode control scheme �see also � #!�� This is the switchingmode analog of

�� Sliding mode control of the Boost converter ��

the unfeasability analysis of Section � which applied to the PWMcontrolled system�

In the sequel� we denote by x� and x� the state variables of the system under idealsliding mode conditions� In other words� x� and x� represent the �average� values ofthe state variables under sliding mode operation� The �equivalent control�� denotedby ueq� represents an ideal� i�e�� a virtual feedback control action that smoothly keepsthe controlled state trajectories of the system evolving on the sliding surface� providedmotions are started� precisely� at the sliding surface itself �see � $ � for de�nitions�The equivalent control� for the case of switch�regulated systems� is not synthetizablein practise due to the discrete�character of the control input� However� its use on thesystem equations provides a convenient way of analysis for the �average� closed�loopsliding motions behavior�

Proposition �� Consider the switching line s � x� � Vd� where Vd � is a desiredconstant capacitor voltage value� The switching policy� given by

u � �� & sgn�s � � �� & sgn�x� � Vd � �$��$

locally creates an unstable sliding regime on the line s � with ideal sliding dynamicscharacterized by

x� � Vd � 'x� �E

L

�� V �

d

RE x�

�� ueq � �� x�

x�

�

Proof� Evidently� the closed loop system �$��$��$ satis�es the well knownsliding mode condition s 's � �see � $ � in the vicinity of the sliding surface s � � provided x� � Vd�R� hence the local character of the sliding mode� The idealsliding dynamics is easily seen to have a unique but unstable equilibrium point atx� � V �

d ��RE� �

B Asymptotic stability of indirect sliding mode control

As discussed in Section � in order to avoid the non�minimum phase behavior we mustproceed to indirectly regulate the capacitor voltage x��

Proposition �� Consider the switching line s � x� � V �d �RE� where Vd � is a

desired constant capacitor voltage value� The switching policy� given by

u � �� sgn�s � � �� sgn�x� � V �

d �RE

�$��

locally creates a stable sliding regime on the line s � with ideal sliding dynamicscharacterized by

x� �V �d

RE� 'x� � � �

RC

�x� � V �

d

x�

�� ueq � �� E

x�� $��


Moreover� the ideal sliding dynamics behaviour of the capacitor voltage variable�described by �� can be explicitly computed as

x��t �hV �d &

�x��th� V �

d

e�

�RC

�t�th�i��

�$�

where th stands for the reaching instant of the sliding line s � and x��th is thecapacitor voltage at time th� �

Proof� Note that the switching policy �$�� creates a sliding regime on s � provided x� � E� which is a well known �amplifying� property of the Boost converter�The ideal sliding dynamics has stable equilibrium points at x� � Vd for all initialvalues of x� which are strictly positive� and at x� � �Vd for all strictly negativeinitial conditions of x�� Note� however� that this second equilibrium point is notachievable when the constant value of the voltage source E is strictly positive� Toshow that the expression �$� is a solution of the di�erential equation in �$�� onesimply di�erentiates �$� � �

C Simulation results

We show below in simulations� the �current�mode� sliding feedback controlled statetrajectories of a typical Boost converter� The circuit parameter values were takenas in the simulation of Subsection �! without the parasitics� Fig� $�# depicts thestate variables evolution of a typical zero initial conditions �start up� response of acurrentmode controlled Boost converter� using the controller of Proposition $�#�

Figure $�#� Typical sliding �current�mode� controlled state responses for the Boostconverter�


Note that the capacitor voltage x� remains at its initial value of zero� while theswitch position u is maintained at the value u � �� The inductor current grows�during this �reaching� phase as a linear function of time given by

x��t �E

Lt�

When the sliding line s � x� � V �d �RE � is reached� at time th � �L�R�Vd�E��

by the controlled inductor current� the voltage capacitor x� rapidly grows from zerotowards its equilibrium value Vd� ideally governed by

x��t � Vd

q�� e�

�RC

�t�th��

Note that x� exhibits an in�nite positive time derivative at the sliding mode reachingtime t � th�

D Performance

The stored error energy of the controlled system is de�ned as

H�t ��

�L

�x��t� V �

d

RE

��

& C �x��t� Vd�

�� $� �

Proposition �� Suppose the Boost converter circuit is initially at the zero state�Then if the switching policy �� is applied to the converter� then during the reachingphase of the sliding mode � t � th the stored energy H�t decreases quadraticallywith respect to time� After the sliding mode is achieved �t � th � then the storedenergy of the circuit further decreases to zero� as the ideal sliding motions of thecapacitor voltage asymptotically approach the equilibrium value� �

Proof� The control dependent time derivative of the stored stabilization errorenergy H�t is given by

dHdt

� �� uVd

�x� � Vdx�

ER

�&

�x� � V �

d

RE

�E � �x� � Vd

x�R

�$�

During the sliding mode reaching phase� with x��t � � t � th and u � �� the aboveexpression reduces to

dHdt

�

�x� � V �

d

RE

�E

and the stored error energy H�t decreases from the value

H� � ��LV �d ��R

�E� & CV �d �


towards the value

H�th � ��CV �d

At the reaching instant� th� the time derivative of H�t is also zero�

During the reaching stage� the stored error energy� is a quadratically decreasingfunction of time� given by

H�t � ��fL��E�Lt� V �d �RE�� & CV �

d g

Its time derivative� at time t � � is given by 'H� � �V �d �R�

Right after the sliding mode is reached the time derivative of the stored stabiliza�tion error energy is obtained in an average sense� by substituting in �$� the switchposition function u by the equivalent control expression found in �$�� After somealgebraic manipulations� we obtain that the time derivative of H�t for all t � th� isgiven by

dHdt

� � �

R

�� &

Vdx�

��x� � Vd

� � �$� !

Since at time t � th� the capacitor voltage� x��th � x��th� is zero� according to�$� !� the stored error energy instantaneously exhibits an in�nitely large negativetime derivative� The total stored error energy thus asymptotically decreases to zeroas the average voltage approaches its desired equilibrium value Vd� �

A measure of the performance of the sliding mode controlled system� describedabove� is obtained by using the integral of the stored stabilization error energy� Thisquantity is given by

IB �

Z

�

H�d �

Z

�

�

�L

�x�� V �

d

RE

��

& C �x�� Vd�

�d� �$� #

Such a performance criterion can also be regarded as a weighted integral square statestabilization error for the state vector� We simply address such an index as the�WISSSE� index�

Proposition �� The WISSSE index� computed along the sliding mode controlledtrajectories of the Boost converter� is unbounded for all initial conditions of theconverter� �

Proof� We compute the WISSSE index by �rst evaluating the integral during thesliding mode reaching phase in the time interval t � th� For the evaluation ofthe index during the sliding mode phase� we consider the ideal sliding dynamics forthe variable x�� For the sake of simplicity� to perform the calculation we assume that


the initial states of the converter are set to zero� Nevertheless� the result is valid forarbitrary initial conditions� including starting the converter at the equilibrium values�

IB�t �

Z th

�

�

��E

L � V �

d

RE

��

& CV �d

�d &

Z t

th

�

CV �

d

�q�� e�

�RC

��th� � �

��

d

�LV �

d

RE

�C &

�LV �d

R�E�

��

#

��#t� th

RC� e��

t�thRC � #

q�� e��

t�thRC

� ln

!"� & e�

��t�th

RC &

q�� e�

��t�th

RC

e��t�th

RC

#A

��RC�V �

d �$� �

It is easy to verify that

limt�

IB�t ��

This limit operation requires using L�H(opital�s rule in the last term in �$� ��

Figure $�� Weigthed integral square stabilization error behavior for the Boost con�verter�

It is easy to see that the above result is independent of the initial conditions� Infact� the transient towards the sliding regime contributes with a �nite quantity to the

� Ch� $� Passivity�based control of DC�to�DC power converters

WISSSE index behaviour� The sliding regime phase� reached always in �nite time�contributes with the in�nite portion of the WISSSE index value�

Fig� $�� depicts the behaviour of the total stored stabilization error energy� itstime derivative� and its time integral for the sliding mode controlled state vector ofthe converter� started at zero initial conditions�

The unbounded growth of the WISSSE index is hardly remarkable for such a smalltime interval� As evidenced from �$� �� this is due to the fact that the dependenceon t� of the above integral� is scaled by a constant factor which is of the order of CV �

d �For the parameters used in our simulated Boost converter� previously de�ned� thisvalue is of the order of � ��

We compute� also for the purpose of comparison� the integral square error asso�ciated with the di�erence between the applied control input u�t and its constantsteady state value� denoted by ueq�� E

Vd� Since during the reaching phase the

input u is at a �xed value� either u � � or u � � the integral square error of thecontrol input� during this transient� is �nite� For this reason we concentrate on thecomputation of the performance index during the sliding phase� We again assume�just for simplicity� that the converter initial states are set at zero�

W �t ��

Z

�

�u�� ueq�� d

�

Z th

�

� ��sgn�x�� V �

d

E� � &

E

Vd

��d &

Z t

th

�ueq�� &

E

Vd

��d

� E

Vd

Z t

th

�� Vd

x��

��d �

E

Vd

Z t

th

�� p

�� e��RC

��th�

��

d

�E�RC

V �d

�

RC�t� th &

�

ln

h�� e�

�RC

�t�th�i

& ln

�� & e�

�RC

�t�th� & �� e�

�RC

�t�th��

e��RC

�t�th�

��&'(

It is also easy to verify that the limit of this last term is unbounded as t � �therefore� we conclude that

limt�

W �t ��

�� Passivity�based sliding controller

In the following developments we introduce an auxiliary state vector� denoted by xd�The basic idea is to take xd as a �desired� vector trajectory for the converter statevector x� This auxiliary vector variable will be determined on the basis of the energy

�� Passivity�based sliding controller �

shaping plus damping injection considerations of the passivitybased approach� Thefeedback regulation of the auxiliary state xd� towards the desired constant equilibriumvalue of the state x� will in fact result in the speci�cation of a dynamical outputfeedback controller for the original converter state� We will be using a sliding modecontrol viewpoint for the regulation of xd towards the desired equilibrium value of x�

A Asymptotic stability

We rewrite the Boost converter equations �$�� $�� in matrix�vector form as

DB 'x� �� uJ x &RBx � EBwhere DB� J � RB� EB were de�ned in �$� �

Proceeding analogously to the case of the averaged model of Section we de�nethe desired closedloop storage function

Hd ��

�x� xd

�DB�x� xd�� x�DB�x

where xd satis�es the following controlled di�erential equation

DB 'xd � �� uJ xd &Rxd �R��x� xd � EB �$� �

Notice that� as done in Section � we have added damping via R� � �

We now concentrate our e�orts in regulating from u� by means of a sliding modecontrol policy� the auxiliary system �$� � towards the desired equilibrium state ofthe converter� This is summarized in the following proposition�

Proposition �� Consider the switching line s � x�d � V �d �RE� where Vd � is

a desired constant capacitor voltage value for the auxiliary variable x�d and for theconverters capacitor voltage x�� The switching policy� given by

u � �� sgn�s � � �� sgn�x�d � V �d �RE � �$� $

locally creates a sliding regime on the line s � � Moreover� if the sliding�modeswitching policy �� is applied to both the converter and the auxiliary system� theconverter state trajectory x�t converges towards the auxiliary state trajectory xd�tand� in turn� xd�t converges towards the desired equilibrium state� i�e��

�x�� x� � �x�d� x�d ��V �d

RE� Vd

��

The ideal sliding dynamics is then characterized by

x�d �V �d

RE�$� �

'x�d � � �

RC

�x�d �

�V �d

E

�E &R��x� � V �

d �RE

x�d

��$� �

ueq � �� E &R��x� � V �d �RE

x�d�$�!


where x� is the converters inductor current under sliding mode conditions� primarilyoccurring in the controller�s state space and induced� through the control input u onthe controlled system state space�

Furthermore� the ideal sliding dynamics for x��t for t � th� can be explicitlycomputed in terms of the inductor current error signal �x��t� V d��RE as�

x�d�t �

�e�

�RC

�t�th�x��d�th & V �

d

RC

Z t

th

e��RC

�t�� &

R�

E�x�� x�d��

�d�

��$�!�

�

Proof� That the state trajectory x�t converges towards xd�t follows from theprevious proposition� The sliding mode locally exists on s � provided x�d � E &R��x��V �

d �RE � � Note that since the same switching policy u is being applied toboth the converter and the auxiliary system� the state error x� � V �

d �RE is actuallydecreasing� in absolute value� during the sliding mode reaching phase� The positivevalue E will eventually overcome the term R��x� � V �

d �RE and the hitting of thesliding line s � is thus guaranteed in �nite time� The fact that x�d converges in �nitetime V �

d �RE implies that x� will also converge towards V �d �RE� Hence� the feasible

steady state equilibrium for x�d from �$� �� is given by x�d � Vd� By virtue of theprevious proposition� x� also converges towards Vd� After all transients have elapsed�the auxiliary system is just a copy of the converter dynamics� Since� under suchsteady state conditions it is always true for the Boost converter� that the �ampli�er�relation x� �� x�d � E is valid� the sliding mode existence on s � is inde�nitelyguaranteed� �

The validity of �$�!� as a solution of �$� � readily follows upon time di�erenti�ation of the proposed solution�

Fig� $�� represents the passivity�based sliding �current�mode� control scheme forthe Boost converter� The auxiliary system is regarded as a dynamical feedback con�troller synthesizing a suitable sliding line for the converter dynamics�

Fig� $�$ depicts some simulations of the closed loop state behaviour of the regulatedconverter system for several initial conditions of the dynamical controller� When theinitial states of the converter are chosen to be zero� in coincidence with the initialstates of the controlled converter� the closed loop response of the PBC coincideswith that of the traditional sliding mode controller� As the initial conditions for thecontroller are chosen �closer� to the desired equilibrium state of the controlled plant�the state responses of the plant become smoother with slightly larger settling timesbut with a much better behaved transient shape�

Remark �� We remark that the sliding mode created on the controller�s state space�on the basis of output feedback� is also induced at the same instant on the controlled

�� Passivity�based sliding controller ��

system state space by the actively switching control input u� also shared by the plant�An analytic expression for the three�dimensional integral manifold of the redundantfourth order closed loop ideal sliding dynamics� using as control the nonlinear equiv�alent control input de�ned in �$�! � is too di�cult to obtain� The form of theintersection of this integral manifold with the two�dimensional controlled converterstate space would help in explaining the obtained smoothness of the controlled systemstate responses�

sliding mode

controller

auxilary

system

DC-to-DC

(boost)

power converter

x1

x1d

Id

u

u

Figure $�� SM&PBC scheme for regulation of the Boost converter�

Figure $�$� Controller and plant state responses for di�erent controller initialconditions� �� x�d� � x�d� � � � � �� x�d� � x�d� � �� x�d� � x�d� � �!�� !$��


If initial conditions for the dynamical controller at t � � are chosen preciselyat the desired equilibrium values of the controlled plant state� �x�d� � x�d� ��V �

d �RE� Vd� a sliding regime is immediately created on s � x�d � V �d �RE � from

time t � on� The ideal sliding dynamics for x��t is then simply given by

x�d�t � Vd

�e�

�RC

�t�th� &

RC

Z t

�

e��RC

�t�� &

R�

E

�x�� V �

d

RE

��d�

��$�!

Note however that under these circumstances� at time t � the time derivative ofx�d�t is actually negative� as it can be inferred from �$� � or from �$�! itself�As a consequence� if the controller initial states are set at the equilibrium value�V �

d �RE� Vd� the value of x�d�t remain constant at the value of V �d �RE� but the

values of x�d�t actually decrease from Vd to later on recover and converge towards Vdagain� Moreover� from the nature of the single�mode exponential nature of x�d�t� noovershoot of Vd ever occurs� This fact allows us to conclude that during the transientof x�d�t� Vd x�d�t for all t�

B Performance

Consider again the stored stabilization error energy H�t de�ned in �$� �� Thecontrol�dependent time derivative of the stored stabilization error energy H�t isidentical to the expression already given in �$� � except for the fact that the controlinput u is now synthesized on the basis of the switching line s � x�d � V �

d �RE�de�ned on the controller�s state space� rather than on the basis of the switching lines � x� � V �

d �RE� de�ned on the systems state space�

dHdt

� �� uVd

�x� � x�Vd

ER

�&

�x� � V �

d

RE

�E � �x� � Vd

x�R� �$�!!

A second fundamental di�erence between time derivative �$�!! and that of �$� isthat now the instant th� beyond which it is valid to perform the substitution of u bythe �equivalent control�� ueq� depends quite heavily on the initial conditions of thedynamical feedback controller �x�d� � x�d� �

Suppose one sets the initial conditions of the dynamical PBC to be exactly thesame as those of the converter� Then� it can be easily seen that the term R��x��x�din the controller dynamics �$� � is identically zero during the reaching phase sincethe controller dynamics becomes the same as the plant dynamics and the same inputu is being applied to both the �plant� and the controller from identical initial states�As a consequence� as in the traditional sliding mode controller of Subsection !� � alarge discontinuity is to be expected in the time derivative of H at the hitting of thesliding line� occurring at time th�

Suppose now that the initial conditions of the controller are set� precisely� at thedesired equilibrium value of the converter state� i�e�� at �x�d� x�d � �V �

d �RE� Vd�


while the converter initial states are placed� as customarily� at zero� The passivity�based switching policy immediately creates a sliding regime on the controller�s �in�ductor current� variable x�d�t� As a consequence� the �equivalent control� �$�! can be substituted in the time derivative expression of H� right from the very initialinstant t � th � � The corresponding average value of the stored stabilization errorenergy satis�es then dH

dt ��H� and we can prove the following result�

Proposition �� The passivity�based sliding current�mode controller described inProposition �� yields a �nite WISSSE index �� provided the initial states of theconverter satisfy x�d� �� x�� and x�d� Vd� �

Figure $�� Comparison of total stored stabilization error energy for traditional andSM&PBC Boost converter�

Fig� $�� depicts the behaviour of the total stored error energy H�t for di�erentinitial states of the passivity�based dynamical sliding mode feedback controller� Asit is customary� zero initial conditions were taken for the �start�up� phase of theconverter in all four computational runs� Note that as the controller initial con�ditions were chosen closer to the desired constant equilibrium value of the plant�the performance index becomes smoother and similar to an exponential behaviour�The controller initial states in Fig� $��a through $��c are �x�d� � x�d� � � � ��x�d� � x�d� � �� x�d� � x�d� � �!�� !$�� respectively�

Another performance indicator of the closed loop system is the total energy ab�sorbed from the voltage source by the system at time� � t � � that is�

R t

�Ex��sds�


Now� the inductor current in the standard sliding mode scheme converges to its �nalvalue in �nite time� while in the new controller the convergence is only exponentialwithout overshoot� Consequently� it is clear that the latter outperforms the slidingmode scheme also from the energy consumption viewpoint�

C Simulation results Robustness to unmodeled parasitics

We have performed some simulations in order to test the robustness and performanceof the proposed passivity�based sliding current�mode control scheme for a realisticmodel of a Boost converter� The derived controller� obtained from a non�perturbedversion of the converter model� was directly applied to a power converter �plant�containing typical modeling errors and stochastic perturbations� In Fig� �� we showthe rather realistic Boost converter model including parasitic resistances and para�sitic voltage sources commonly considered in modeling the transistor�diode switchingarrangement� This converter� originally proposed in ��#�� was also additionally per�turbed by including a �computer generated stochastic noise source� denoted by �� ofsigni�cant voltage amplitude �approximately ) in relation to the constant voltagesource value E�

Our controller was applied then to the following converter model

'x� � � �

L�rL & urDS & �� u �RF & rC jjRL� x� � �� u

�

L

�RL

RL & rC

�x�

&E & �

L� �� u

VFL

�$�!#

'x� � �� u�

C

�RL

RL & rC

�x� � �

�rC &RLCx� �$�!�

where as before x� and x� represent the Boost converter state variables� The parasiticresistance of the inductor is denoted by rL� The parameter rDS represents the �ON�resistance for the transistor integrating the transistor�diode switching arrangement�The parasitic resistance RF represents the �forward� resistance� or �ON� resistance�of the diode in the switch realization� VF is a parasitic voltage source associatedwith the diode operation� and rC is the capacitor�s parasitic resistance� The signal� represents the stochastic perturbation input associated with the constant voltagesource E�

The parameters of the perturbed version of the Boost converter were set to

rL � �� * � rDS � � ! * � RF � �� * � rC � � � * � R � ! * � VF � �$V�

In Fig� $�� we depict the controlled inductor current and the controlled capacitorvoltage responses of the perturbed Boost converter �$�!#� �$�!� regulated by meansof the passivity�based sliding �current mode� controller� derived on the basis of thenon�perturbed model�


Figure $�� Closed loop state response of the passivity�based controlled perturbed�Boost� converter�

The obtained equilibrium value for the output capacitor voltage was x�� !�� V which represents a steady state error� due to unmodeled perturbations andparasitics� of !�$ ) The proposed controller may be regarded to be reasonably ro�bust with respect to unmodeled parasitics elements� parasitic voltage sources andstochastic inputs�

Remark �� It is worth pointing out that even though traditional static slid�ing mode controllers are obviously simpler in nature than the proposed dynamicalpassivity�based sliding �current�mode� regulators� our results indicate that there isa de�nite advantage in the use of the more complex controller� This advantage isrelated to their superior performance� as measured by a weighted integral square sta�bilization error performance index and the energy consumption� While the traditionalsliding mode controller has an unbounded weighted integral square stabilization error�that of the PBC can be made to be �nite for a wide range of the controller initialconditions� as long as they do not exactly coincide with those of the regulated plant�Furthermore� this is achieved consuming less energy from the external voltage source�

Remark �� This study has only explored the implications of the passivity basedapproach in the sliding mode control of a particular class of DCtoDC power convert�ers� namely the Boost converter� Similar developments can� in principle� be workedout for more complex types of converters� such as the Buckboost converter and thepopular �Cuk converter�


Adaptive stabilization

A frequent assumption in the design of feedback regulators for DCtoDC powersupplies is that the converter loads and the parameters associated with the variouscircuit components are perfectly known� In practice� however� lack of precise knowl�edge about these parameters arises from inescapable measurement errors� unavoidableaging e�ects and imperfectly modeled loads� These facts motivate the adoption of anadaptive feedback approach for the design of regulation loops in DCtoDC powersupplies�

Adaptive control of DCtoDC power supplies has been treated from an approx�imate linearization viewpoint in � !�� Their approach relies on Lyapunov stabilityand passivity considerations for the linear feedback controller design� A full adaptivefeedback inputoutput linearization viewpoint for DCtoDC power supplies was pro�posed in � #�� An adaptive feedback design technique that suitably combines inputoutput linearization� through generalized observability canonical forms as developedby Fliess in �� and the backstepping design procedure� was recently presented in� #��

In this section we extend our previous developments by exploring the viability ofapplying passivitybased control for the adaptive stabilization of a class of averagemodels of PWM regulated DCtoDC power converters� We treat the two basic typesof switched power supplies� the Boost and Buckboost converters�

�� Controller design

The procedure to construct the adaptive versions mimics the one used for mechanicalsystems in Sections ��#� and �� Namely� we use a certainty equivalent versionof the control law� and taking into account the fact that the uncertain load enterslinearly in this signal� we generate the parameter error which appears as an additivedisturbance to the error dynamics� Since by de�nition� the error dynamics de�nes anOSP operator� the construction is completed with a gradient estimator� which in itsturn is passive�

A The Boost converter

Proposition �� Consider the averaged dynamics of the Boost converter� whereC � � L � � E � are known constants representing the capacitance� inductanceand external voltage respectively� and R � is the unknown load charge resistance�

Let �z�� z� � z�d and de�ne an adaptive nonlinear dynamic state feedback controller

�� Controller design ��

as�

'z�d � �(�

C

�z�d � z��

Ez�d

�E &R�

�z� � (�

z��E

�& L

z��Ez�d ��z�

��$�!�

� � ��

z�d

�E &R�

�z� � (�

z��E

�& L

z��Ez�d�z�

��$�!$

'(� � �z�d�z� �$�!�

where the dynamical controller initial condition is chosen so that� z�d� � and(�� The constant reference value for z�� denoted by z�� is a strictly positivequantity� The quantity (� denotes the estimate of �

R� The parameter R� is a designer

chosen constant with the only restriction of being strictly positive� Under theseconditions� it is always possible to chose the controller�s initial state z�d� and (�� such that the closed loop system �� has an equilibrium pointgiven by�

�z�� z�� z�d� (� �

��

R

z��E� z�� z��

�

R

��$�!�

which is asymptotically stable� �

Proof� It can be veri�ed� by direct substitution� that �$�!� represents an equilib�rium point for the closed loop system�

Let us de�ne the certainty equivalent version

z�d � (�z��E

�$�#

which is the same than that considered for the known parameter z�d in Section �Hence� z�d and z� coincide at the equilibrium point� Let� again� z � zd stand for theerror vector �z� In terms of the error signals� ��!�� !� are rewritten as�

DB'�z & �� JB�z &RBd�z � �

where

� � EB ��DB 'zd & �� JBzd &RBzd �

�R��z�

��$�#�

and RBd is a positive de�nite matrix given by�

RBd �

�R� ��R

�� R� �


Expression �$�#� is explicitly written as

�� L 'z�d � �� z�d &R��z� & E

�� C 'z�d & �� z�d � �

Rz�d�

Using �$�# and �$�!��$�!� we have �� and �� z�d� where �� (� � �R�

The resulting stabilization error system is then given by the following perturbeddynamics�

DB'�z & �� JB�z &RBd�z �

�

��z�d

�� $�#

Using as a Lyapunov function the storage function of the error system plus thestandard term to handle the parameter estimation error�

Hd�t ��

��z�DB�z & ��

�we verify that the relation

'Hd�t � ��z�RBd�z & ��h'�� & z�d �z� � z�d

iis satis�ed along the trajectories of �$�# � Using �$�!� and the fact that '��

'(� weobtain

'Hd�t � ��z�RBd�z ��jj�z�jjwhere � may be taken to be min

�R��

�R

�� We conclude that �z and �� are bounded and

that �z is square integrable� To actually show that �z � asymptotically� it must beveri�ed that �z is uniformly continuous� For this� it su�ces to show that '�z is bounded�From the perturbed error dynamics �$�# � and the established boundedness of �� and�z� it follows that '�z is bounded if� and only if� z�d is bounded� In order to prove thatz�d is bounded� note �rst that its associated zero�dynamics� given by

'z�d � �(�

C

�z�d � z��

z�d

�� $�#!

is asymptotically stable towards the equilibrium point located at z�d � z�� for allinitial conditions satisfying z�d� � � provided (� � � t� The dynamics �$�#! isalso asymptotically stable towards a second equilibrium point� located at z�d � �z��for all initial conditions satisfying z�d� � � provided (� � � t�

Take as a Lyapunov function candidate for the controller dynamics� V� �C��z�d�

z�� The time derivative of V� along the trajectories of �$�!��$�!� results in thefollowing expression�

'V� � �(� �z�d � z��z�d � z��

z�d

�� &

R�

E�z�

�& L

z��Ez�d�z�

� �� $�##

�� Controller design ��

Then� by virtue of the boundedness of �z�� z�� and (�� and the fact that initial conditionsfor such variables can be entirely chosen at will and� also� provided that (� � � t�it follows that given positive constants � and �� with�

� � �E

R��

E�

Lz��

rR�

E� �$�#�

such that initial conditions for the error vector components satisfy� j�z�j � � � j�z�j �� then� the time derivative of V� given by �$�##� is strictly negative outside the closedinterval �Zm� ZM � of the real line� containing in its interior the equilibrium point� z��for z�d� where

Zm � z��

� �� R�

E� &

L�z��#E�

��

�� Lz��

E��

�

ZM � z��

� �� &

R�

E� &

L�z��#E�

��

��&Lz�� E�

�

��

We conclude that �z is absolutely continuous and hence limt� �z�t � � Moreover�given that z� asymptotically converges to the same equilibrium point of z�d� given byz�� it follows that z� converges to its corresponding equilibrium value� z��RE� Sincez� and z�d asymptotically converge to the same equilibrium point it follows thatnecessarily� (� � ��R�

�

B The Buck�boost converter

The following proposition summarizes the properties of a passivitybased nonlinearadaptive dynamical controller for the Buckboost converter� The proof follows usingsimilar arguments to those used in the proof of the previous proposition�

Proposition �� Consider the averaged dynamics of the Buck�boost converter cir�cuit� where C � � L � � E � are known constants representing the capacitance�inductance and external voltage respectively� and R � is the unknown load chargeresistance�

De�ne an adaptive nonlinear dynamic state feedback controller as

'z�d � �(�

C

�z�d & z��

�z��E

& ��

�E & Lz��

�z��E

& ��z�d�z� &R�

�z� � z�� z��E & �(�

�E � z�d�t

��$�#�

��t �z�d�t & Lz��

�z��E

& ��z�d�z� � zd &R�

�z� � z�� z��E & �(�

�z�d�t� E

�$�#$


where �z�� z� � z�d and the dynamical controller initial condition is chosen so that�

z�d� � E and (�� The constant reference value for z�� denoted by �z�� is astrictly negative quantity� The quantity (� denotes the estimate of �

R� The parameter

R� is a designer chosen constant with the only restriction of being strictly positive�Under these conditions�� it is always possible to chose the controller�s initial statez�d� and (�� such that the closed loop system �� has anequilibrium point given by�

�z�� z�� z�d� (� �

�z��R

�z��E

& ��z��z��

R

�which is asymptotically stable� �

C The multivariable Boost converter

The following proposition summarizes the properties of a passivitybased nonlinearadaptive dynamical controller for the cascaded Boost converter that� indirectly� regu�lates the average output capacitor voltage towards a desired value z�� by regulating theaverage inductor currents towards a given set of desired steady state values uniquelydetermined from the arbitrarily chosen sequence of growing steady state capacitorvoltages�

Proposition �� Consider the averaged dynamics of the multivariable cascadedBoost converter circuit� where Ci� Li � � i � �� n� and E � are known con�stants representing the capacitances� inductances and voltage of the external source�respectively� The constant parameter� RL � � is the unknown load charge resistance�

Consider the adaptive nonlinear dynamic state feedback controller

��

z�d

�E &R�

�z� � (�

z��E

�& L�

z��Ez�nd�z�n

��$�#�

��

z�d

�z�d &R�

�z� � (�

z��z�

�& L�

z��z�

z�nd�z�n � z�nd

��

�n � ��

z�nd

�z��n��d &R�n��

�z�n�� (�

V �d

z�n��

�& Ln

V �d

z�n��z�nd�z�n

�

'z�d � �(�

C�

�� z�

Ez�d

�E &R�

�z� � (�

z��E

�& L�

z��Ez�nd�z�n

� �z��z�

&�

R�C��z�

'z�d � �(�

C�

�� z�

z�z�d

�z�d &R�

�z� � (�

z��z�

�& L�

z��z�

z�nd�z�n

� �z��z�

&�z�

R�C�

�� $�#�

�� Simulation results ��

'z�nd � �(�

Cn

�z�nd � �

z�nd

�z��n��d &R�n��

�z�n�� (�

V �d

z�n��

�

&LnV �d

z�n��z�nd�z�n

�V �d

z�n��

�'(� � �z�nd�z�n �$��

where the dynamical controller initial conditions are chosen so that� z��i�d� � � i �

�� n� and (�� Vd� the constant reference value for the output voltage z�nd�is chosen to be a strictly positive quantity� The set of constants� z�� z�� z�n�� z�n�satisfy the restriction

E � z� � z� � � � � � z�n�� z�n � z��

but they are� otherwise� completely arbitrary� The scalar variable (� denotes theestimate of �

RL� The parameters R�� R�� R�n�� are designer chosen constants

with the only restriction of being strictly positive� Under these conditions� it isalways possible to choose the initial state of the controller z��j�d j � �� n and(�� such that the closed loop system �� has an equilibrium point givenby�

�z�� z�� z�� z�� z�n�� z�n� z�d� z�d� � � � � z��n��d� z�nd� (� ��V �d

RLE� z��

V �d

RLz�� z��

z��RLz�n��

� z�� z�� z�� z�n��z��

RL

�which is asymptotically stable� �

Remark �� A passivitybased adaptive controller corresponding to the single stageBoost converter case can be immediately obtained as a particular case of Proposition�$��# For this� one simply sets n � � and z� � E in �$��


Simulations of the closed loop behaviour of the average Boost converter and thepassivity based indirect adaptive feedback controller were performed on the followingperturbed version of the Boost converter circuit�

'z� � ��

Lz� &

E & �

L

'z� � ��

Cz� � �

RCz�

where � represents an external stochastic perturbation input a�ecting the systembehaviour directly through the external voltage source value� Note that this pertur�bation input is of the �unmatched� type� i�e�� it enters the system equations through


an input channel vector �eld� given by ��L �� which is not in the range spaceof the control input channel� given by the vector �eld �z��L z��C�� The mag�nitude of the noise was chosen to be� approximately� � ) of the value of E� Thecircuit parameter values were taken as in the simulation of Subsection �!� and weset z�d � !�� Amp�� with a steady state duty ratio of � � �� This correspondsto a nominal average output voltage� z� � z�� !$�� V� Fig� $�� shows the closedloop state trajectories corresponding to the feasible adaptive duty ratio synthesizerderived for the Boost converter� This �gure also presents the trajectory of the dutyratio function� the trajectory of the parameter estimation values and a realization ofthe computer�generated stochastic perturbation signal �� addressed to as the �sourceperturbation noise��

Figure $�� Performance evaluation of the indirect adaptive controller in a perturbedaverage �Boost� converter�

The simulations show that the proposed controller achieves the desired indirectstabilization of the output voltage around the desired equilibrium� while exhibitinghigh robustness with respect to the external stochastic perturbation input�

� Experimental comparison of several nonlinear controllers ��

Experimental comparison of several nonlinear

controllers

DC�to�DC power converters are regulated in applications by means of simple linearleadlag compensators designed using the averaged linear approximation of the model�This induces an obvious hard constraint on the achievable performance for this classof controllers� It is then natural to ask if performance can be improved by means ofnonlinear control� For most of the nonlinear control algorithms� so far proposed in theliterature� we can establish stability properties by means of appropriate theoreticalanalysis� However� it is di�cult to assess the merits and drawbacks of a particularcontrol scheme� not to mention the potential performance improvement with respectto the linear designs� based solely on such theoretical analysis�

In this �nal section of the chapter we present some of the experimental results on aBoost converter reported in �$�� We compare �ve controllers� including the controllerbased on a linear approximation of the average PWM circuit� in terms of their easeof implementation and their closedloop performance� For all these algorithms� localasymptotic stability of the desired equilibrium is guaranteed� The motivation of thepresent study is not just to illustrate the validity of the corresponding theoreticalderivations� but to test the performance of the various schemes when confronted witha real physical application where unpredicted situations inevitably arise�

The performances of the various schemes are compared in the light of the followingbasic criteria� transient and steady state response to steps and sinusoidal references�attenuation of disturbances a�ecting the power supply and sensitivity to unknownloads� Particular emphasis is placed throughout on the �exibility provided by thetuning parameters in the shaping of the circuit responses� Even though this issue isnot always appreciated in theoretical studies� we have found it to be of primordialimportance in actual experimentation�

�� Feedback control laws

We now present �ve control laws whose performance was experimentally comparedin a laboratory facility� The �rst controller is constituted by a simple statefeedbackpoleplacement scheme based on a �rst order linear approximation of the averagePWM model� This control scheme may be regarded as the industry standard �see��$�� The other control schemes are nonlinear and have been studied in the previoussections� They rely� respectively� on linearization � ##�� passivation � #�� slidingmode control � $�� and a combination of sliding modes and passivity � �� In theabsence of external disturbances and parameter uncertainty� they all achieve �localasymptotic stabilization� that is� they guarantee that for suitable initial conditions�the capacitor voltage converges towards a desired prespeci�ed constant value withinternal stability� We refer the reader to the above references for further details on


the theoretical background of the schemes�

We consider the wellknown �Boost� circuit model in its switched�mode represen�tation �$�� and �$�� or the averaged PWMcontrolled model �$�� Two pertur�bations are considered� an unknown �timevarying disturbance �� which satis�esj�j � E is added to the voltage source� and uncertainty in the output resistance 4R�

The control laws that we consider are classi�ed into two groups� depending onwhether they directly generate the switching signal u� or they generate duty ratiofunction� �� that is fed to an auxiliary PWM circuit� This means also that for thecontrol design� they use the continuous averaged model or the switched� exact� model�respectively� For lack of a better terminology we use these quali�ers to classify theschemes below�

A Continuous control laws

A�� Linear Averaged Controller �LAC�

This controller is based on the linearization of the averaged model around anequilibrium point

'�z � A�z &B��

where A and B are given by

A �

� ��

L��C

� �RC

��

� � E

VdLEVdC

� �RC

�

B �

� z�L

� z�C

��

�VdL

� V �d

REC

�

and �� denotes the deviation with respect to the equilibrium value�

Some simple calculations show that the pair �A� B is controllable� Hence� thepoles of �A � BK can be located arbitrarily with a suitable choice of the statefeedback gains K � �k� k��

Taking the averaged error voltage as the circuit output we obtain the followingtransfer function

H�s �N��s

D�s�

K�s� Z�

�s� P��s� P��$��

As expected from the discussion of Section ��! and the commutativity of theoperations of linearization and zero�dynamics extraction �� the linearized system

�� Feedback control laws ��

transfer function has a zero in the right hand side of the complex plane� given byZ� �

RE�

LV �d

� The two stable poles are located in

P��

RC

��

�� #

E�R�C

V �d L

��

On the other hand� if we take as an output the averaged error current �z�� then weobtain a transfer function as in �$�� but with a left half plane zero located inZ � � �

RC�

A�� Feedback Linearizing Controller �FLC�

In � # � and � ! � we proposed the following nonlinear static state feedback con�troller that linearizes the input�output behaviour of the system� with linearizing out�put taken to be the circuit total energy�

� �z�

EL& �z�

RC

��

R�C� a�

R&a�C

�z�� &

�a�E &

a�L

z�

�z� &

E�

L� a�Hd

��$��

where a�� a� � are the design parameters� and

Hd��

V �d

�C &

L

R�E�V �d �

More precisely� it is shown in � # �� that the converters total energy satis�es thelinear equation

-H & a� 'H & a�H � a�Hd

Notice that Hd is the energy level required to ensure that as H � Hd we havez� � Vd� as desired� Since the dynamics is now linear� the convergence rate can be�xed arbitrarily with a suitable choice of the controller parameters a�� a��

The advantage of having a linear closedloop dynamics� expressed in some physi�cally meaningful variables� can hardly be overestimated� It allows us to easily predictthe e�ect of the tuning parameters and simplify the controller commissioning� How�ever� as it will be shown by our experiments� the existence of unmodeled nonlinearities�and in particular input saturation� limits the validity of these predictions�

A�� Passivity�Based Controller �PBC�

�The system will display an initial undershoot because it has an odd number of real zeros in theopen right hand plane�


In Section # the following nonlinear dynamic controller that preserves passivity ofthe closed loop circuit was proposed in slightly di�erent but equivalent terms�

� � � �

z�d

�E &R�

�z� � V �

d

RE

��where the controller dynamics is given by

'z�d � � �

RC

�z�d � V �

d

Ez�d

�E &R�

�z� � V �

d

RE

�� z�d� � �$��!

where R� � is a design parameter� As shown in Section # the desired storage

function Hd��

��z�D�z� where �z � z � �

V �d

RE� z�d�

�� satis�es

'Hd � ��z

�R� ��R

��z ��Hd� �

��

min�R�� R

max�L�C�

we see that R� injects the damping required for asymptotic stability� and that theconvergence rate of �z to zero is improved by increasing R�� From these observationsone might be tempted to try a high�gain design� but a more careful analysis andexperimentation reveals this not to be convenient� To see this� notice that �z� � does not imply that z� � Vd as desired� unless z�d � Vd as well� To study the

behaviour of the latter consider the signal �� z��d � V �

d � which satis�es

RC '� � � �&R�V�d �E�z�

This equation clearly shows two important limitations of the scheme� �rst� that thespeed of convergence is essentially determined by the natural time constant of theconverter� We remark that even if �z� converges to zero very fast� z�d �and consequentlyz� evolves according to this time constant� Second� increasing the damping willinduce a �peaking� in �� and consequently a slower convergence of z�d � Vd�

Furthermore� we have that the convergence rate of �z is bounded from below bythe undamped dynamics� i�e� � � �

RC� The sluggishness of this scheme has been

observed in our experiments� To overcome this drawback we have tried to add somedamping in the subsystem associated with z�d� that is� we modify �$��! to

'z�d � � �

RC

�z�d � V �

d

Ez�d

�E &R�

�z� � V �

d

RE

�&G��z� � z�d

��with z�d� � and R� � the new damping coe�cient� This gives a closedloop ofthe form

D '�z � �� J �z &Rd�z �

�� Feedback control laws ��

but with a new damping matrix �R� �

R&G�

�

Hence the convergence of �z to zero can be made arbitrarily fast� Unfortunately� thisdoes not change signi�cantly the dynamics of z�d� since � now satis�es

RC '� � � �& V �d

E�R��z� &G��z�

Another possibility is to use cross terms in the damping matrix Rd� All thesemodi�cations were tried experimentally but no signi�cant improvement was obtained�

B Switched control laws

B�� Sliding Mode Controller �SMC�

The indirect sliding mode controller of Proposition $�# was also tested� From theanalysis given there we see that when the sliding line s � x��V �

d �RE � is reachedat time th � �L�R�Vd�E�� the voltage capacitor x� grows from zero towards itsequilibrium value Vd� ideally governed by

x��t � Vd

h�� e�

�RC

�t�th�i��

Notice that� similarly to PBC� it is the openloop time constant that regulates thisdynamics� Furthermore� we will show in our experiments that this remarkably simpleapproach is� unfortunately� extremely sensitive to parameter uncertainty and noise�Finally� as usual with sliding mode strategies� the energy consumption is very high�

B�� Sliding mode plus PBC � SM�PBC�

We tried also the sliding mode plus PBC controller of Proposition $�$ was also tested�Consistent with the analysis carried out in that section� our experiments will showthat as the initial conditions for the controller are chosen �closer� to the desired equi�librium state of the controlled plant� the state responses of the plant become smootherwith slightly larger settling times but with a much better behaved transient shape�Unfortunately� it su�ers from the same drawback as PBC of providing no freedomto shape the response of the output voltage�


C Adaptive schemes

All the control strategies presented above are of the indirect type� where we controlthe capacitor voltage via regulation of the inductor current and invoke the one�to�onecorrespondence between their equilibria to achieve the output regulation objective�This strategy is clearly very sensitive to parameter uncertainty� in particular loadresistance changes� To overcome this drawback we have tested adaptive modi�cationsof the control laws� As done already in Section # some stability analysis can beperformed�

C�� Adaptive PBC

First� we tried the adaptive PBC of Proposition $�� in Section #�

C�� Adaptive SMC

To add adaptation to the basic SMC we propose to modify the switching line as

s � x� � (�V �d

E

with the parameter (� estimated by

'(� � ��Vd �x� � Vd � � �E�

V �d L

�$��#

To understand the main idea behind this scheme� note that in the switching line�the term ��R has been replaced by its estimation (�� Moreover� the adaptation law�$��# was motivated by the form of the corresponding adaptive version for the casePBC where z�d has been substituted by Vd� because there is no auxiliary dynamics�as before�

C�� Adaptive SM�PBC

An adaptive version for the SM&PBC can be similarly obtained considering thesame switching line as above� but using the estimator

'(� � ��x�d �x� � x�d

�� Experimental con�guration ��

The controller auxiliary dynamics is modi�ed accordingly to

'x�d � � �

L�� ux�d &

R�

L�x� � x�d &

E

L

'x�d ��

C�� ux�d �

(�

Cx�d

Again� the idea behind the above proposed scheme is to substitute ��R by itsestimated value (� whenever it occurs� and take the form of the adaptive law to estimatethis value as in the PBC case� but now using x�� x�d instead of z�� z�d� respectively�

�� Experimental con�guration

PCComputer

ConverterPower Boost

CurrentSensor

Control Signal

Mode Switch

Duty Ratio

0v

10v

PWM

in RPerturb.

SourceDC Power

D/A

DA

/D

DSpace CardNoisew(t)

Output Voltage

Inductor Current

ExperimentalBoost CircuitCard

Figure $�� Experimental setup�

The experimental card was assembled using low cost commercial electronic elementsplaced on a card designed in the laboratory� In Fig� $�� we show the experimentalset�up consisting of the Boost circuit card that receives control signals from a D%Aconverter of a dSpace card placed in a PC� The dSpace card acquires� using an A%Dconverter� the output voltage and inductor current signals previously conditionedfrom the Boost card� Two DC power supplies are necessary to operate the wholesystem� one to provide energy to the Boost system �we will refer to it as the powersupply in the rest of the section� and the other one to feed the electronic part of thecard�


Element Value Units

Capacitance � �FInductance �$ mHResistance � *

Power supply � V

Table $�� Parameters for the power converters experimental setup�

12345678

Vrefe2-e2+

ConVccC2E2E1

e1+e1-VsDtcCR

VssC1

+10 V+10 V

S1

Mes

S2

LEM100A

MBR1045

IRF540 C1

1000

uF63

V

CIRCUIT BOOST

D1

T1

3.3K

33K

R3 2K2

C3 47uF

R5 6K8

C4 4uF4R7 10K

P2 10K

P1 2K

C2 100uF

-15V -15V

+15V+15V

R2

3K3

R8

330 P3

10K

R6

15K

R1

10K

2N22

222N

2907

R9 100

L

16151413121110

9

TL

494

u(IN)

U(I

N)

X1(OUT) X2(OUT)

u (IN) r

R9 100

+10 V

w(t)

Mode Switch

IRF540

170mH

Figure $�� Boost circuit card

In Fig� $�� we show the main circuit which is constituted by a Boost circuit� aPWM circuit� and some signal conditioners� This design is very close to that of ��

The Boost circuit is basically composed by an inductor� a capacitor� a resistivecharge and a switch� the last one is implemented by interconnecting a FET transistorand a rapid diode in a suitable manner� all this elements fed by a DC power supply�

�� Experimental results ��

The values of its elements are shown below�

A current sensor together with a current�to�voltage converter are introduced toobtain a measurement of the inductor current x�� In this way� a voltage signal isgenerated which we can feed it into the dSpace card to be used in the control law�In the case of the output voltage x� we put a voltage divisor so that we can reducethe level of this signal in such a way that its �nal value is always in the �� Vrange� To compensate the nonlinear characteristic of the current sensor and someo�set introduced by the linear ampli�er circuits� we computed their characteristicfunctions and implemented their inverse�

In this card we have the choice between controlling the Boost circuit by means ofa PWM generated signal or by directly introducing a switching signal coming fromthe dSpace card� This selection is done depending on the position of the mode switch�If a PWM control is selected� then we should feed a continuous signal that representsthe duty ratio in the range �� V� corresponding to �� )� The PWM circuitwas designed using a commercial integrated circuit and it was programmed to have asampling rate of � KHz� On the other hand� if switch control�mode is selected� thenwe should provide a switching pulse signal with amplitudes of and � V�

Another interesting feature of the card is the possibility to connect or disconnecta second resistive charge placed in parallel to the output load� This is done by meansof a digital output signal� obtained from the dSpace card� which commands the gateof a MOSFET acting as a switch�

To study the e�ect of disturbances � in the power source� a driver circuit has beeninterconnected between the power supply and the Boost circuit� In this way we canintroduce disturbance signals from a signal generator or even from the PC�

Programs were written for a PC using C language� The programs containing thedescription of the controllers are translated and downloaded into the dSpace memoryas assembler programs by means of a suitable software� Time derivatives in some ofthe control laws are discretized using a rectangular approximation with a samplingperiod of � x � �� sec�

�� Experimental results

The �ve control laws described in the previous subsection have been implemented inthe above Boost circuit card� Their behaviour is compared with the following basiccriteria�

i� Transient and steady state response to steps and sinusoidal output voltagereferences�

ii� Attenuation of step and sinusoidal disturbances in the power supply�

iii� Response to pulse changes in the output resistance�


To gain some further insight into the behaviour of the converter� and motivate theneed for closedloop control� we also present the responses of the openloop system�

Unless indicated otherwise� in all the experiments we consider� as desired outputvoltage� the value Vd � V� This corresponds to a desired inductor current V �

d �RE � �# Amp� and to an equivalent duty ratio of � � �� E�Vd � ��

A Response to output voltage references

A�� Step references

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

t (sec)

x1 (

Am

p)

Inductor current

0 0.05 0.1 0.15 0.210

15

20

25

x2 (

Vol

t)

t (sec)

Capacitor voltage

Figure $��!� Open loop step response

In Fig� $��! we show the typical behaviour of the open loop system introducing astep in the duty ratio � of �� As we can see� the behaviour of the output voltagex� is quite good� it is fast and not too oscillatory� On the other hand� the current x�through the inductor has a large overshoot that exceeds the limit of available currentin the power source for a considerable time� This behaviour is not desirable becauseit could trigger the safety elements that disable the power source� We also observethat there exists a small undershoot due to the nonminimum phase character of theaverage system�

In Fig $��# we show a family of step responses of the system under LAC for di�er�ent pole placements� namely �P��P� � �� #$$�� #�� ! ��$�� etc� As expected� for faster poles we obtain faster responses� However�due to the presence of nonlinearities� specially the saturation of the inductor current�the time responses do not correspond to the proposed pole placement� In particular�


0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

t (sec)

x1 (

Am

p)

Inductor current

0 0.05 0.1 0.15 0.210

15

20

25

x2 (

Vol

t)

t (sec)

Capacitor voltage

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

mu

t (sec)

Duty ratio

Figure $��#� Step responses for LAC

the response of x� is quite slow� and we can not obtain oscillatory responses that arepredicted by the linear approximation theory� The former can be explained using rootlocus analysis which reveals that for large k� one pole approaches � while the otherapproaches �� so that the time response is dominated by the slowest pole� We alsoobserved that for relatively small gains a signi�cant steady state error appears� whilethe undershoot amplitude increases for faster responses�

In Fig� $�� we present typical responses of the system under FLC for di�erentvalues of a� and a�� namely �a�� a� � �! � � � �$�� ! � �� etc� Thesecorresponds to di�erent pole locations of the closedloop system described in thecoordinates �H� 'H�� Again� faster responses in x� are obtained with faster poles�which yields� in turn� higher peaks in x�� This limits the speed of convergence of x�due to controller saturation� Notice� however� that for comparable convergence ratesthere is a signi�cant reduction on the peak size with respect to LAC�

In Fig� $�� we show the responses of the PBC for di�erent values of R�� namely


0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1Inductor current

t (sec)

x1 (

Am

p)

0 0.05 0.1 0.15 0.210

15

20

25Capacitor voltage

t (sec)

x2 (

Vol

t)

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1Duty ratio

t (sec)

mu

Figure $�� Step responses for FLC

R� � �� !�� etc� From the plots we see that the design parameter R� a�ectsonly the behaviour of x�� while x� remains almost invariant� Actually� for smalldampings� the current x� varies slowly� with a large overshoot� but� as we increase thedamping x� converges faster �with no overshoot�� Finally� for large values of R� theresponse exhibits a fast peaking� As explained above� damping determines the speedof convergence of x� towards z�d� the oscillatory responses in Fig� $�� correspond tosmall damping� However� z�d remains essentially invariant with respect to R�� Weshould note that the peaking in z�d� predicted by the theory� was not observed� Areason for this is that it is actually �ltered by the dominant pole�

In Fig� $��$ we present the response of the system controlled with a SMC� Aswe can see� the sliding regime is reached almost instantaneously� thus x� reaches itsdesired value very fast� From the equations describing the sliding dynamics �$��we know that the response of x� is governed by the natural time constant �

RC� which

makes the response slow� Notice that there are no tuning parameters in this control


0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1Inductor current

t (sec)

x1 (

Am

p)

0 0.05 0.1 0.15 0.2

10

15

20

25Capacitor voltage

t (sec)

x2 (

Vol

t)

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1Duty ratio

t (sec)

mu

0 0.05 0.1 0.15 0.210

15

20

25Desired voltage

t (sec)

z2d

(Vol

t)

Figure $�� Step responses for PBC

law�

In Fig� $�� we present the responses of SM&PBC for di�erent values of the designparameter R�� Again� as in the PBC� the voltages x� and x�d remain almost invariantwith respect to R�� Only the currents x� and x�d are a�ected by this parameter andboth exhibit a peaking phenomenon during the transient part of the response� Thispeaking becomes higher as R� is taken to be larger�

As a conclusion for this part it is found that only LAC and FLC provide some�exibility to shape the step response� Two important advantages of FLC over LAC isthat it achieves the same convergence rates with smaller inductor currents� Hence� lessenergy is consumed� Furthermore� as reported in � �!�� the steadystate error becomessystematically smaller� As expected� the predictions of the theory are accurate onlyup to the point that the saturation levels are reached�


0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

t (sec)

x1 (

Am

p)

Inductor current

0 0.05 0.1 0.15 0.210

15

20

25

x2 (

Vol

t)

t (sec)

Capacitor voltage

Figure $��$� Step response for SMC

Strategy Cut�o� frequency range �Hz

LAC � � ��FLC � � � � ��PBC !��SMC !�

SM&PBC � �� !�

Table $� � Comparison of cut�o� frequency ranges�

A�� Sinusoidal references

Even though the controllers are designed for a speci�c regulation objective� a veryimportant characteristic to study in the controlled circuit is its ability to follow a timevarying desired output signal Vd�t� Obviously� this characteristic is closely relatedwith the circuits frequency response� Speci�cally� it is highly dependent upon thecircuit bandwidth� In Fig� $�� we show the closedloop frequency responses for the�ve control laws� Due to reasons related to the actual physical construction of thecircuit we can only follow reference signals of the form Vd � Vd�&AV dsin� �ft � E�The frequency responses were obtained assuming that we were placed at the equilib�rium point corresponding to the DC�component Vd� � V� As an approximation� weproceeded to take only the alternating part of the response� To generate the gain plotone divides� for each frequency value� the amplitude of the alternating output signalby the amplitude AV d � � V� In LAC� we observe that there appears a problem ofsteady state gain� that is� for higher values of k�� for instance k� � �� k� � � ��the steady state gain almost reaches � which is the expected value� but which corre�


0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

t (sec)

x1 (

Am

p)

Inductor current

0 0.05 0.1 0.15 0.210

15

20

25

x2 (

Vol

t)

t (sec)

Capacitor voltage

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

x1d

(Am

p)

t (sec)

Desired current

0 0.05 0.1 0.15 0.210

15

20

25

x2d

(Vol

t)

t (sec)

Desired voltage

Figure $�� Step responses for SM&PBC

sponds to a bandwith of � Hz� On the contrary� for lower values of k�� for instancek� � �� k� � � the steady state gain decreases to �#�� with a bandwith of Hz�

This problem is due to the fact that the system has been linearized around anequilibrium point� and now we are carrying the system far from this point putting inevidence its nonlinearities that are not considered in the control law�

In the case of FLC there is no problem of variations in the steady state gain andin this case the bandwidth can be enlarged choosing a� and a� such that the corre�sponding damping coe�cient is small and the natural frequency high� For example�choosing a� � � � a� � !� we have a bandwidth of � �� Hz and for a� � � � a� � � we get a bandwidth of Hz�

In PBC this frequency response is not a�ected when R� is changed� so the band�with is �xed to !�� Hz� This is also the case for SMC� For SM&PBC the bandwithvaries slightly depending on the value of R�� for an R� � � we can enlarge the band�


10−1 100 101 102 1030

0.5

1

1.5A

x2/A

Vd

f (Hz)

LAC

10−1 100 101 102 1030

0.5

1

1.5

Ax2

/AV

d

f (Hz)

FLC

10−1 100 101 102 1030

0.5

1

1.5

Ax2

/AV

d

f (Hz)

PBC

10−1 100 101 102 1030

0.5

1

1.5

f (Hz)A

x2/A

Vd

SMC

10−1 100 101 102 1030

0.5

1

1.5

Ax2

/AV

d

SM+PBC

f (Hz)

Figure $�� Frequency responses to periodic reference voltage� �Vd�t �� x��t

with to ! Hz� and for R� � � we reduce it to �� Hz� We have observed also someresonance phenomena in the circuit for low values of R�� this is manifested in theform of peaks in the frequency response which disappear for higher values of R�� Wedo not have at this point a physical or theoretical explanation of this phenomenon�

The results of these experiments are summarized in Table $� �

Besides the bandwidth we are� of course� also interested in the phase shift intro�duced in the loop� To assess this characteristic we show in Fig� $� some typicaltime responses of the output voltage for the various control strategies and a referencesignal of Hz� We have chosen the tuning that gives the largest bandwidth� We cansee that the smallest phase shift is achieved with FLC� which also provides the bestachievable bandwidth� We should underscore the poor performance of LAC in thisrespect� The remaining responses behave quite similar�


0 0.2 0.4 0.6 0.8 115

20

25LAC

t (sec)

x2 (

Vol

t)

0 0.2 0.4 0.6 0.8 115

20

25FLC

t (sec)

x2 (

Vol

t)

0 0.2 0.4 0.6 0.8 115

20

25

t (sec)

x2 (

Vol

t)

PBC

0 0.2 0.4 0.6 0.8 115

20

25

t (sec)x2

(V

olt)

SMC

0 0.2 0.4 0.6 0.8 115

20

25

t (sec)

x2 (

Vol

t)

SMPC

Figure $� � Time responses to a periodic reference signal Vd�t

B Disturbance attenuation

In this section we study the behaviour of the control laws in the face of a disturbancein the power supply� We consider two classes of disturbances steps and sinusoids�

B�� Step disturbance

In this experiment we propose to add a pulse disturbance w to the power supply�obtained from a signal generator of amplitude ! V and duration �� sec� In Fig�$� � we show the behaviour of the output voltage x� for each control law and di�erenttunings�


0 0.1 0.2 0.3 0.4 0.518

20

22

24

26

28

t (sec)

x2 (

Vol

t)LAC

0 0.1 0.2 0.3 0.4 0.518

20

22

24

26

28

t (sec)

x2 (

Vol

t)

FLC

0 0.1 0.2 0.3 0.4 0.518

20

22

24

26

28

t (sec)

x2 (

Vol

t)

PBC

0 0.1 0.2 0.3 0.4 0.518

20

22

24

26

28

t (sec)x2

(V

olt)

SMC

0 0.1 0.2 0.3 0.4 0.518

20

22

24

26

28

t (sec)

x2 (

Vol

t)

SMPC

Figure $� �� Response to a pulse disturbance w�t

We can see that FLC is quite sensitive to step disturbances� while SMC isalmost insensitive to it� To obtain some quantitative measure we evaluated the energyampli�cation of the circuit� that is we calculated the ratio

� �jj�x�jj�jjwjj� �

jj�x�jj�!p ��

where k � k��

R ��dt�

This number provides a lower bound to the L��gain of the operator Tw�x� � w �� x��See � !!� for some theoretical evaluation of bounds on this norm for FLC and PBC�

In LAC� � can be reduced using higher values of k� which implies that thedominant pole is slow� so for a � � �� we have chosen k� � �� and k� � � �which results in a dominant pole near �


Strategy Ranges of �

LAC � ��$�� FLC � �� $�$�PBC � �� SMC �#$!�

SM&PBC � ��# ��

Table $�!� Comparison of ampli�cation ratios�

Strategy Steady�state gain range Cut�o� frequency range �Hz

Open loop �#LAC �� FLC �!�� #� �#� �PBC ��!$�� $�� $� � � ��SMC ��

SM&PBC ��!� �� $�� $� � � �

Table $�#� Comparison of gain and cut�o� frequency ranges�

In FLC� � can be reduced proposing high values of a� and small values of a�� thiscorresponds to poles with high real and imaginary parts and damping coe�cients lessthan ��

In PBC� big values of R� reduce �� for instance� a value of R� � � correspondsto a � � �#$!�� This is consistent with the theoretical results reported in � !!��The same behaviour was observed for SM&PBC� For example� taking R� � � weobtain a � � ��# �� The range of the gains that we obtained in our experiments issummarized in Table $�!�

B�� Sinusoidal disturbances

In these experiments we obtain the frequency responses of the output voltageunder periodic perturbations introduced in the power supply� To this end� we add tothe voltage source E a perturbation w � Awsin� �ft where Aw � ! V and we scandi�erent values of f �

The magnitudes of the Bode plots �w �� x� for the openloop system and thoseof the controllers �for di�erent tuning parameters are given in Fig� $� � We see thatin all cases the closedloop behaves like a low pass �lter and the question is what isthe e�ect of the tuning gains on the steady�state gains and on the bandwidth and


10−1 100 101 102 1030

1

2

Ax2

/w

f (Hz)

LAC

10−1 100 101 102 1030

1

2

3

Ax2

/w

f (Hz)

FLC

10−1 100 101 102 1030

1

2

Ax2

/w

f (Hz)

PBC

10−1 100 101 102 1030

1

2

f (Hz)

Ax2

/w

SMC

10−1 100 101 102 1030

1

2

Ax2

/AV

dSM+PBC

f (Hz)

10−1 100 101 102 1030

1

2

Ax2

/w

f (Hz)

Open Loop

Figure $� � Open loop frequency response� Magnitude Bode plots of w �� x�

rollo� of the frequency responses�

Note that in these curves for each controller� variations in the parameters implyvariations in both� steady state gain and cut�o� frequency� Of course the best curveis that with the minimum steady state gain and cut�o� frequency�

In open loop the cuto� frequency is of �#Hz� hence extremely bad disturbanceattenuation� For the LAC controller� the steady state gain can be reduced if thegain k� is decreased� this corresponds also to a very small increasing in the cut�o�frequency� For high values of k� and k� �k� � �� k� � � � the steady state gaincould arrive up to �� with � Hz of cut�o� frequency and for smaller values�k� � �� k� � this gain is �� with � �� Hz of cut�o� frequency�

In the case of FLC controller� depending on the pole locations� we can havesteady state gains that go from �� for fast poles �a� � � � a� � !� until !�� for a


dominant slow pole �a� � � � a� � �� These pole locations correspond to cut�o�frequencies of �# Hz and �# Hz respectively� Thus appearing a compromise betweensteady�state gain and cut�o� frequency depending on the pole locations�

For PBC we have smaller steady�state gain and cut�o� frequency for bigger valuesof R�� For instance� a steady�state gain of ��!$$� with a cut�o� frequency of $ Hz�correspond to an R� � � � For an R� � �� the corresponding values are ��$�� Hz�

In SMC� since there is no design parameter� the steady�state gain and the cut�o�frequency are unique and they take respectively the values �� and � Hz � Whichmeans that disturbance rejection in this controller is quite good� For SM&PBC� asin the case of PBC� we have larger steady�state and cut�o� frequencies for biggervalues of R�� For instance� with an R� � � we have an steady�state gain of ��!� � anda cut�o� frequency of $ Hz� and for R� � � we have ��$� and � � Hz� respectively�

In Table $�# we show the steady�state gain and cut�o� frequency ranges we ob�tained experimentally for each control strategy�

It�s important to remark that this cuto� frequencies are relatively small comparedwith the possible perturbations caused by the natural line frequency noise �� %� Hz�so the rejection of this kind of natural perturbations is assured�

C Robustness to load uncertainty

In this experiment we introduce a load change that reduces the e�ective resistancefrom its nominal value of R � � * to R & 4R � � * during the interval � �� sec� To implement this e�ect a digital signal generated in a dSpace card is sent tothe gate of a MOSFET transistor to turn it on or o�� This transistor is actuatingas a switch that connects or disconnects a resistance of � * placed in parallel withthe nominal load� which is also of � *�

The openloop response is shown in Fig� $� !� As we can see� there appears asteady state error in the voltage output x�� that even if it�s small� there is no way toreduce it�

As discussed at the beginning of this section the fact that all control strategiesare indirect makes them extremely sensitive to this kind of disturbance� introducingin particular a large steadystate error� To remove this error we tried an heuristicapproach of adding an integral loop around the output voltage error �for continuouscontrol laws as well as the adaptive versions� Notice that we do not dispose of anadaptive scheme for FLC� however for simple step changes in the load the integralaction corrected the steadystate error� It is an interesting open question how toprovide FLC with adaptation capabilities to track timevarying parameters� as donefor PBC�


0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

t (sec)

x1 (

Am

p)

Inductor current

0 0.5 1 1.510

15

20

25

x2 (

Vol

t)

t (sec)

Capacitor voltage

Figure $� !� Openloop response to a step change in the output resistance

C�� Non adaptive versions

We show in Fig� $� # the behaviour of the output voltage signal x� of the systembeen controlled for each strategy when there appears a pulse change in the outputresistance� which is clearly inadmissible�

C�� Adaptive versions

The adaptive version of PBC with � � � and R� � � is shown in Fig� $� ��Observe that the parameter estimate converges very close to the true values� i�e� �� and � *�� remember that the algorithm estimates ��R� but that this smalldiscrepancy induces an steady state error both in the inductor current �which shouldbe x� � �� for the new load resistance and the output voltage� The error howevervanishes when we come back to the nominal resistance� where now the estimateexactly converges to the true value� Given the proof of asymptotic stability of thedesired equilibrium� the existence of this steadystate error for higher currents isparticularly distressing� The only explanation we have is that since more current ispassing through the electronic elements their parasitic losses become more signi�cant�It may also be that the additional computations demanded by the adaptation lawinduce numerical errors� This critical issue of numerical sensitivity will be discussedwithin the context of induction motor control in Chapter ��

As usual in adaptive control� even though we started the experiment with theright value of the load resistance� the estimate moves initially away from it� This� insome way� speedsup the step response of the output voltage�

In SM&PBC we took again � � � and observed a behaviour very similar to PBC�


0 0.5 1 1.50

5

10

15

20

25LAC

t (sec)

x2 (

Vol

t)

0 0.5 1 1.50

5

10

15

20

25FLC

t (sec)

x2 (

Vol

t)

0 0.5 1 1.50

5

10

15

20

25PBC

t (sec)

x2 (

Vol

t)

0 0.5 1 1.50

5

10

15

20

25SMC

t (sec)x2

(V

olt)

0 0.5 1 1.50

5

10

15

20

25SMPC

t (sec)

x2 (

Vol

t)

Figure $� #� Output voltage behaviour for a disturbance in the output resistance

with the addition of the high frequency oscillations in the current mentioned above�

For adaptive SMC we take again � � � and observe the same phenomenon of lackof parameter convergence when the load is reduced� In this case� however� there is nosteady state error in the output voltage� this because as we see from the equations�adaptation introduces an integral term in the output voltage error� The inductorcurrent exhibits very high frequency components due to the low frequency used togenerate the switching signal u�

C�� Adding an integral term

In Figs� $� � $� $ we present for the three laws employing a PWM� LAC� FLC andPBC� the responses to a pulse disturbance in the output resistance� In all cases thesteady state error vanishes in a relatively short time with small overshoot� There is


0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

Inductor current

t (sec)

x1 (

Am

p)

0 0.5 1 1.510

15

20

25Capacitor voltage

t (sec)

x2 (

Vol

t)

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1Duty ratio

t (sec)

mu

0 0.5 1 1.50

0.005

0.01

0.015

0.02

0.025Parameter estimated

t (sec)

tet (

1/O

hm)

Figure $� �� Response to a pulse disturbance in the output resistance for adaptivePBC

however some degradation in the quality of the �rst step response� a large overshoot�that could not be reduced via tuning without seriously degrading the transient andsteadystate performances�

It is worth noting the oscillatory behaviour of LAC and PBC for both load values�To dampen the oscillations the integral term had to be considerably reduced with theensuing increase in the settling time� While for PBC this destabilizing e�ect of theintegral action is not a serious problem� because we dispose of an adaptive version�for LAC it casts some doubts for its practical application�

�� Conclusions

The following conclusions of our experimental study are in order�

�� Conclusions ��

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1Inductor current

t (sec)

x1 (

Am

p)

0 0.5 1 1.510

15

20

25Capacitor voltage

t (sec)

x2 (

Vol

t)

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1Duty ratio

t (sec)

mu

Figure $� �� Response to a disturbance in the output resistance for LAC & integralterm

� Nonlinear designs provide a promising alternative to classical leadlag con�trollers� In particular� the linear scheme LAC performed very badly in trackingtimevarying references� and exhibited an undesirable oscillation when an inte�gral term was added to compensate for load uncertainty�

� FLC performed very well in output regulation and tracking but exhibited ahigher sensitivity to voltage disturbances than the other schemes� Incorporatingan integral action e�ectively compensated for a step change in load resistance�even though no theory is available to substantiate this� To handle other type ofload changes �e�g�� slowly timevarying the integral action will not be su�cientand some kind of adaptation should be incorporated into the controller�

� The main drawback of PBC� which is shared also by SMC and SM&PBC�is the inability to shape the output response� which evolves according to theopenloop dynamics� This� of course� stems from the fact that we cannot inject


0 0.5 1 1.50

0.2

0.4

0.6

0.8

1Inductor current

t (sec)

x1 (

Am

p)

0 0.5 1 1.510

15

20

25Capacitor voltage

t (sec)

x2 (

Vol

t)

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1Duty ratio

t (sec)

mu

0 0.5 1 1.510

15

20

25Desired voltage

t (sec)

z2d

(Vol

t)

Figure $� $� Response to a disturbance in the output resistance for PBC & integralterm

damping to the voltage subsystem without nonlinearity cancelation� On theother hand� PBC achieved a better disturbance attenuation� hence it may bea viable candidate for applications where rise time is not of prime concern� Weshould stress that� as shown in motor control� this is not a limitation intrinsicto passivitybased designs� rather it pertains to our ability to inject �pervasivedamping to the controlled variable�

� SMC and SM&PBC proved very robust to source disturbances but highly sen�sitive to parameter uncertainty� The latter could be alleviated incorporating anovel adaptation mechanism� The lack of �exibility of SMC is somehow allevi�ated in SM&PBC� at least to shape the disturbance attenuation characteristic�Unfortunately� both schemes su�er from the main drawback mentioned above�

� In contrast to the study carried out in Section ��# �see also��! � we were notinterested here in issues pertaining to numerical sensitivity and computational

�� Conclusions ��

complexity� this might prove important in a lowcost implementation�

� Average models of DCtoDC power converters have been known to be di�er�entially �at �see �� i�e�� all system variables are di�erential functions of thetotal energy of the system� which then quali�es as a linearizing output� As such�an interesting line of research can be proposed which exploits the di�erential�atness property of the system in connection with the possibilities of energyshaping and damping injection� i�e�� passivity� controller design techniques�

Part III

Electromechanical systems

#�

Chapter

Nested�loop passivity�based

control An illustrative example

We start with this chapter the third part of the book which is dedicated to PBCof electromechanical systems� Particular emphasis will be given to AC electricalmachines� to which Chapters �� are devoted� Chapter �� treats robots with ACdrives� hence connecting the material of the next three chapters with our previousdevelopments on mechanical systems of Chapter ��

We will show throughout this third part that for electromechanical systems thePBC approach can be applied in at least two di�erent ways� leading to di�erentcontrollers� In the �rst� more direct form� a PBC is designed for the whole electrome�chanical system using as storage function the total energy of the full system� Thisis the way we designed PBCs for mechanical and electrical system in the previouschapters� and we will refer to it as PBC with total energy shaping�

Another route stems from the application of the passive subsystems decomposi�tion of Section �� # to the electromechanical system� Namely� we show that �undersome reasonable assumptions we can decompose the system into its electrical andmechanical dynamics� where the latter can be treated as a �passive disturbance��We design then a PBC for the electrical subsystem using as storage function onlythe electrical part of the systems total energy� An outerloop controller �which asshown in Chapter �� can also be a PBC� but here is a simple poleplacement is thenadded to regulate the mechanical dynamics� The sodesigned controller will be callednested�loop PBC� There are at least three motivations for this approach� �rstly� us�ing this feedbackdecomposition leads to simpler controllers� which in general do notrequire observers� Secondly� typically there is a timescale separation between theelectrical and the mechanical dynamics� Finally� since the nestedloop con�gurationis the prevailing structure in practical applications� we can in some important casesestablish a clear connection between our PBC and current practice�

#!

�� Ch� �� Nested�loop passivity�based control� An illustrative example

� Introduction

To facilitate the presentation of the new concept of nestedloop PBC we considerin this chapter the simple problem of stabilization of a magnetically levitated ball�However� for the sake of continuity and to establish the connection with the previ�ous material� we present �rst the PBC with total energy shaping� and afterwards wedesign the new nestedloop PBC� Similarly to the case of electrical and mechanicalsystems for electromechanical systems the EL model derived in Section ��#� is in�strumental to guide us in the design of PBC� We use this model to derive the totalenergy shaping PBC� On the other hand� in this simple example it is possible to ex�hibit a passivity property of the system without invoking the Lagrangian formalism�Therefore� we use a standard statespace description to design the nestedloop PBC�Our motivation to adopt this notation is twofold� �rst by simplifying the notationwe believe the reader can concentrate better on the structural aspects of the design�Second� expressing the PBC in these coordinates permits to establish some interest�ing connections and similarities with classical feedback linearization and the widelypublicized backstepping designs� with which the reader is probably more familiar� Infact� an additional objective that we pursue in this chapter is to compare in a simpleexample the three controller design techniques�

Application of these� seemingly unrelated� methodologies will typically lead to thede�nition of di�erent control schemes� For some speci�c examples these di�erencesblur and some interesting connections and similarities between the controller designtechniques emerge� The careful study of such cases will improve our understandingof their common ground fostering crossfertilization� In this chapter we investigatethese questions� both analytically and via simulations� for the simple problem of sta�bilization of a magnetically levitated ball� Our motivation in choosing this particularexample stems� not only from the fact that due to its simplicity the connections be�tween the controllers are best revealed� but also that such an equipment is availablein many engineering schools� hence experimental work can be easily carried out tocomplement our studies�

Before closing this introduction a word on style is in order� Since the main ob�jectives in this chapter are the introduction of the basic idea of nestedloop PBC�and the clari�cation of its connections with backstepping and feedback linearization�we have adopted an informal format of presentation without theorems and proofs�For this reason we have also favored simpler� as opposed to more powerful or novel�solutions to the problem�

� Model and control problem ��

� Model and control problem

In Section ��#� we have derived the model of the magnetically levitated ball of Fig� �#� We repeat here for ease of reference the EL model

D�qm-q & C� 'qe� qm� 'qm 'q &R 'q & G �Mu ��

where the generalized coordinates are q � �qe� qm�� and we have de�ned

D�qm��

�c

��qm

m

�� C� 'qe� qm� 'qm �� c�

�� qm�

�'qm 'qe� 'qe

�

R ��

�Re

�� G �

�

�

�mg

�� M �

�

��

�

Notice that we have added a constant c � in the de�nition of the inductanceL�qm �

c��qm � As pointed out in Section ��#� this model can also be written in state

space form as

. �

��

'� � �Rec�� qm�& u

F � ��

m-qm � F �mg��

where� � L�qm 'qe �

c

�� qm'qe

is the �ux in the inductance L�qm� and we have de�ned the force of electrical originF � which will be extensively used later�

The control objective is to track a bounded reference signal qm��t �with knownand bounded �rst� second and third order derivatives with all signals bounded� Wewill assume in the �rst instance that the full state is available for measurement� Thenwe will show that for the nestedloop PBC we only require the measurement of qmand 'qm�

Remark �� There is a huge body of literature on control of magnetic bearing sys�tems with one or several electromagnets� We refer the readers to the special issue ofSeptember �� in the IEEE Trans� Control Systems Technology for an exhaustivelist of references� It has� in particular� been studied from the perspective of �atnessand feedback linearization in ��#� and with a backstepping approach in � !�� Anadaptive backstepping controller for the levitated ball studied here� albeit with a sim�pler model with linear electrical dynamics� has been reported in ��# �� The schemedoes not require the measurement of 'qm� it requires however qm and 'qe� In � �$�several PBCs are developed for this system� In particular� we present a PBC that


measures instead qm� 'qm� but obviates the need for 'qe� Here� we adapt for our simpli��ed example the schemes of ��#� for feedback linearization� of � !� for backstepping�and of � �$� for PBC� We refer the reader to �� $� for background material onfeedback linearization� and to ��# � for an exhaustive coverage of backstepping�

Remark �� Writing the model �� in terms of � yields the di�erentialalgebraicrepresentation

'� �Re

c'qe & u

� � L�qm 'qe

F ��

�L

�qm�qm 'q

�e

m-qm � F �mg

This model clearly reveals the passivity properties of the feedback decomposition ofProposition �� That is� the operators .e � �u�� 'qm� �� 'qe� F � and .m � F �� 'qm are�locally passive with storage functions �

�L�qm 'q

�e and

��m 'q�m&mg��qm� respectively�

� Passivity�based control with total energy�shaping

In this section we derive a PBC mimicking the approach taken for mechanical andelectrical systems� In particular we follow closely the developments carried out tosolve the tracking problem of robots with �exible joints in Section ��!� That is�motivated by the passivity of the operator u �� M� 'q with storage function the totalenergy

H ��

'q�D�qm 'q &mg�� qm

we propose to assign to the closedloop a storage function Hd ��s�D�qms� where�

as in Section ��!� we de�ne

s � '�q & 2�q� '�q�� 'q �

�'qed'qm�

�� 2

��

� 2m

�Notice that� since we are not interested in controlling the charges qe� but only thecurrents 'qe and the positions qm� we have chosen 2 with just one nonzero term� Also�remark that the �rst error signal '�qe is de�ned in terms of a signal 'qed to be de�nedbelow� This signal plays the same role as q�d for �exible joint robots in Section ��!�We now write the perturbed desired error dynamics as

D�qm 's& �C� 'qe� qm� 'qm &Rd� s � /

� Nested�loop passivity�based control ��

where Rd�� R &Kd� Kd � diagfkde� kdmg � is the damping injection matrix� and

the perturbation term is

/ �Mu&Kds� �D�qm-qr & �C� 'qe� qm� 'qm &R 'qr � G� ��!


'qr �

�'qed'qm�

�� 2�q

and 2 � � Setting / � we obtain the controller equations as

u �c

�� qm-qed &

�

c

�� qm�'qm 'qed &

�

c

�� qm�'qe� 'qm� � 2m�qm &Re 'qed � kde '�qe

m�-qm� � 2m'�qm�

�

c

�� qm�'qe 'qed � kdm� '�qm & 2m�qm &mg � �

To obtain an explicit realization of the PBC we solve the second equation as�

'qed � f�� 'qe� 'qm� qm� t� Then� we calculate u from the �rst equation� which requires-qed and consequently the measurement of the full state� Observe that� as in the case ofthe simpli�ed model of the �exible joint robots� the diagonal structure of D�qm allowsus to calculate -qe and -qm from knowledge of the state q and 'q� Another importantpoint is that we have to ensure that there are no algebraic loops in the calculationof u �like in the controller of �� This is �locally the case here because one canshow that u satis�es an equation of the form�

� &'qed'qe

�u � f�� 'qe� 'qm� qm� t

hence� it is de�ned in a neighborhood of the operating point�

The stability of this PBC can be established along the� by now standard� linesshowing that 'Hd � �s�Rds and chasing the signals �see e�g� Chapter �� In anycase� it is clear that this controller is extremely complicated for the purposes of thisapplication and furthermore it requires measurement of the full state� an assumptionthat we want to avoid in PBC�

� Nested�loop passivity�based control

In this section we present a controller structure which is motivated from a physicalpartition of the system and allows us to overcome the drawbacks of the PBC above�

�Notice that there is a division by �qe in this expression that introduces a singularity�


�� Control structure

Notice that the system . can be decomposed as a feedback interconnection of anelectrical subsystem .� � u �� and a mechanical subsystem .� � �F �mg �� qmas depicted in Fig� �� We will prove below that .� is �locally passive� hence ��and consequently F � see �� are �easy� to control� On the other hand� since.� is just a double integrator it is reasonable to concentrate our attention on theproblem of controlling F � in the understanding that if it is is suitably regulated thenposition qm will be easy to control with classical linear techniques� e�g�� PI control�An additional motivation for this approach stems from the fact that typically there isa natural timescale decomposition between the electrical and mechanical dynamics�For these reasons the socalled nestedloop control is the prevailing structure inpractical applications of electromechanical systems which will be adopted throughoutthis chapter�

u

qm -mg

F

Σ

Σ

1

2

λ

Figure �� Feedback decomposition�

The nestedloop control con�guration is shown in Fig� �� where Cil� Col are theinner and outerloop controllers� respectively� The innerloop controller is designedto achieve asymptotic tracking of the desired force Fd� The outerloop controllertakes care of the position tracking� and essentially generates a signal Fd such thatif F � Fd then position tracking� i�e�� qm � qm�� is ensured� In applications Col istypically a simple PI around position errors� �In some cases� for instance if .� is notLTI as in � �$�� an interlaced design of Cil and Col is needed�

C C Fil

-mg

qΣ Σ1 2

λ

ol

Fdq

mm*

Figure �� Nestedloop controller structure�

�� Passivity�based controller design ��

Remark �� As pointed out in Remark �� above it is easy to check that the La�grangian of the levitated ball model satis�es the conditions of Proposition �� forfeedback decomposition into passive subsystems� This decomposition� though similarto the one introduced above� is not de�ned in terms of the same operators� Thatis� .��.� above� and .e�.m in Proposition �� are not the same� In the passivesubsystems decomposition� that we will use for the PBC of electrical machines� theessential property is that both operators are passive� This feature� which is crucial forthe successful design of PBC of underactuated electrical subsystems� is not requiredin this simple example with fully actuation�

�� Passivity based controller design

As discussed above we will adopt the nested loop �i�q� cascaded controller structureof Fig� �� to design our PBC� The innerloop controller is designed following theprinciples of energy shaping and damping injection as applied to the electrical sub�system� The outerloop controller Col is chosen to place the poles of the mechanicalsubsystem�

The procedure to design the cascaded PBC consists of the following steps� �Choice of u to ensure �ux tracking� i�e�� d � Choice of �d so that �� d �F � Fd � ! Prove that F � Fd � qm � qm� with all signals bounded� These stepsare detailed below�

A Flux tracking

As usual in PBC we �rst prove that the electrical subsystem satis�es some passivityproperties� This fundamental step will identify the output which is �easy to control�and its corresponding storage function� An incremental version of this function willbe the desired storage function used to enforce passivity to the closedloop�

This step is usually carried out invoking the Lagrangian structure of the systemhowever� by simple inspection in this example we propose as storage function candi�date�

H� ��

��

whose derivative along the trajectories of �� satis�es

'H� � �Re

c�� qm�

� & �u

�� & �u

�Note that this function is some kind of scaled magnetic energy in the inductance�


with �� Re

c� � � Integrating the last inequality from to t and using the fact that

H� � yields Z t

�

u�s��sds � �

Z t

�

��sds&

where we have set �� H� � This proves �local output strict passivity of the map

u ��

The passivity property above suggests to try to assign to the closedloop thedesired storage function

Hd ��

�� #

where �� d� with �d the desired �ux that induces on the mechanical subsystem

the force required to regulate qm� This signal is de�ned in the next step� With someabuse of notation this step is commonly referred to as energy shaping�

It is easy to see that if we set u � uPB with

uPB � '�d &Re

c�� qm�d & v ��

then we get the error dynamics

'�� Re

c�� qm��& v ��

Proceeding as above we can establish the bound

'Hd �� & ��v ��$

which proves that the map v �� is output strictly passive as desired� Of course� if weset v � we have that �� exponentially fast� Given that the rate of convergence� may be small to enhance performance it is convenient to add a socalled dampinginjection term into the electrical subsystem� setting for instance

v � �RDI��

with RDI � � It is clear that this additional term will speed up the electrical transient�However� as will be shown in the simulations later� increasing RDI does not necessar�ily lead to transient performance improvement� This stems from the fact that theoverall closedloop system under PBC is still nonlinear and we are confronted withunpredictable peaking e�ects which may unstabilize the system �see e�g� � !$��

�� Passivity�based controller design ��

B From �ux tracking to force tracking

The force of magnetic origin F can be written in terms of the �ux error �� and thedesired �ux �d as

F ��

c��d &

��& �d� ��

From the analysis above we have that �� hence it is reasonable to choose �d asthe solution of

Fd ��

c��d ��

where Fd � is some desired force� whose derivative is assumed to be known� Wecan now write the control �� in terms of Fd and 'Fd as

uPB �

rc

Fd'Fd &R�� qm

r Fdc

& v ��

If Fd is bounded we can conclude from �� and �� that the force trackingcontroller �� ensures F � Fd� The problem is that in position tracking problemsFd will not be a priori bounded� since it is generated by Col� This leads to the thirdstep of our design�

C From force tracking to position tracking

In this �nal step we have to design Col and prove convergence of the position error tozero� We start by de�ning Fd as

Fd � m�-qm� � k� '�qm � k��qm � k�

Z t

�

�qm�sds� ��

where we have de�ned the tracking error �qm�� qm� qm�� and we choose k�� k�� k� �

to ensure

d�p�� p� & k�p

� &&k�p& k� ��

is a Hurwitz polynomial� Notice that 'Fd is used in �� From the de�nition of Fdabove� this implies that -qm �or equivalently � or F is required for implementation ofthe PBC �� Remark also that we have introduced in �� an integralterm� This takes care of the steady�state error due to the constant disturbance mg�but it could be removed and simply replaced by g� As we will see in the simulations�and is well known in applications� the inclusion of the integral term robusti�es the


loop� Furthermore it simpli�es the comparison of the various schemes because in thisway the closedloop mechanical dynamics will be the same order�

Let us now analyze the stability of the overall system �� withu � uPB and v � �or v � �RDI

�� We know already from the analysis of the �uxtracking step that �� exponentially fast� To study the mechanical dynamics letus replace �� and �� in �� to get the error equation

G�p�qm ��

mc��& �d� g

where G�p��

pd�p� and d�p as in �� This system admits a statespace realiza�

tion of the form�

'x � Ax &B�

mc��& �d ��!

with x�� qm� '�qm� z��

�� 'z� � �qm� B � IR�� and the characteristic polynomial ofA � IR�� equal to d�p� Now� �� is an exponentially decaying term and from �� and boundedness of -qm� we see that �d satis�es the bounds

j�dj �� & ��jFdj �� & ��jxj

for some suitable constants �i � � i � �� #� Consequently the forcing termsatis�es the bound

j��& �dj �� & �jxj�� #

for some �� The proof of asymptotic stability of ��! follows consideringthe quadratic function

V� ��

x�Px ��

with P the symmetric positivede�nite solution of

A�P & PA � �Q� Q � Q� �

and invoking standard arguments for exponentially stable systems perturbed by lin�early bounded terms multiplied by an exponential� see� e�g�� $��

Remark �� The proof of stability above proceeds in two steps and exploits the factthat the closedloop is the cascade connection of two exponentially stable systems�It is actually possible to obtain a proof in one shot with a single Lyapunov function

�Notice the presence of the zero at the origin that takes care of the constant term g�

� Output�feedback passivity�based control ��

that establishes the stronger exponential stability property�� To this end� we considerthe Lyapunov function

V � V� &a

�� &

b

#��

with a� b � � Its derivative along the solutions of ��!� �� satis�es the bound

'V ��

x�Qx� ��a�� & b�� &

�

mcx�PB��& �d

The key observation here is that� using Young�s inequality and a suitably weightedtriangle inequality� the sign inde�nite term can be bounded as

jx�PB��& �dj dkxk� & �� & ��

for some � � and an arbitrarily small d � � Hence� with a suitable choice of a� bwe get 'V ��V for some ��

Remark �� The controller �� has a singularity when Fd crosses through zero�hence it is only locally de�ned� This problem is� of course� related with the poorcontrollability properties of levitation systems with only one electromagnet� Sincethe focus of the present chapter is on the nestedloop PBC design procedure andstructural similarities of various controllers we will concentrate on local propertiesand assume throughout that the system operates away from singularity regions� In� �$� we propose a controller that takes into account the presence of the singularity�Another interesting way to avoid the singularities is to add to the PBC �or for thatmatter to any other controller a trajectory planning stage using the fact that qm isa �at output� The utilization of this fundamental concept to enhance performance isa topic of current research� which is illustrated for the magnetic levitation system in��#��

Output�feedback passivity�based control

As we have discussed throughout the book� one of the main advantages of PBC isthe possibility of avoiding full�state measurements� In this subsection we develop aPBC that requires only the measurement of qm and 'qm while preserving asymptoticstability� To this end� we recall that -qm is needed only in the evaluation of 'Fd� hencewe propose to replace Fd in �� by

Fd � m�-qm� � k�z� � k��qm � k�

Z t

�

�qm�sds� ��

�This proof� as many other nice re�nements scattered throughout the book� was suggested byLaurent Praly�


where z� is the approximate derivative of �qm� that is

'z� � �az� & b '�qm ��$

with a� b � � This modi�cation does not a�ect the �ux tracking property �� nor the control signal �� The key point is that now uPB can be implementedfeeding�back only qm and 'qm�

The stability analysis of this new scheme mimics the developments above� Thatis� we obtain a state space realization of the mechanical dynamics of the form ��!

but with the augmented state x�� qm� '�qm� z�� z��

�� The new A matrix is

A �

��

� �k� �k� �k� b �a �

��

and the forcing term still satis�es the bound ��#� To study the stability of theunforced equation let us introduce the partition

A��

�� A�

�� B�

� � � � � � ��C�

�

��

��

with A� � IR�� B�� C� � IR�� de�ned in an obvious manner� We can then provethat

det�sI � A � det�sI � A��C�� sI � A�

��B� & s�

and det�sI�A� is a Hurwitz polynomial for all k�� k�� a� b � � Hence it only remainsto choose the integral gain k� to ensure stability of A� Stability of the forced equationfollows verbatim from the analysis of the full state feedback case�

Comparison with feedback linearization and back�

stepping

Among the various controller design techniques for stabilization of nonlinear sys�tems that have emerged in the last few years we can distinguish three which areprobably the most widely applicable and systematic� Feedback Linearization Control�FLC� Integrator Backstepping Control �IBC and Passivitybased Control �PBC�The three techniques have evolved from di�erent considerations and exploit di�er�ent properties of the system� While in FLC we aim at a linear system� in PBC weare satis�ed with assigning a certain passivity property to a suitably de�ned map�consequently enforcing a certain storage function to the closedloop dynamics� In

�� Feedback�linearization control ��

IBC we aim at a somehow intermediate objective �more ambitious than passivationbut less demanding than linearization of assigning a strict Lyapunov function to theclosedloop�

It is clear then that the three techniques will typically lead to the de�nition of dif�ferent control schemes� For some speci�c examples these di�erences blur and some in�teresting connections and similarities between the controller design techniques emerge�It is our belief that the detailed study of such cases will improve our understandingof their common ground fostering crossfertilization� With this motivation in mindwe have carried out in Section ��! a comparative study of PBC� cascaded and back�stepping control for �exible joint robots� In this section we compare PBC� FLC andIBC design techniques in the levitated ball example� which is simple enough to revealthe connections between the controllers� but yet requires the application of nonlinearcontrol to achieve a satisfactory performance�

�� Feedback linearization control

We start our study with FLC since it is clearly the best wellknown and� in thisparticular example� the simplest to derive and understand� To this end� we �ndconvenient to introduce a change of coordinates and we write the system dynamicsin terms of force� position and velocity as

'F � ��Rc�� qmF &

q�Fcu

m-qm � F �mg

��

Di�erentiating once more -qm we get

mq��m � 'F

� �R�� qm F

c&

r F

cu

It is clear from the equations above that� taking as output qm and provided F � � thesystem has relative degree ! and no zero dynamics� Thus� an FLC can be immediatelyde�ned as u � uFL with

uFL �

rc

FmvFL�qm� 'qm� F &R�� qm

r F

c��

which yieldsq��m � vFL�qm� 'qm� F

A suitable choice for the outer loop signal is then

vFL�qm� 'qm� F � q��m� � k��

mF � g� �z �qm

�-qm�� k� '�qm � k��qm ��


The FLC �� is a static state feedback law which ensures in closedloopthe third order linear dynamics

d�p�qm �

with d�p as in ��

Remark �� Since in closedloop 'F � mvFL�qm� 'qm� F � it is useful for the sake ofcomparison with the other controllers to think of mvFL�qm� 'qm� F as a desired valuefor 'F �

�� Integrator backstepping control

As pointed out in Section ��! of ��# �� where a simpli�ed version of our model isconsidered� the system �� is not in any of the special forms required for applicationof integrator backstepping� However� as shown in � !� �see also ��# � this does notpreclude a backsteppinglike design� Since almost any design can be interpreted asbacksteppinglike we point out from the outset that we will takeo� from the approachused in � !� for a magnetic bearing system as applied to our simple levitated ball�Motivated by our �nal objective of comparison of techniques we will also presenta variation of the basic scheme of � !� that incorporates features of the PBC� Asimilar combination of PBC and IBC has been reported in �� where ideas of � ��are added to an IBC to remove singularities in induction motor control�

In IBC we start from the mechanical equation of �� and� assuming F is thecontrol� de�ne an Fd that stabilizes this subsystem� For the sake of comparison wechoose Fd as in FLC and PBC� that is �� Adding and subtracting 'Fd tomq

��m � 'F

yields the �rst error equation

'x � Ax &�

mB�F � Fd

where x� A and B are as in ��!� We start constructing our Lyapunov function withthe quadratic function �� whose derivative yields

'V� � ��

x�Qx &

�

mx�PB�F � Fd

� ��

x�Qx &

�

mcx�PB��& �d

where we have used the de�nitions of F and �d in �� to get the last equation� Letus now look at the dynamic equation of �� that is

'�� R

c�� qm�& u� '�d�

At this stage we decide a control u that stabilizes �� plus a term to compensate for thecross term in 'V� above� We have here the choice of adopting an FLClike approach

�� Comparison of the schemes ��

cancel the term �Rc�� qm� and add a damping term �RDI

�� or follow the PBCroute� use the existing damping� and set u � uIB with

uIB � uPB ��

and uPB as in �� We can complete our Lyapunov function with Hd ��# as

V � V� &�

Hd

where we have added a tuning parameter � � Di�erentiating V and using ��$we get

'V ��

x�Qx� �

�� & ��

�

v &

�

mcx�PB��& �d��

The IBC design is completed setting

v � �

mcx�PB��& �d ��

which removes the cross terms and yields the desired strict Lyapunov function

'V ��

x�Qx� �

��

Remark �� Following the line of reasoning used in Remark ��# to generate a strictLyapunov function for PBC� we can think of dominating instead of cancelling thecross term in the bound of 'V � Although this idea is theoretically very attractive�our experience in applications is quite disappointing because the new control involvehigher order terms of the error signals which amplify the noise and saturate theactuators� In our opinion� this undesirable phenomenon also present in FLC is animportant drawback of IBC which is clearly illustrated in Sections �� and ��# forthe induction motor and in �� for robots with AC drives�

�� Comparison of the schemes

We will now compare FLC �� PBC �� without damping in�jection� i�e�� with v � � and IBC �� To this end� we notice that�� which we repeat here for ease of reference

Fd � m�-qm� � k� '�qm � k��qm � k�

Z t

�

�qm�sds�

is related to �� as'Fd � mvFL�qm� 'qm� F �


Hence� the three control inputs can be written as

uFL �

rc

FmvFL�qm� 'qm� F &R�� qm

r F

c

uPB �

rc

FdmvFL�qm� 'qm� F &R�� qm

r Fdc

uIB � uPB � p mc

��qm� '�qm�

Z t

�

�qm�sds�PB�pF &

pF d

The following remarks are in order

� FLC and PBC di�er only on the utilization of the forces in some terms� desiredforces Fd for PBC instead of the actual forces F for FLC�

� One additional di�erence comes from the generation of Fd� which involves theinclusion of an additional state� Hence� PBC and IBC are dynamic state feed�back controllers while FLC is static�

� The IBC equals the PBC plus the signal that achieves the cancelation of thecross terms in 'V � The two schemes approach as � �

� In contrast with FLC� the closedloop equations for PBC and IBC are nonlinear�In PBC we get a cascade structure where the �rst subsystem is exponentiallystable and some growth conditions on the input to the second subsystem allowsus to complete the stability proof� In IBC we get a closedloop system of theform

�'��'x

��

��R

c�� qm � p

�c�pF &

pF dB

�P�p�c�pF &

pF dB A

� ��x

�

�� ADI & ASK

��x

�

where the blockdiagonal matrix ADI is �stable� and ASK satis�es the skewsymmetry property �

�

P

�ASK � �A�

SK

��

P

��

This pattern is present in all IBC designs�

� Given that their closedloop dynamics are nonlinear� predicting the e�ect ofthe tuning parameters for PBC and IBC is less obvious than for FLC�


� While there is an output feedback version of PBC� the presence of the extra termin IBC hampers this possibility� However� if we want to add damping to thePBC we need the full state�

� As explained in the previous section an FLClike version of IBC may be obtainedas

uIB �

rc

FdmvFL �qm� 'qm� F &R�� qm

r F

c

� p c

��qm� '�qm�

Z t

�

�qm�sds

�PB�

pF &

pF d�RDI�

pF �

pF d�

Notice that it combines terms from PBC and FLC plus the damping injectionand the additional �decoupling� term�


In this section we present some simulation results of the levitated ball model �� in closedloop with FLC� PBC and IBC� To facilitate the comparison� and unlessotherwise stated� we have in all cases �xed d�p � �p&� � and show the plots of po�sition and current� We tried a series of small step references in position� As expected�when we applied larger steps we cross through controller singularities� We have topoint out that for the feedback linearization controller the �admissible� referenceswere larger than in the other cases� Away from singularities the control signals donot di�er signi�cantly� therefore we show in all cases only the position and currentresponses�

To motivate the utilization of nonlinear control we designed �rst a pole placementlinear controller for the linearized approximation of �� around the zero equilibrium�In Fig� ��! we show the responses for various �� m step references� As expected theperformance is degraded as we move away from the domain of validity of the modelingapproximation� The response of FLC �� which is shown for the sake ofreference in Fig� ��# is of course the same for all reference levels� It is interesting tonote that we observed very little stability degradation when the actual parametersR and c were replaced by estimates within practically meaningful ranges of �� )�Of course� some steady state error was observed� Comparable robustness propertieswere observed for the other schemes�

PBC was tried in di�erent variations� Fig� �� shows the response of �� without damping injection� i�e�� with v � and without integral action k� � � Inthis case the mechanical dynamics is order two and we �xed the characteristic poly�nomial at �p&� �� Even though the errors eventually converge to zero� the transientresponse is extremely poor even when the poles of the mechanical dynamics werepushed farther left� Fig� �� shows the e�ect of adding an integral action� while for


Fig� ��$ we included also the damping injection� For the latter we have observed that�for large values of RDI� the response overshoots� due to the peaking phenomenon� Theresponse of the outputfeedback version of PBC �� $ �with the bandwidthof the approximate di�erentiator �ve times faster than the mechanical dynamics isessentially indistinguishable from the statefeedback case� hence is not shown herefor brevity� Finally in Fig� �� we present the responses of the IBC �� for a value of � � ��

In summary we can say that in most of the cases� and as long as we kept awayfrom the singularity region� the observed responses were consistent with the theo�retical predictions� Also� despite the signi�cant �analytical di�erences of the threecontrollers that we pointed out above we could not observe big discrepancies in thesimulation responses� We believe this stems from the fact that the system is ratherbenign �it is passive with respect to �� and y is a �at output� particularly for thechoice of parameters given here�

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time [t]

posi

tion

[m]

Linearized model based controller

Figure ��!� Linear controller�


−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.1

00.10.20.30.40.50.60.70.80.9

time [s]

Feedback−linearization control

posi

tion

[m]

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.2

00.20.40.60.81

1.21.41.61.8

Feedback−linearization control

time [s]

curr

ent [

A]

Figure ��#� Feedback�linearization controller�

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.1

00.10.20.30.40.50.60.70.80.9

posi

tion

[m]

time [t]

Passivity−based control without integral term

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.2

00.20.40.60.81

1.21.41.61.8

time [s]

curr

ent [

A]

Passivity−based control without integral term

Figure �� Passivity�based controller without integral term�

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.1

00.10.20.30.40.50.60.70.80.9

time [s]

disp

lace

men

t [m

]

Passivity−based control with integral term

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.2

00.20.40.60.8

11.21.41.61.8

time [s]

curr

ent [

A]

Passivity−based control with integral term

Figure �� Passivity�based controller with integral term�


−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.1

00.10.20.30.40.50.60.70.80.9

time [t]

posi

tion

[m]

PBC with integral term plus damping injection

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.2

00.20.40.60.81

1.21.41.61.8

PBC with integral term plus damping injection

curr

ent [

A]

time [s]

Figure ��$� Passivity�based controller with integral term plus damping injection�

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.1

00.10.20.30.40.50.60.70.80.9

time [s]

posi

tion

[m]

IBC with integral term

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−0.2

00.20.40.60.81

1.21.41.61.8

IBC with integral term

time [s]

curr

ent [

A]

Figure �� Integrator backstepping�based controller�

�� Conclusions and further research

The following concluding remarks concerning general aspects of FLC� PBC and IBCare in order

� While in FLC we aim at a linear system� in PBC we are satis�ed with assigninga certain passivity property to a suitably de�ned map� consequently enforcinga certain storage function to the closedloop dynamics� In IBC we aim ata somehow intermediate objective �more ambitious than passivation but lessdemanding than linearization of assigning a strict Lyapunov function to theclosedloop�

� Both� in IBC and PBC� the main question is �Which storageLyapunov functionyou �can want to assign"� In PBC� at least of EulerLagrange systems� the

�� Conclusions and further research ��

answer is provided by the systems energy function �either of the full system oronly a part of it� The power of IBC is that� for a certain class of cascadedsystems� the Lyapunov function can be recursively constructed� The exampleabove shows that IBC can be pro�tably combined with PBC to answer thisquestion�

� It is often argued and reasonable to expect� though hard to prove rigorously� thatavoiding cancelation of nonlinearities enhances the robustness of the scheme inthe face of parameter uncertainties�� Our simulation evidence in the presentexample� however� showed that all schemes are highly insensitive to these un�certainties�

The three schemes will in general exhibit di�erent transients and posses di�erentrobustness properties� a challenging research problem is to establish some commonframework to compare their robustness and performance properties� One particularlyinteresting question is to assess the degrees of freedom provided to the designer toenhance the systems response�

The skewsymmetry property of IBC is related� though not so clear how� with thesocalled worklessforces of EulerLagrange systems� In PBC we essentially disregardthese forces and concentrate on the damping injection to increase the convergencerate� In this class of systems these coupling terms are related with the transformationof energy from one form to the other� e�g�� from electric to magnetic in electricalcircuits or from potential to kinetic in mechanical systems� In the recent interestingpaper �� the possibility of �shaping� also these forces for performance improvementis explored� This is an intriguing� and certainly quite reasonable� proposition whosee�ect cannot be captured with our classical convergence rate analysis�

�See Section �� for an induction motor example where the superiority of PBC is unquestionablyestablished� both theoretically and experimentally�

Chapter �

Generalized AC motor

�Man mu immer generalisieren��

C� G� J� Jacobi

In the second part of the book we pursue our research on development of PBC forEL systems as applied to electromechanical systems� In this chapter we restrict ourattention to the practically very important class of the generalized rotating electricmachines ��$�� The main contribution is the de�nition of a class of machines forwhich the output feedback torque tracking problem can be solved with PBC� Roughlyspeaking� the class consists of machines whose non�actuated �rotor dynamics is suit�ably damped� and whose electrical and mechanical dynamics can be partially decoupledvia a coordinate transformation� Machines satisfying the latter condition are knownin the electric machines literature as Blondel�Park transformable ��$�� In practicalterms this requires that the air�gap magneto motive force can be suitably approxi�mated by the �rst harmonic in a Fourier expansion� These two conditions� stemmingfrom the construction of the machine� have clear physical interpretations in terms ofthe couplings between electrical� magnetic and mechanical dynamics� and are satis�edby a large number of practical machines�

� Introduction

� AC motors

The importance of the class of physical systems chosen in this chapter can hardly beoverestimated� The �eld within motion control called mechatronics has become a veryimportant technology for industrial automation� This technology merges mechanics�precision mechanics� coupled electromechanical systems and electronics �microelec�tronics� power electronics� using among other tools control theory� and is of high

��

�� Ch� �� Generalized AC motor

economic importance� Control of electrical drives holds a central position within thistechnology�

Due to their simplicity from a control point of view� DC motors have been thetraditional choice where high dynamic performance� i�e� extremely rapid and accuratetorque� speed or position control in all four quadrants and for a wide speed range�including zero speed� is required �machining tools� robots� These machines areexpensive and di�cult to construct for high power%speed applications� even if slot�lessarmature designs have increased their power range� High torque standstill operationis also di�cult� due to brushes being welded to the commutator� DC motors are heavywith high rotor inertia and large dimensions� and they have failure prone brushes�

which are exposed to corrosion and wear� hence regular maintenance is required�Because of the brushes they are not suited for hazardous environments where electricsparks are not allowed �oil and gas industry� unless they are especially encapsulatedin material or by inexplosive gases under higher pressure than the surroundings� Tosummarize� the DC motor can be thought of as an expensive device� but with a cheapcontroller�

AC machinery has been the choice for high power constant speed industry ap�plications �compressors� fans� mills� and pumps� or in assembly lines with severalmachines connected to the same power supply� These machines do not su�er fromthe typical disadvantages associated with DC motors �no brushes� less complicatedrotor construction� but due to their inherent nonlinear dynamics� they have beenconsidered di�cult to control and not suited for high dynamic performance appli�cations� Compared with DC motors� they can be thought of as cheap devices� butwith expensive controllers� The recent years advances in power electronics and mi�croprocessor technology have enabled implementation of advanced nonlinear controlschemes using DSPs� and AC machines have replaced DC machines in a large varietyof low and medium power applications� leading to higher reliability and lower costs�

The new advances in AC motor technology have also led to a re�examination of thecontrol schemes used in the traditional �uncontrolled constant speed high power in�dustry drives� due to demands on higher product quality �tighter speed control� fasterrecovery from disturbances� In addition� these advances have led to an increasingnumber of new applications� Examples of high power� high dynamic performanceapplications are ships� where conventional diesel�%turbine�mechanical propulsion hasreached the top of its evolution cycle� and is now being replaced with diesel�%turbine�electric propulsion �� It is even claimed that in future permanent magnet ACmotors will take over in application areas where hydraulic and pneumatic actuators�which are bulky and failure prone� have been the traditional choice� Such applicationsinclude robotics� aircrafts� and spacecrafts� In many of these applications� and alsofor ships and vehicles� which carry their own fuel �e�g� batteries� economy is of high

�Tesla pointed out already in �� that the commutator is a highly complicated device which isthe source of most of the problems experienced with DC machines�

� AC motors ��

importance� and the concept of power e�ciency could be equally important to highdynamic performance�

It is estimated that in the United States more than � ) of the generated energyis converted to other forms by electrical drives �!$�� and only �) by DC drives�With such amounts of energy being converted� power e�ciency of electrical drivesin general has become an area of increasing research interest� In fact� it is pointedout in a book from the American Council for an Energy�E�cient Economy� that�adjustable speed drives and other controls are the largest potential source for motorsystem energy savings� � �� This does not only include large industrial drives� wherea ) improvement of speed regulation can give signi�cant long term cost reductionsdespite higher initial investments� It includes all kinds of electrical drives� in a hugenumber of applications� including households�

To some extent high dynamic performance and power e�ciency �typically morethan � ) at rated conditions of a drive can be obtained by the design of the motoritself� However� for motor drives with a highly varying range of operating conditions�motor design alone cannot ensure high performance and e�ciency for all conditions�and advanced control schemes must be used together with power electronics�

From a control point of view� AC drives pose the following research problems�

� Transfer of electromagnetic into mechanical energy is essentially described bynonlinear models� making standard linear control theory inferior to nonlinearschemes for control of such systems�

� They are multivariable systems� with several input voltages or currents and oneor more outputs �torque%speed� �ux level to be controlled�

� Varying parameters �e�g� temperature dependent resistances and friction pa�rameters� inductances depending on �ux level�

� Unknown load disturbances�

� Only partial state measurement �e�g� unmeasurable �uxes and rotor quantities�

� In some cases failure prediction and prevention is also needed �supervisory con�trol system to monitor defects in windings or bearings�

Because of these factors AC drives� and especially induction motors� have becomeinteresting benchmark problems for testing of new nonlinear control schemes �$ ��The challenging control problems and the rapidly growing number of applications�are also highly motivating factors for working with this class of physical systems�


� Review of previous work

Due to the importance of electrical drives in industry� thousands of papers and nu�merous textbooks presenting research in this �eld� have been published during thelast ! years� It would be a time consuming and di�cult task to go into the detailsof all these approaches� and that is not the intention of this summary� In this sectionthe most important control approaches are explained� with emphasis on recent resultswithin the nonlinear control theory direction this research has taken� For additionalinformation about this direction� the reader should consult � �!� and the referencestherein� A recent summary of results along the more application oriented branch canbe found in �!��

A Classical stationary control �Scalar control�

The classical methods for control of AC machinery have been based on linearizationsof the nonlinear equations at steady state operating points� This approach has theadvantage that classical linear theory can be used for controller design� Typically�this has resulted in schemes where amplitude and frequency of sinusoidal stator volt�ages or currents are the basic control variables� Such designs give varying dynamicperformance when applied to nonlinear systems� depending on to which degree theunderlying small�signal assumption is ful�lled� i�e� depending on how far from anominal operating point the system is driven� Another disadvantage of applying suchmethods to multivariable systems� is the problem of coupling between inputs and out�puts� making independent control of outputs di�cult� For instance� in a voltage fedinduction motor� both torque and air gap �ux are functions of voltage amplitude andfrequency� giving considerable coupling and slow response when trying to control onlytorque �!�� Even with a well tuned scheme it is di�cult to match the performanceof a DC drive�

B Vector control

Some of the de�ciencies of classical linear methods when applied to control of ACmachinery� were overcome by the vector control methods introduced in the period�� $ � These methods aimed at making the AC motor behave like a DC motor�with asymptotic decoupling of torque and �ux control� To achieve this� the nonlinearmodel of the motor had to be used in the design�

The vector methods� consist essentially of a nonlinear coordinate transformation�a rotation� followed by a nonlinear decoupling feedback� At the time of introduction�implementation of the schemes �especially the rotations was computationally heavyand di�cult with analog electronics� These schemes were therefore considered to be

�Generically also known as �eld oriented control �FOC� methods�

�� Review of previous work ��

rather �academic� �� and did not gain their wide popularity until the introductionof the digital microprocessor around �� The basic control variables were now theindividual components of the rotated two�dimensional stator current vector� eithercontrolled directly� or indirectly through a nonlinear feedback voltage control law�Vector control in its various implementations is now the de facto standard for highdynamic performance control of AC drives� and its superior dynamic performance ascompared to the use of classical linear methods is widely accepted ��

One of the most common implementations of vector control� rotor��ux�orientedcontrol� is discussed in greater detail in Section �� with application to inductionmotors� The basic drawback of these schemes is the assumption of full state measure�ment ��ux measurement� Sometimes it is also claimed that it would be desirable toachieve exact� instead of only asymptotic� decoupling between torque and �ux control�This can be achieved with linearizing controllers� which are discussed below�

See � $�� for the theoretical part of vector control� and �� for implementa�tion aspects�

C Modern nonlinear control

During the last � years� there have been signi�cant advances in nonlinear controltheory� and among many other applications for which linear theory cannot give satis�factory solutions� its application to electric machines has gained widespread interest�Of course� these schemes are also based on nonlinear models and end up with a spec�i�cation of the current or voltage input vector� However� since they are derived fromdi�erent and sometimes purely mathematical nonlinear control theories� aiming atformal proofs of stability� they are named nonlinear schemes in this book instead ofvector control schemes�

The following review of modern nonlinear control theory applied to electric ma�chines is based on � �!�� with the addition of recent results� To limit the number ofreferences� only the recent theoretical and experimental applications of these methodsto induction motor control are included here� since this machine will be particularlyemphasized in Chapters �� and �� of this book�

According to Taylor�s recent overview� the methods can be broadly classi�ed intothe three following categories�

�� Exact linearization design

� Backstepping and manifold designs

!� Passivitybased design�

�In Taylor�s survey paper� this approach is referred to as �Energy Shaping Design�� We preferto use the more general PBC term since� as we have seen throughout the book �energy shaping� isjust one step in the whole procedure�


Each of these approaches are explained in the following text� and some comments aregiven to methods not belonging to these classes�

C�� Exact linearization design

The goal of these schemes is to transform the system into a linear input�outputrelation between external inputs and controlled outputs� using an inner nonlineardecoupling loop� Controllers can then be designed to ensure stability and performanceof the resulting linear system� by use of conventional linear theory� Fundamental tothis approach is the choice of coordinates for the representation of the system� togetherwith the design of the inner loop decoupling control in these new coordinates� Incontrast to �eld�oriented control� which also has a coordinate transformation and aninner decoupling loop� the decoupling between outputs is no longer only asymptotic�The coordinate transformations used are also generally more complicated than therotation used in �eld�oriented control�

Disadvantages of these schemes are that measurement of the full state is needed�exact cancelation of dynamics is necessary� and controller singularities are introduced�typically for zero rotor �ux norm� It also seems to be di�cult to apply this methodin a general way to a broader class of machines� and then derive controllers for eachmachine in particular from a general controller� The dynamic equations for statorand rotor quantities must be transformed to a common frame of reference �typicallythe stator �xed frame before the di�erential geometric tools can be used� Symbolicsoftware is often necessary to answer the question of whether the system� with a givenset of inputs and outputs� can be transformed into a linear system or not� It is wellknown that the development of these software tools has enjoyed very limited success�hence this remains an important stumbling block for the application of this theory�

In the feedback linearization technique� all nonlinearities of the system are can�celled to obtain in closedloop a linear input�output relation� There are two draw�backs to this approach� �rst the cancelation� if not exact� brings along very seriousrobustness problems�� Second� even from the mathematical point of view it is clearthat not all of the nonlinearities are harmful to the closed loop dynamics�� Thesearguments have systematically been invoked in PBC designs as a critique to feedbacklinearization� see in particular Sections �� and the experimental results in Sec�

�It is claimed that adaptation may alleviate� to some extent� the lack of robustness of feedbacklinearizing schemes� However� the issue of robustness of adaptive systems in general is far from beingsettled�

�In fact� by use of Lyapunov theory� it can be shown that some nonlinearities are useful for systemstabilization� and hence they should not be cancelled by the controller� For instance� consider thesystem �x � �x��cosx� u� For stabilization around zero� it is of interest to cancel the cosine termwith the input u� but not the third order nonlinearity� which is helpful� See also Example �� in��


tion ��#� They have recently been fervently adopted in the socalled backsteppingliterature ��# � !$� which we review below�

For the �fth order induction motor model case� recent contributions belonging tothis class may be summarized in chronological order as follows�

� The exact input�state linearization under the assumption of a slowly varying speedpresented in ��$�� which essentially was a linearizing controller design for the reducedfourth order electromagnetic model� This early paper� though not very often cited�triggered a lot of interest in the community and should be considered as an importantprecursor to later developments�

� The exact input�output linearization from stator voltages to torque or rotor speedand square of rotor �ux norm proposed in ��#!�� for both the rotor��ux�oriented modeland the stator �xed model� This is a fundamental paper for this approach� whichwas later substantially clari�ed and extended in ��$ � to handle unknown constantrotor resistance and load torque by the use of adaptation� local stability results werealso derived� The resulting zero dynamics with these outputs� is the dynamics ofthe rotor �ux angle� which is weakly minimum phase� actually periodic� In thispaper it was also shown that the �fth order induction motor model is not input�stateexactly linearizable� and that the largest input�state feedback linearizable subsystemhas dimension four�

� In �� it was shown that the extended sixth order system obtained by augment�ing the �fth order model by an integrator in one of the inputs �or the higher ordersystem obtained by one integrator in one input and two in the other� can be ex�actly input�state linearized� but only for speed� not position control� The result isonly locally valid� and the control structure is computationally heavy and requiresswitching between two transformations to avoid singularities� A nonsingular feedbacklinearizing transformation was shown to exist only for nonzero torque� The result hasrecently been extended in �� for the case of a third order rotor��ux�oriented reduceddq�model� for which stator dynamics is neglected and stator currents are consideredas inputs� while rotor �ux angle� amplitude� and rotor speed are states� This model isnot input�state exactly feedback linearizable� In the work by Chiasson it was shownthat if one integrator is added to the input in the d�axis� a single feedback lineariz�ing transformation and controller exist� The controller has a dynamic singularitycondition� and if �ux is kept constant� it must be nonzero to avoid this singular�ity� Otherwise� the dynamic condition sets a limit on how fast it can be decreased�This limits the �ux tracking capabilities of the scheme� The controller structure ishowever computationally feasible� Adding instead one integrator to the q�axis input�

�In this model the two rotor currents �or uxes�� two stator currents� and rotor speed are con�sidered to be the states of the system� while the two stator voltages are the inputs� The outputs tobe controlled are torque or rotor speed and rotor ux norm� In the results reported here� the stator�xed frame of reference has been used for model representation� unless something else is explicitlystated� The model of the induction motor is also described in Section ��


results in the �fth order system with position included being feedback linearizable�Unfortunately the required controller is singular for zero torque� and another con�troller structure is needed when the torque is required to change sign� This limits thepractical usefulness of the result�

The results in �� clearly reveal how fragile are the geometric properties uponwhich feedback linearization is based� Namely� they change with coordinate transfor�mations� hence care has to be taken when model representation� inputs and outputsare to be chosen� Otherwise the linearizing approach can result in controllers havingsingularity conditions which are di�cult to interpret and avoid�

� Experiments validating the practical importance of the feedback linearizing ap�proach have been presented in �� ! � and �! �� Results from extensions to non�linear magnetics with saturation and speed observers� have also been presented in�!�� ! ��

A more basic property than feedback linearizability is the property of �atness�

Roughly speaking� we say that a system has a �at output if its relative degree isthe same as the systems degree� hence it has no zero dynamics� A �rst immediatecorollary is that systems with �at outputs are feedback linearizable� but this is byfar not the must important consequence of this feature� In the authors� opinion thisis a fundamental concept that induces� in a natural way� a classi�cation of nonlinearsystems into �at and non��at� It has been shown in� e�g� ��!� that many physicalsystems are �at with� furthermore� physically meaningful �at outputs� For instance�we have seen already in Chapter � that the total energy is a �at output for the boostconverter� while the joint velocity vector is a �at output for robots with �exible joints�A lot of research has been devoted recently to further elucidate this characterization�In the important paper ��$�� it was shown that the induction motor is �at� with�at outputs the slip angle of the �ux and the rotor position� In the same paper aninteresting extension of the �ux observer of � � � is given and a predictive controllerlike approach is proposed� A FOClike controller and some experimental results alongthis line of research have been reported in ��

Quite a lot of e�ort has been devoted to the estimation problem� and linear andnonlinear observer theory � � �� extended Kalman �lters ��#�� and more physicallybased adaptive observers �� have been proposed as solutions� Recently� an observerwhich is adaptive with respect to rotor resistance have been proposed in ��$ �� In thiswork exponential convergence of �ux and rotor resistance errors to zero� is provedusing a Lyapunov type argument� under reasonable assumptions on persistency ofexcitation�

�We say that a system is at if there exists certain special outputs� called the at outputs� equalin the number to the inputs� which are functions of the state vector and of a �nite number of itsderivatives� Additionally� the at outputs are such that every variable in the system can� in turn�be expressed as functions of the at outputs and a �nite number of their time derivatives� See ��for a detailed explanation of this concept and its application to a wide variety of physical systems�


The incorporation of estimated states in the control law is however in most casesbased on a �nonlinear separation principle�� Even if there are rigorous theoreticalproofs of stability in the case of full state measurement� generally no proofs are givenfor stability of the closed loop system when ad hoc estimates have been substitutedfor real states in the controller� As explained in Section �� see also ��! �� it isimportant to consider the e�ect of the convergence rate of the observation error onthe performance of the total system� since there is only exact decoupling after theestimation error has converged to zero�

� For the reduced third order model of the induction motor� an interlaced outputfeedback controller and observer design has been reported in �� Rotor speed isthe only measurement� and the controller provides singularity free �under the assump�tion of suitable initial conditions for the �ux estimates speed and rotor �ux normtracking� The algorithm is also adaptive with respect to a constant load torque� Animplementation taking advantage of using the rotor �ux reference frame for digitalcalculations� has been reported in ��

� Much of the work along this direction has now evolved� via adaptive extensions offeedback linearizing controllers� into adaptive output feedback controllers� which arederived using nonlinear tools like Lyapunov theory� An important result is the glob�ally de�ned adaptive output feedback controller presented in ��$�� The controller isadaptive with respect to constant load torque and rotor resistance� and gives asymp�totic �ux and speed tracking� with convergence of estimated parameters and statesto true values� under an assumption of persistency of excitation�

C�� Backstepping and manifold designs

Backstepping is a recursive Lyapunov procedure for controller design whose ori�gins� as pointed out in Chapter ! of ��# �� date back to the � year old work ofTsinias� Byrnes and Isidori� and Sontag and Sussmann� The �rst step in this ap�proach is to choose the output to be controlled and derive its dynamic equation� Actitious control signal is then chosen from this equation� Using a �rst Lyapunovtheory approach� a desired function for this �ctitious control is found� such that thethe control objective of the �rst subsystem can be asymptotically achieved� In theoriginal versions of backstepping the control was chosen with cancelation of dynamicsin mind� however some elements of PBC have been recently incorporated at this stepto develop more robust designs� If the �ctitious control is the real input to the system�which can directly be speci�ed to be the desired function� then the design ends here�This is generally not the case� and there will be a deviation between the �ctitiouscontrol and its desired behavior� The dynamic equation for this error is then derived�an integrator is added� and the design above is repeated with the aim of forcing theerror to zero by the use of a new �ctitious control� Stability and convergence of this


new subsystem can be proved by adding a term square in the error to the previousLyapunov function� The procedure is then repeated until �nally the real control canbe speci�ed to be a desired function� and the desired control properties can be provedusing a �nal Lyapunov function� which is a sum of the previous functions� The num�ber of steps needed� is equal to the relative degree between the output to be controlled�and the input of the system� For multivariable systems� the design is done separatelyfor each of the outputs� This results in a linear combination of inputs being equalto desired functions� and inversion of a matrix is necessary to specify the real inputs�hence control singularities are likely to occur�

There are now several results from the application of backstepping to inductionmotor control� In all of the results listed below� the �fth order stator �xed modelrepresentation has been used� and unless something else is explicitly stated� observershave been used to avoid �ux measurement�

� In �� the �rst application of backstepping to speed and rotor �ux norm trackingfor induction motors was presented� Exact model knowledge was assumed� and aproof of exponential stability was given for the total system with an observer basedon the rotor �ux equations� The stability result was only regional in the sense thatthe invertibility of a matrix needed to calculate the controls� restricted the initialconditions to be in a set which could be estimated a priori� The basic idea used for theinterlaced design� was to add nonlinear damping terms in the controller to compensatefor interactions due to observation errors� These terms made it possible to dominatecross�terms containing observation errors in the derivative of the Lyapunov function�and it could consequently be made non�positive�

The advantage of this scheme� as compared to conventional schemes� which donot compensate for observation errors� was demonstrated by simulations� and showedthat not compensating for observation errors could have signi�cant negative impacton transient responses�

� This result was extended in �� to compensate for unspeci�ed modeling imperfec�tions and external disturbances� by addition of PI�loops for speed and rotor �ux normtracking� The regional property of stability was still present� imposing restrictionson initial conditions and reference functions� In the observer structure used in thatpaper� additional nonlinear terms were introduced� as compared to the previous de�sign� In the stability analysis� it was shown that with a proper choice of these terms�they could be used for eliminating instead of dominating cross terms stemming fromobservation errors� in the derivative of the Lyapunov function�

The introduction of nonlinear damping terms in the observer%controller� gavea systematic method for handling of estimation errors and other disturbances instability proofs� Later designs have taken advantages of these ideas�

� The above results were changed to semi�global uniform ultimately bounded positiontracking error and rotor �ux norm regulation in �� $�� To avoid controller singularities


for zero �ux estimates� a control parameter had to be made su�ciently large relativeto initial conditions�

� Departing somewhat from the previous results with interlaced controller and ob�server designs� adaptive and robust controllers� which could compensate for para�metric uncertainty in all parameters �robust case also had additive norm boundeddisturbances� were presented in �� There were controller singularities for zero ro�tor �ux norm� and proofs of asymptotic position%speed tracking �adaptive case anduniform ultimate boundedness of position tracking error �robust case� were derivedunder the assumption of full state measurement�

� While all the above approaches required speed measurement� a design with bothspeed and �ux observers was proposed in �� Local exponential rotor �ux norm andposition tracking was proved� with measurement of only stator currents and position�The local nature of this result was again due the the invertibility of a matrix needed forcontrol calculations� To avoid the singularity� certain restrictions had to be imposedon rotor �ux norm reference and initial conditions� The result was novel in the waythat speed and �ux observers were both taken into the stability analysis�

� An adaptive controller which could compensate for parameter uncertainty in bothrotor resistance and the mechanical subsystem� was presented in �� #�� The observerwas based on the adaptive observer from ��$ �� but to obtain an interlaced designof observer and controller� additional terms were added to the observer� The proofof asymptotic position%speed tracking and norm of estimated �uxes converging to adesired function� was only locally valid� due to a singularity in the controller for zero�ux� A drawback of this scheme is that asymptotic convergence of �ux estimates toreal values was not proved �not even for exactly known rotor resistance� hence rotor�ux norm tracking cannot be claimed� However� this was the rst asymptoticallystable output feedback scheme with adaptation of rotor resistance� which has beenreported�

� We have shown already in Chapter how we can combine backstepping ideas withPBC to robustify the design� Other examples are reported in � $$�� where globalresults were achieved by incorporation of some of the ideas presented in this bookinto a backstepping design� The clue to avoid singularities was to use� as in � �� thedesired rotor currents and �uxes instead of the actual ones in the controller design�

In the �rst paper a result on global asymptotic position%speed tracking and rotor�ux norm regulation was presented� The scheme included adaptation of mechanicalparameters� The result in the second paper was a scheme with globally exponentialvelocity tracking and rotor �ux norm regulation� Experimental results were includedin both papers�

Drawbacks of these schemes are that they do not provide rotor �ux norm track�ing� electrical parameters are not compensated for� and they are computationallyheavy� This last point limits the value of the experimental results� since responses


are restricted to be very slow because of the high sampling period needed�

� Some of the above problems have been solved in � $�� where an adaptive singularity�free controller for asymptotic rotor position and rotor �ux tracking has been proposed�Rotor �ux estimates were calculated by integration of voltage and current measure�ments using Stanley�s equations � �� This allows for adaptation of rotor �ux resis�tance and mechanical parameters� but relies on zero initial conditions and use of openloop integration for �ux calculation� This approach is not numerically robust� and anad hoc integration method with a forgetting factor has to be used for implementation�The performance of the controller was demonstrated by experiments� Unfortunatelythe same points as above regarding response times were present� It is believed thatthis will be solved in future work� either by use of controller simpli�cations or specialhardware �ASICs�

It must be pointed out that this scheme �as well as many other backsteppingschemes has been derived with the stator voltages as basic control variables� andit is not clear how the controller can be applied to the reduced order model� withcurrents as inputs�

In the above schemes the full order model has been used for controller design� andthere are no approximations or implicit assumptions about time scale separationsbetween interacting parts of the dynamics�

The rationale behind manifold designs based on singular perturbation or integralmanifold theory� is to take advantage of inherent time scale separations betweendi�erent parts of dynamics in a system� Such a separation exists for instance betweenthe high frequency current dynamics and its low frequency average dynamics� becauseof small inductances or high�gain current control� Another example is the relativelyslow mechanical dynamics for high rotor inertias� as compared to the fast electricaldynamics� See ��$�� for a nice presentation of these practical considerations�

These methods essentially consist in �rst designing a fast control� which steersthe fast dynamics to the manifold of the slow dynamics �i�e� makes it attractive�and is inactive once the fast states hit the slow manifold� A second slow control isthen designed to give the desired behavior of the slow dynamics� assuming that thereduced order slow model is a su�ciently good approximation of the system dynamics�These methods can be applied in combination with other approaches� to analyze andimplement approximations of for instance schemes based on feedback linearization�

Another related approach is to force the systems dynamics to evolve on a manifoldcalled a sliding mode with discontinuous controls� having analogy to classical bang�bang control� The behavior on the sliding mode is speci�ed to be the desired one�giving for instance zero speed and �ux tracking error� Once on the sliding mode� thesystem states will stay on it� due to the controls being discontinuous across it� Anysmall deviation from the sliding mode will activate controls to force the states backto it� Theoretically this is equivalent to in�nitely high gain� The inherently high


frequency discontinuous on�o� switching of controls have made this approach veryappealing for control of power converters and electric machines� where thyristors orother switches are used�

For the design of sliding mode motor controllers� the converter is taken into themodel� and the switching pattern follows directly from the controller� not from aPWM block with outputs from the controller scheme as reference inputs� This setsrequirements to speed of computation� if high switching frequencies are needed forsatisfactory control� Unfortunately� analysis of di�erential equations with discontin�uous right�hand sides is technically di�cult� due to the fact that classical theoremsof existence and uniqueness are not necessarily satis�ed for such systems� This hasmotivated the development of special tools for rigorous analysis of sliding mode sys�tems�

Examples of these schemes and their combinations for motor control can be foundin � $�� and � �� Experimental results have been reported in � � �� The applicationof backstepping for control of electric machinery is explained in detail in ��

C�� Passivity�based design

As pointed out in Chapter � �see also � �!�� this approach has evolved from con�siderations of physical properties like energy conservation and passivity� This shouldbe contrasted with feedback linearization� which results from purely mathematicalconsiderations� Brie�y� in PBC design it is aimed at reshaping the energy of thesystem in a way leading to the desired asymptotic output tracking properties� Themain goal is to drive the system to a desired dynamics� leaving the closed loop systemnonlinear� without cancelling dynamics or introducing controller singularities�

To give some historical perspective to the developments of PBC as applied toelectrical machines� we brie�y summarize them here in a chronological order� Cela vasans dire that these results are the core of this chapter� and will therefore be presentedin full detail here�

� In � �� the controller design method used in robot motion control to solve theoutput tracking problem for a class of underactuated Euler�Lagrange systems� wasextended to torque regulation of the induction motor� with all internal states bounded�There were no controller singularities� but exact model knowledge and full state mea�surement had to be assumed� It was also indicated how to follow sinusoidally varyingtorque references� A model representation in a dq�frame of reference was used� andthis model became the basis for later designs�

� The previous design was extended to a globally stable controller for torque regulationwithout measurements of rotor variables in � �� This globally de�ned and globallystable interlaced design of controller and observer� was the rst such result reported in


the control literature � �!�� Exact model knowledge was assumed� but it was indicatedhow to compensate for unknown rotor resistance and load torque� unfortunately underthe assumption of full state measurement� The torque reference was restricted to bebelow a certain upper limit depending on motor and controller parameters� but againit was shown how this could be avoided in the case of full state measurement� This isa foundational paper for PBC that �rst illustrated the application of this techniquefor electromechanical systems�

� Torque regulation with a globally de�ned and stable controller without measure�ments of rotor variables� was extended to torque tracking with adaptation of unknownlinearly parameterized load torque in � #��

� In �$$� the passivity�based controllers were extended to include the important caseof rotor �ux norm regulation without rotor variable measurements� The coordinateindependent properties of this approach were also rigorously explained� It followsthat PBC can be derived in any frame of reference chosen for model representation�hence clarifying some erroneous claims made in ��

� Recently� a new approach to the induction motor control problem was presented in�$�� where it was shown that global torque tracking and rotor �ux norm regulationcould be done without �ux measurement or estimation� This was accomplished by thefundamental observation that the mechanical part of the induction motor dynamicsde�nes a passive feedback around the electrical subsystem� which is also passive�Hence� instead of shaping the energy of the total system as in previous designs� thecontrol goal could be achieved by shaping only the energy of the electrical subsystem�with the mechanical subsystem as a passive disturbance� It was also shown in thispaper how to extend the controller for speed tracking with adaptation of a constantload torque�

Drawbacks of this scheme are that it is open loop in speed� and that the con�vergence rate of the speed tracking error is bounded from below by the mechanicaltime constant� relying on a positive damping of the mechanical system� However�this paper gave a �rst rigorous solution to the longstanding problem of avoiding ro�tor �ux estimates in induction motor control� still with global stability results� butunfortunately under the assumption of known parameters�

� The problems with the convergence rate and the speed controller were solved in� �� In this paper mechanical damping was injected into the closed loop by use oflinear �ltering of the speed tracking error� giving a globally stable observer�less speed�or position tracking controller with �ux regulation�

� In �$�� it was shown how the controller controller could be extended from regulationto tracking of rotor �ux norm� an important result for power e�cient operation ofinduction motor drives�

� The results cited above are speci�c to the induction motor� and it was of interest tosee if these results could be extended to other types of electric machines� An answer to

�� Outline of the rest of this chapter ��

this question was given in the seminal paper ��!�� where it is shown that passivity�based controllers can be designed for a large class of electric machines� includingsynchronous�� stepper� and reluctance motors�

� In � �� the �rst globally stable discrete�time induction motor PBC was presented�The controller was based on the exact discretetime model of a currentfed inductionmotor�

� Experimental results from the application of passivity�based controllers� have beenpresented in several publication� e�g� �#�� ! � ��

Other results along the line of PBC for electromechanical systems will be presentedin Chapters ��

C�� Other approaches

It might be tempting to include also a fourth class� consisting of all those schemesbased on other approaches in nonlinear control theory which do not �t into the aboveframework� Among these we �nd the so�called �intelligent� schemes� based on expertsystems� fuzzy logic and neural networks� Common for the schemes are that they donot provide formal proofs of stability for the resulting system� not even under theassumption of full state measurement� Such proofs are important goals for the otherthree classes� In lack of theoretical results� the issues of performance and stabilityare instead demonstrated by simulations or experimental results� An overview ofthese techniques is given in �!�� and examples of recent applications can be found in� �#� and �� Some of the problems with the derivation of proofs for stability andperformance of �intelligent� schemes� are due to their model free structure� whichalso seems to be their advantage when the model is �fuzzy� or missing� It is not yetclear if these nonlinear function approximation techniques have some advantages overthe above approaches for high�performance control of electromechanical systems witha structured model but uncertain varying parameters�

�� Outline of the rest of this chapter

The outline of the rest of this chapter is as follows� In Section the Lagrangian modelof the generalized electric machine is presented� and the control problem is de�nedfor AC machines� Some remarks to the dynamic model are also given in this section�An approach for PBC design is proposed in Section !� and in Section # a torquetracking controller is derived� In Section � we revisit the PBC of electric machinesfrom a geometric perspective� Section � presents examples of controller design forsome typical AC machines� Finally� concluding remarks to the PBC design is givenin Section $�


� Lagrangian model and control problem

Rotor windingStator windingri

ui

ri

ui

i � �� ns

'qi

Li

Vi Vi

'qi

Li

Vi voltage induced due to mutual �ux couplings

Stator �xed axisRotor �xed axis

i � ns & �� ne

�qm

qm

Figure �� Cross�section and schematic circuits of generalized electric machine�

In this section we apply the variational technique to derive the model of the general�ized rotating machine considered in ��$�� see also � �� and present the torquetracking problem� The advantages of using variational principles� particularly for thepurposes of PBC design� have already been thoroughly discussed in previous chapters�As pointed out in these chapters the model is� of course� the same one would obtainby applying �rst principles� see also Remark ��

It is widely recognized that the fundamental control problem in electrical machinesis to regulate the torque ��!�� !�� Therefore we concentrate in this chapter on PBC

�� The Euler�Lagrange equations for AC machines ��

of torque� A good behavior of the mechanical subsystem can be expected by useof a simple �e�g�� PI outer speed loop due to the fact that PBC ensures passivityin closed loop� These observations are rigorously formalized in the remaining of thechapter� and in Chapter �� where speed and%or position are actually controlled forthe induction motor� Further� in Chapter �� we show that this approach is stillapplicable for the far more challenging problem of controlling robot manipulatorsactuated by AC drives�

� The Euler Lagrange equations for AC machines

The machine consists of in all ne � ns&nr windings on stator and rotor� as shown inFig� �� Even if it is not shown in the �gure� there could also be permanent magnetsor a salient rotor in the machine�

Ideal symmetrical phases and sinusoidally distributed phase windings are assumed�The permeability of the fully laminated cores is assumed to be in�nite� and saturation�iron losses� end winding� and slot e�ects are neglected� Only linear magnetic materialsare considered� and it is further assumed that all parameters are constant and known�

Under the assumptions above� application of Gauss� law and Ampere�s law leadsto the following a�ne relationship between the �ux linkage vector � and the currentvector 'qe

� � De�qm 'qe & ��qm ��

with � � �� ne�� 'qe � � 'q�� 'qne �

� � qm � IR the mechanical angular positionof the rotor� and De�qm � D�

e �qm � the ne ne multiport inductance matrixof the windings� The vector ��qm represents the �ux linkages due to the possibleexistence of permanent magnets� Both being bounded and periodic in qm with period ��N� N � N �

If the generalized coordinates of the system are de�ned as the total amounts ofmoving electric charge that has passed any point on the di�erent phase windings� qi�i � �� ne� and the angular position of the rotor qm� the magnetic��eld coenergy�with � denoting the variable of integration can be computed as ��$��

T �e � 'qe� qm �

neXi �

Z �qi

�

�i� 'q�

id 'q�

i ��

'q�e De�qm 'qe & ��qm 'qe

and the mechanical kinetic coenergy as T �m� 'qm � �

�Dm 'q�m� where Dm � is the

rotational inertia of the rotor�

Neglecting the capacitive e�ects in the windings of the motor� and consideringa rigid shaft� the potential energy V of the system is only due to the interactionsbetween the magnetic materials in stator and rotor� i�e� V � V�qm� This energy


contribution is zero if there are magnetic materials in only one part �stator or rotor�and the reluctance properties of the other part is uniform� The Lagrangian consiststhen of the sum of the electrical

Le� 'qe� qm��

�

'q�e De�qm 'qe & ��qm 'qe � V�qm ��

and the mechanical Lagrangian

Lm� 'qm��

�

Dm 'q�m ��!

leading to

L� 'qe� 'qm� qm ��

'q�D 'q & ��qm 'qe � V�qm ��#

where we have de�ned q�� q�e � q

�m�� and D

�� diagfDe�qm� Dmg�

To model the external forces� it will be assumed that the dissipative e�ects arelinear� time�invariant and only due to the resistances in the windings ri � � i �� ne� and the mechanical viscous friction coe�cient Rm � � Therefore thecorresponding Rayleigh dissipation function takes the form

F� 'q��

�

'q�R 'q

where R�� diagfr�� rne� Rmg � �

The control forces are the voltages applied to the windings u � IRns� ns ne�In this work fully actuated as well as underactuated machines� that is� machineswhere the voltages can be applied only to stator windings �e�g� induction motor�or to both stator and rotor windings �e�g� synchronous motor with �eld windingswill be considered� Hence� it is convenient to partition the vector of generalizedelectrical coordinates as qe � �q�s � q

�r �� IRne� qs � IRns � qr � IRnr � nr � ne � ns�

where the subscripts s� r are used to denote variables related to windings with andwithout actuation respectively�� In the case of underactuated machines the partitioncoincides with stator and rotor variables as well� Notice however� that there arealso machines� like the PM synchronous� PM stepper and variable reluctance motors�where qe consists only of stator variables which are directly actuated by the statorvoltages� see the examples in Section �#� Finally� it is assumed that the load torqueL in the mechanical subsystem is of the form�

L�qm� 'qm � �c� & c� 'q�m�tanh�

'qm� ��

�See De�nition �� in Chapter ��The presence of a load torque �L of this form ensures that to every bounded � there exists a

bounded �qm� Except from this� as will be shown below� the load torque �L plays no role in thetorque tracking problem� Also� as shown in �� it can be treated as an external disturbance for thespeed tracking problem�

�� Control problem formulation ��

with a scaling parameter � � and c�� c� � IR��

With the considerations above� and by applying the EL equations � �� of Chapter� to ��#� the equations of motion of the generalized machine are derived as

De�qm-qe &W��qm 'qm 'qe &W��qm 'qm &Re 'qe � Meu ��

Dm-qm � � 'qe� qm &Rm 'qm � �L ��$

where� without loss of generality� we have assumed rs�� r� � � � � � rns and rr

��

rns�� rnr � and we have de�ned

W��qm��

dDe�qm

dqm� W��qm

��

d��qm

dqm� Re

�� diagfrsIns� rrInrg� Me

��

�Ins

�

Here is the generated torque� which as discussed in Chapter � couples the electricaland mechanical subsystems according to

��L�qm

�q� 'q

which in our case reduces to

��

'q�e W��qm 'qe &W�

� �qm 'qe & ��qm ��

where

��qm�� dV

dqm�qm

which is also bounded and periodic in qm�

Notice that� as pointed out in Chapter �� the machine is fully characterized bywhat we called the EL parameters� that is the quadruple fT �q� 'q�V�q�F� 'q�Mg�where M �

�

�Me

��

� Control problem formulation

It will be assumed here that the currents of the actuated windings 'qs� rotor positionqm� and velocity 'qm are available for measurement� Also� the basic regulated variableis taken to be the generated torque � which is however unmeasurable since it dependson the variables 'qr� Notice that the motor speed is related to torque via a simple LTIsystem ��$� The choice of torque control is rationalized in terms of passive operatorsin Section !�

The torque control problem is therefore formulated as follows�


De�nition �� Output feedback tracking problem�� Consider the ne & di�mensional machine model �� with state vector � 'q�e � qm� 'qm�

�� inputs u � IRns�regulated output � measurable outputs 'qs� qm� 'qm and smooth disturbance L � L �Find conditions onD�R� ��qm� ��qm such that� for all continuously di�erentiable ref�erence output functions � � L with known derivative '� � L � global torque track�ing with internal stability is achieved� i�e� limt� j � �j � with all internal signalsbounded� Further� for underactuated machines� asymptotic �ux amplitude trackingwill be required� that is� for a given twice di�erentiable positive and bounded function��t � � � � with known and bounded '��t� -��t� limt� j k�rk � ��t j � must hold�

�� Remarks to the model

Remark �� Lumped system and Euler�Lagrange formulation�� Magnetic andelectric �elds are distributed phenomena� which are naturally modeled with partialdi�erential equations to give a boundary value problem� This is usually done duringthe construction phase� when e�ects of di�erent materials and geometric shapes areto be studied� Even in the case of two�dimensional �elds� this results in problemswhich must be solved numerically� for instance by �nite�element analysis�

A distributed model is too complicated for control purposes� and a lumped modelis generally considered to give satisfactory results� In this chapter it is assumed thata lumped model of the system in terms of inductance and resistance matrices is al�ready available� The lumped parameters are generally derived as functions of materialconstants� turns and span of distributed windings� air�gap parameters and approxi�mations� by integration of surface�current densities over rotor and stator periphery��# � �$�� From a lumped description� the dynamical equations of motion are derivedusing the Euler�Lagrange procedure�

As discussed in � �� see also Chapter �� it could be argued that insight intothe physical process is lost when the variational approach to modeling is used insteadof basic force laws� even though the model equations are equivalent� Against this�it could be argued that physical insight is gained because coupling terms betweenvarious subsystems are derived in an analytically formal way� due to the generality ofthe method� This property has previously been exploited in numerous examples ofcontroller designs for purely mechanical systems in Chapters �� see also �� forsome recent developments� and it is also the main argument behind its use here� Inparticular� the coupling between electrical and mechanical subsystems is highlighted�and storage and dissipation functions for subsequent PBC design are easily obtained�These properties are obscured if the EL equations are translated into a state�spaceformulation of the dynamical equations�

Remark �� Voltage balance equations�� As pointed out in Chapter � we canalternatively represent the electrical part of an electromechanical system in terms of

�� Remarks to the model ��

�uxes �� and currents �which are the generalized electrical momenta and velocitiesof the Hamiltonian formalism � �� as

'� &Re 'qe � Meu ��

This is a voltage balance equation� Particularly useful for further developments is thefollowing relationship between rotor �uxes and rotor currents for motors where therotor windings are short circuited �e�g� induction motors

'�r &Rr 'qr � ��

where �� s � �

�r �� with Rs � rsIns � IRns�ns� Rr � rrInr � IRnr�nr � The

importance of �� is that it de�nes a dynamic relationship -qe � f�qe� 'qe� qm� 'qmthat is invariant with respect to the control action that will have to taken into accountwhen de�ning a �desired behavior� for the machine� This relationship will play afundamental role in the explicit derivation of the PBC as explained in Sections # and��

Remark �� Ignorable coordinates�� An interesting property of the machineand other magnetic �eld devices in which currents are of main interest� is that theelectrical charges are ignorable � �� also known as cyclic in mechanics �� Thatis� the Lagrangian of the system does not contain qe� although it contains the corre�sponding currents 'qe� It must be pointed out that when choosing the form � �� ofChapter � of the EL equations� it is crucial that the currents be expressed in theirnatural frames� where electrical charges can be obtained by integration of currents�to avoid the introduction of quasi coordinates� See also � ��

Remark �� Mechanical commutation�� For a class of machines with mechan�ical commutation� the relation between the physically applied port currents and therotor currents introduces non�holonomic constraints � �� and the dynamic equa�tions cannot be obtained directly from � �� with the given Lagrangian� Insteadquasi coordinates could be introduced� and the dynamic equations derived by use ofthe Boltzmann�Hamel � �� or Gaponov �� form of the EL equations� These proce�dures are however quite involved� and the dynamic equation for this class of machinesis therefore usually not derived from variational principles� but by the use of basicforce laws as Faraday�s law� Ohm�s law and Euler�s law� As pointed out in Section ��motors with mechanical commutation are of less interest for nonlinear control design�and will not be considered in this text�

Remark �� Parameters�� It is well known that the lumped parameters are notconstant� but that they change due to temperature variations� skin e�ects� currentdisplacement and magnetic hysteresis and saturation� In some cases� like for theswitched reluctance machine� magnetic nonlinearities must be taken into account inthe modeling procedure for satisfactory performance� In other cases the assumption


of linear magnetics leads to controller designs which works satisfactory� at least ifthe machine is forced to operate at �ux levels below the saturation limit� A controllerdesign based on a model including magnetic saturation is however highly desirablefor performance improvements� For instance� while older uniform air�gap inductionmotors often were robustly designed to operate under linear magnetic conditions�constant inductances� there is now an increasing interest of operating these undermagnetic saturation �current%�ux dependent inductances� saving cost and weight�

The EL procedure is based on energy properties of the system� and the dynamicequations for machines with nonlinear magnetics can also be derived by this proce�dure� However� it is believed that the simpler problems of constant parameters andlinear magnetics must �rst be rigorously solved using passivity�based methods� beforethe formal framework can be used to incorporate and solve the more complex prob�lems stated above� Hence� magnetic nonlinearities� time�varying resistances� or statedependent inductances will not be considered in this text� However� some robustnessresults to uncertain parameters� as well as tuning and adaptation rules for the con�troller gains are presented in Chapter �� For the induction motors case� work alongthe line of extending the PBC to handle nonlinear magnetics has been reported in��

Remark �� Complex versus real notation�� In this book a real representationof the machine�s model is used� meaning that rotations of vectors will be presentedby matrix exponentials� allowing for easy manipulation of the equations� This is acommon approach which have been widely used for design of nonlinear controllers� atleast within the control theory community� The complex notation� which is closelyrelated to the transfer function language� has been widely used for analysis of station�ary operation and presentation of two�axis theory using space phasors� but has notbecome the common approach for the application of recent nonlinear control theoryto electric machines� Of course� the two notations are equivalent�� and nonlinearanalysis could be done in either of the two settings� Recent results from nonlinearcontrol and analysis of induction motors reported in � �#� and ��$�� have taken ad�vantages of using a complex formulation of the induction motor model because of itscompactness�

Remark � �Load torque�� The model in �� is su�ciently general to includemechanical friction� windage� and pump or compressor loads� The scaling of thehyperbolic tangent function is done to mimic the signum function� replacing thediscontinuity in the friction model with a curve of �nite slope� This is done to avoidintroduction of additional problems with di�erential equations having discontinuousright hand sides in the stability analysis� However� this model does not provide a true

�To state it more mathematically� this stems from the fact that the metric space of complexnumbers is homeomorphic to the metric space of matrices on the form �I� � �J � where �� IRand J is a skew�symmetric matrix� i�e� there exists a continuous and invertible mapping from onespace to the other� with a continuous inverse�


stiction mode� something which may be important if the period of the stick�slip limitcycle is long �� The issue of adaptive friction compensation is addressed in Chapter�� Also� in Chapter �� we address the far more challenging problem of controllingrobot manipulators with AC drives�

�� Examples

In ��$� several examples of electric machines modeled by �� $ are given� Inthis section two examples of fully actuated machines� i�e� machines with ns � ne andMe � Ine � are presented�

Example � �PM synchronous motor�� The !� PM synchronous motor ��# �has ne � ! and the parameters

De�qm �

�� Lls & LA � LB cos npqm ��

�LA � LB cos �npqm � �

�

��LA � LB cos �npqm � �

� Lls & LA � LB cos �npqm � ��

�

��LA � LB cos �npqm & �

� ��

�LA � LB cos �npqm & �


�


Lls & LA � LB cos �npqm & ��

��

��qm � �m

�� sinnpqm

sin�npqm � ��

sin�npqm & ��

��

where Lls� LA� LB are inductance parameters� np is the number of pole pairs and �mis the amplitude of the �ux linkage established by the permanent magnet� �

Example �� PM stepper motor�� The � PM stepper motor has ne � andthe following parameters � ��

De �

�L L

��!

��qm �Km

Nr

�cos�Nrqmsin�Nrqm

��#

L is the self�inductance of each winding� and Km is the torque constant� Nr is thenumber of rotor teeth of same polarity� In this case the torque has a term due to theinteraction between the permanent magnet and the magnetic material in the stator�detent torque� and ��qm � �KD sin�#Nrqm� KD � �� ) of Kmi�� where i� isa rated current� �

� Ch� �� Generalized AC motor

� A passivity�based approach for controller design

We have seen already in the simple levitating ball example of Chapter that forelectromechanical systems the PBC approach can be applied in at least two di�erentways� which lead to di�erent controllers� In the �rst� more direct form� a PBCis designed for the whole electromechanical system using as storage function thetotal energy of the full system� Another way to design PBC is to �rst decomposethe system into its electrical and mechanical dynamics� Then� observing that themechanical dynamics can be treated as a �passive disturbance�� we design a PBConly for the electrical subsystem using as storage function the electrical total energy�One important advantage of the second feedbackdecomposition approach is that itleads to simpler controllers� which as will be shown below do not require observers�For these reasons we follow in this chapter the second route to solve the torquetracking problem posed above� In Section ��# we illustrate how the �rst approachcan be applied to the induction machine� with the addition of an observer�

�� Passive subsystems feedback decomposition

In this subsection it is shown that the dynamic model of the generalized electricmachine satis�es the conditions of Proposition �� in Chapter �� and consequentlyit can be decomposed into the feedback interconnection of two passive subsystems�This result formalizes our approach that concentrates in the electrical dynamics andtreats the mechanical subsystem as a �passive disturbance��

�

��

�

�

� �.e

'qs

L.m

'qm

u

Figure �� Passive subsystems decomposition of generalized electric machine�

Proposition �� Passive subsystems decomposition�� The system �� can be represented as the negative feedback interconnection of two passive subsystems

�� Design procedure �

�see Fig� ��

.e � Lns��e � Lns��

�e �

�u� 'qm

��

�'qs

�

.m � L�e � L�e � � � L �� 'qm

�

Proof� The proof is a corollary of Proposition �� noting that the Lagrangian ofthe electric machine ��# can be decomposed into L�q� 'q � Le� 'qe� qm&Lm� 'qm withLe� 'qe� qm� Lm� 'qm as de�ned in �� !� respectively�

�

�� Design procedure

The rationale of the design stems from the passive subsystems decomposition givenabove and� �disregarding� the mechanical dynamics� attempts to control the gener�ated torque by imposing a desired value to the currents 'qe� There are thereforethree natural steps to follow�

�� Apply the passive subsystems decomposition of Section !�� to the machine� toview .e as the �system to be controlled�� and .m as a passive disturbance� Toensure the latter does not �destroy� the stability of the loop damping must beinjected to .e to strengthen its passivity property to strict passivity�

� De�ne a set of attainable� currents 'qed� i�e� currents for which it is possible to�nd a control law that ensures limt� k 'qe � 'qedk � � To this end� the energyof the closed loop must be shaped to match a desired energy �storage function�

which is chosen here as Hed��

�'�q�e De�qm '�qe� with current error de�ned as

'�qe�� 'qe � 'qed ��

!� Among the attainable currents choose 'qed to deliver the desired reference torque�� that is� such that if 'qe � 'qed then � �� Finally� give conditions underwhich limt� k 'qe � 'qedk � implies limt� j � �j � with internal stability�

A globally stable torque tracking controller

In Section we presented the model of a generalized electric AC machine and formu�lated the torque tracking problem� In this section we identify a subclass of this model


for which we can design a globally stable torque tracking PBC using the feedbackdecomposition approach advanced in Section !�� As explained in the previous chap�ters the PBC is �rst expressed in an implicit form involving the controller states andits derivatives� the control signal and the external references� In a second stage wemust express these equations explicitly to get an implementable form of the controller�It is at this stage that restrictions on the admissible machines will be imposed� Fromthe analytic point of view these conditions boil down to an invertibility assumption�Interestingly enough this assumption is satis�ed if the machine satis�es a decouplingcondition called BlondelPark �BP transformability which is well�known in theelectric machines literature� We proceed along the BP transformability route� whichis more familiar to practitioners� in the �rst derivation of the controller� and thengive an interpretation of this condition in terms of the geometric concepts of relativedegree and zero dynamics� which are wellknown to control theorists�

�� Strict passi�ability via damping injection

The �rst step of our design procedure� the decomposition of the model into passivesubsystems .e� .m� can be carried out as described in Section !�� In this section itis explained how damping is injected to .e such that the mapping from control inputto measurable output is output strictly passive�

Proposition �� Consider the electrical subsystem .e described by �� Assume

A�� The rotor resistance is positive� that is�� Rr � rrInr � �

A�� The nr nrdimensional � � block of the matrix W��qm �dDe�qm�dqm

is zero�i�e�

W��qm �

��W��qm�� W��qm��W��qm�� nr�nr

�

A�� The non�actuated rotor components of the vector ��qm are independent ofqm� that is

W��qm �d��qm

dqm

��

�W�s�qm

�� W�s�qm � IRns

Under these conditions there is an output feedback of the form

u � v &W�s�qm 'qm �K��qm� 'qm 'qs ��

such that the mapping v �� 'qs is output strictly passive for all qm� 'qm � L�e� �

��This condition is obviously always satis�ed in physical machines� It is stated here as an �as�sumption� to underscore the importance of having a good estimate of Rr� as discussed in Remark��

�� Strict passi�ability via damping injection ��

Proof� The dynamics of .e is described by �� which is repeated here for ease ofreference

De�qm-qe &W��qm 'qm 'qe &W��qm 'qm &Re 'qe � Meu

Closing the loop with �� results in

De�qm-qe & Ce�qm� 'qm 'qe & Res�qm� 'qm 'qe � Mev ��$

where the matrices Ce�qm� 'qm and Res�qm� 'qm have been de�ned as

Ce�qm� 'qm��

�

W��qm 'qm

Res�qm� 'qm�� Re &

�

W��qm 'qm &

�K��qm� 'qm

��

Taking the time derivative of electric part of the total energy of .e� that is He ��'q�e De�qm 'qe� along the trajectories of .e gives

'He � 'q�s v � 'q�e Res�qm� 'qm 'qe

Now� let

K��qm� 'qm � K�� qm� 'qm

� supqm� �qm

f 'q�m

#�W��qm��R

��r �W��qm

��

�

�W��qm�� 'qmg ��

Then� by use of standard matrix results �see Section # in AppendixD� the symmetricmatrix Res�qm� 'qm can be shown to be uniformly positive de�nite in the sense that

infqm� �qm

��Res�qm� 'qm � � � ��

Integration of 'He completes the proof� �

A Remarks to conditions for damping injection

Remark �� Notice that strict passivity is achieved� via the nonlinear dampingtermK��qm� 'qm 'qs� which recovers the positivity of the �damping� matrixRes�qm� 'qm�

Remark �� In the case of full actuation� i�e� ns � ne� the damping matrix can bea full matrix� and the required positivity of Res�qm� 'qm is guaranteed if �see ��

K��qm� 'qm � K�� qm� 'qm � �Re � �

W��qm 'qm ��


Remark �� Assumption A�� is a reasonable condition of damping of the un�actuated dynamics which is satis�ed in all electric machines� The problem is withcondition �� which shows that to overcome the imprecise knowledge of the rotorresistances� high gains will have to be injected into the loop�

Remark �� A�� is a decoupled dynamics condition equivalent to requiring thatthe contribution to the magnetic coenergy of the terms quadratic in 'qr must beindependent of qm� Physically� this translates into the condition that if there are rotorwindings� then the rotor �ux induced by the rotor currents must be independent of therotor position� This means that the stator must have uniform reluctance properties�non�salient and of uniform magnetic material� This assumption is satis�ed by manymachines� e�g� classical Park � � � and polyphase machines� � ��

Remark �� Since the torque �� consists of one component due to the currents�and the other of purely magnetical origin� and since there is no control on the �eldsfrom the permanent magnets� it is reasonable to expect that the e�ect on .e of the �uxlinkages due to the permanent magnets must be eliminated� This explains the needfor Assumption A�� Physically� this assumption also means that if the machinehas rotor windings� then the stator must have uniform reluctance properties i�e�� ifthe machine has permanent magnets� then they can only be placed on the rotor� Ascan be seen from �� the term from the permanent magnets must be cancelledout� The need for this cancellation is a drawback of the scheme� However� the termis generally a vector with periodic functions of a measurable quantity �position� andproportional to a constant which can be precisely identi�ed�

�� Current tracking via energy�shaping

The �attainable� currents 'qed in the second step of the design procedure are now tobe de�ned�

Proposition �� If in �� v� 'qed � Lne satisfy

Mev � De�qm-qed & Ce�qm� 'qm 'qed &Res�qm� 'qm 'qed ��

then �see �� for the de�nition of the current error '�qe � as t�� independentlyof qm� 'qm and the choice of 'qed� Furthermore� when 'qed is bounded then 'qm� 'qe and are also bounded� In addition� boundedness of -qed ensures that -qe and v are alsobounded� �

Proof� Rewriting ��$ in terms of the error signals gives

De�qm-�qe & Ce�qm� 'qm '�qe &Res�qm� 'qm '�qe � � �� !

with

�� Mev � �De�qm-qed & Ce�qm� 'qm 'qed &Res�qm� 'qm 'qed� �� #

�� Current tracking via energy�shaping ��

Then �� implies � � � and the dynamics of the system is fully described by

De�qm-�qe & Ce�qm� 'qm '�qe &Res�qm� 'qm '�qe � ��

Dm-qm &Rm 'qm � � L ��

These equations are locally Lipschitz in state� and under the assumptions on thedesired torque and load torque� they are continuous in t� so there exists t� � suchthat in the time interval � � t� the solutions exists and are unique�

Taking the time derivative of the desired energy function Hed � ��'�q�e De�qm '�qe

along the trajectories of �� ! results in

'Hed � � '�q�e Res�qm� 'qm '�qe� �t � � � t� �� $

It follows from �� $ and the proof of Proposition �� that

'Hed � � '�q�e Res�qm� 'qm '�qe ��k '�qek�� t � � � t�

and it can be concluded that

k '�qe�tk mek '�qe� ke��et� �t � � � t� ��

holds with me �q

��De�qm��De�qm��

and �e ��

��De�qm�� where � is de�ned in �� These

two constants are independent of t��

From this� and since 'qed � Lne � it can be deduced that 'qe is bounded on the openinterval � � t�� Now� it must be proved that it remains bounded also on the closedinterval � � t�� To this end� notice that the right hand side of �� is also boundedon � � t�� thus its solution can not grow faster than an exponential� and consequently'qm� qm remain bounded �t � � � t�� This in its turn ensures the boundedness of Res�and consequently k '�qek cannot escape to in�nity in this time interval�

Since me� �e and � are independent of t�� it is possible to repeat this argument fora new initial condition� to de�ne solutions on the time interval �t�� t�� This showsthat it is possible to extend this procedure to prove existence of solutions for thewhole real axis� and the system cannot have �nite escape time� It follows from �� that

limt�

'�qe �

and it can be concluded that 'qe is bounded� which implies that is also bounded�From this� and the de�nition of the load torque �� it follows that 'qm remainsbounded� Now� it follows from �� and �� # that -qe and v will be bounded if -qedis also bounded� �


�� From current tracking to torque tracking

It is now convenient to take a brief respite and recapitulate the previous derivations�The �rst step of the design procedure described in Section !� was carried out inSection #�� where an inner control loop was designed to ensure that ��$ de�nesa strictly passive mapping v �� 'qs� In Section #� the second step was carried out�that is� a relationship between the control signals v and 'qed �� which implieslimt� k 'qe � 'qedk � � was established� The third step� to which this section isdevoted� demands the de�nition of 'qed from the �attainable� set that delivers the de�sired torque� and the establishment of conditions under which current tracking impliestorque tracking� Notice that these steps are not straightforward since De� Ce� Res

in �� and in �� depend on qm and 'qm� thus some additional conditions onthe couplings between the subsystems must be satis�ed� These conditions are ex�pressed in terms of restrictions on the parameters De�qm� Re�qm� 'qm� ��qm� ��qm ofthe generalized electric machine model�

A Desired current behavior

Motivated by �� it is proposed to de�ne 'qed such that for the given reference torque�� the equation

� ��

'q�edW��qm 'qed &W��qm 'qed & ��qm ��

holds� This gives� using ��

� � ��

'�q�e W��qm '�qe & '�q

�e W��qm 'qed &W�

� �qm '�qe

with the error signal '�qe�� 'qe � 'qed� Since W��qm and W��qm are bounded� it

follows that asymptotic torque tracking will be achieved if limt� k 'qe � 'qedk � �with 'qed � Lne � can be ensured�

� The PBC is implicitely characterized by �� and �� To attain the torquetracking objective we must now solve these equations explicitely and further�more ensure that the resulting 'qed is bounded�

Towards this end� �rst notice that in the case of fully actuated machines Me � Ine �thus there are no restrictions on the set of �attainable� currents� That is� in this case�for any given -qed� 'qed we simply plug in these functions in �� to calculate v �hencemaking � � � The only remaining question is then the solution of �� for a given�� '�� In Section � we show the procedure for the synchronous and the PM steppermotors�

�� From current tracking to torque tracking ��

Now� in the more di�cult case of underactuated machines there are not enoughcontrol actions to directly set � � for any given 'qed� It will be shown in the sequelthat the well known BP transformation conditions which are fundamental in theanalysis of rotating machines ��$� �$�� will allow us to get an explicit expression for�� and ��

B Decoupling conditions

The following de�nition is in order�

De�nition �� BP transformable machine�� The machine �� is BPtransformable if there exists a coordinate transformation for the current of the form

'ze�qm� 'qe � P �qm 'qe � P� e�Uqm 'qe ��!

such that the dynamics of .e in these coordinates are independent of qm �but stilldependent on 'qm � P� is any nonsingular constant matrix� If furthermore the matrixU is of the form

U �

� U��

�� U� �

� �U��

��

�� SS�ne

then the machine is strongly BP transformable�

5From the structure of the matrix U above it can be seen that in strongly BPtransformable machines we can remove the dependence on qm of .e by rotating onlythe rotor variables� As will become clear later� this condition is needed when the rotorcircuits are not actuated� as in the induction motor case�

In the fundamental paper ��$� necessary and su�cient conditions for BP trans�formability are given� Since the de�nition of the BP transformation given above isslightly di�erent from the one given in ��$�� and for the sake of selfcontainment� asimpli�ed version of their theorem is given below� A proof can be found in Section !of Appendix D�

Proposition �� If there exists a constant matrix U � IRne�ne such that

UDe�qm�De�qmU � W��qm ��!�

ReU � URe ��!

UW��qm �dW��qm

dqm��!!

then the machine �� is BP transformable� In this case the dynamics of .e

�see �� in the coordinates 'ze is described by

De� P�� -ze & UDe� P

�� 'qm 'ze &W�� 'qm &ReP

�� 'ze � e �UqmMeu � Meu

� ��!#


while the dynamics of .m �see �� is described by

Dm-qm &Rm 'qm � � L ��!�

� 'z�e P�� UDe� P

�� 'ze &W�

� � P�� 'ze & ��qm

�

Example �� Park�s transformation�� For a class of electric machines with in�ductance matrix as in �� the BP transformation to the dq �frame is given as��# �

P �qm � p�

�� cos�npqm cos�npqm � ��

� cos�npqm & ��

�

sin�npqm sin�npqm � �� sin�npqm & ��

�

��

��

��

�� !�

where p� � �! �orp

�! is a constant� This transformation can also be written as

P �qm � P� e�Uqm

where

U �npp!

��

� ��

�� P� � p�

��

��

�

�p��

p��

��

��

��

��

and U satis�es ��!��!!� The inverse transformation for p� � �! is given as

P��qm � e UqmP��

�� cos�npqm sin�npqm �

cos�npqm � �� sin�npqm � ��

� �

cos�npqm & �� sin�npqm & ��

� �

�� !$

Several slightly di�erent forms can be found in the literature� �� This stems fromthe choice of the constant factor p�� which is sometimes chosen to preserve power inthe transformed phases �p� � �!� or with the objective of making P� orthogonal

�p� �q

��

C Remarks to the BP transformation

Remark �� For the purpose of the present work� the key feature of BP trans�formable machines is that it allows us to get� via a kind of system inversion� anexplicit expression for the PBC �� and �� Notice also that� for constantspeed� the electrical subsystem in ��!# is linear and time�invariant when u� is takenas the new input� This fundamental property has been exploited in the literature todetermine stability properties in stationary operation � $��

�� PBC for electric machines ��

Remark �� The underlying fundamental assumption for the machine to be BPtransformable� is that the windings are sinusoidally distributed � � �� giving a sinu�soidal air�gap magnetomotive force �MMF and sinusoidally varying elements in theinductance matrixDe�qm� For a practical machine� this means that the �rst harmonicin a Fourier approximation of the MMF must give a su�ciently close approximation ofthe real MMF� Examples of machines in which higher order harmonics must be takeninto account� are the square wave brushless DC motors in ��#�� and machines withsigni�cant saliency in the air gap � �!�� The squirrel�cage induction machine is anexample of a machine where the squirrel�cage rotor with non�sinusoidally distributedMMF is replaced by an equivalent �ctitious sinusoidally wound rotor for analyticalpurposes� without introducing detrimental e�ects to controller design�

Remark �� It is interesting to remark that the BP transformation can not bederived from a canonical transformation ��$� z � Z�q of the generalized coordinatesand momenta� This fact is presented in Section ! of Appendix D�

Remark �� Since the matrix U is real and skew�symmetric� it follows that e �Uqm

is an orthogonal transformation� and the transformation P �qm is bounded�

Remark �� It is worth to point out that the passivity properties� being input�output properties� are invariant under a change of coordinates on the tangent bundleof the con�guration manifold� hence they are preserved for the transformed systems.e and .m� This can be proved by evaluating the time derivate of

H� 'ze� qm ��

'z�e P

�� De� P

�� 'ze & V�qm

along the trajectories of ��!#� and from the fact that the transfer function between � L and 'qm is positive real� and the mapping is passive� Thus it is possible todesign PBC also for the transformed system as done in � �� for the case of inductionmotors� See also �$$� for further discussion in this respect�

�� PBC for electric machines

The property of BP transformability will now be related with the problem of realiz�ability of the implicit PBC �� For the sake of clarity we discuss separatelyunderactuated �ns � ne and fully actuated machines �ns � ne�

A Underactuated machines� ns � ne

Proposition �� Desired currents for underactuated machines�� Assume thatthe machine �� is strongly BP transformable� ��qm � ��qm � � the � � �block of De�qm is nonsingular� and that ��t is a bounded strictly positive twice

� Ch� �� Generalized AC motor

di�erentiable function with known �rst and second order derivatives� Under theseconditions� the following de�nition of 'qed satis�es �� and �� for any given �

'qed �

�� 'qsd

'qrd

��

��

�De�qm��

hInr & �De�qm��

n�

��t��U��

�� &��t��t�

R��roi

�rd

�h

��t�

�U�� &

��t��t�

R��ri�rd

��

��!�

where �rd is the solution of the di�erential equation

'�rd ��

��t

h�RrU

�� & '��t��t

i�rd ��!�

with initial conditions such that k�rd� k � �� Furthermore�

k�rd�tk � ��t� �t �

�

Proof� The last statement of the proposition follows immediately by taking thetime derivate of �

�k�rdk�� substituting ��!�� and using the fact that strong BP trans�

formability implies RrU�� & �RrU

��

� � �

To simplify the notation of the rest of the proof� it is convenient to introduce thedesired �ux �d � ��sd� �

�rd�

� as

�d�� De�qm 'qed ��#

Some simple calculations using �� #� �� and ��# show that��

�r � � '�rd &Rr 'qrd � ��#�

Now� notice that BP transformability of the machine implies that �� can berewritten as �see ��!�

� � 'q�edUDe�qm 'qed

which� in terms of the desired �uxes and currents looks like

� � 'q�edU�d

If further the machine is strongly BP transformable then

� � 'q�rdU��rd

� � '��rdR��r U��rd ��#

��See �� and Remark �� in Section �� for the physical motivation behind this choice of relationsbetween desired uxes and currents�

�� PBC for electric machines �

where �r � and ��#� has been used in the last equation� From this it can beseen that for �� to hold� '�rd must be de�ned such that ��# always holds� Itis straightforward to verify that this is the case when '�rd is de�ned as in ��!��using U��

�� U�� see De�nition �� the symmetry of Rr and the fact that

k�rd�tk � ��t�

The proof is completed using ��#� to obtain 'qrd and the de�nition of �d tocalculate 'qsd�

�

The main result for underactuated machines is contained in the proposition below�

Proposition �� PBC for underactuated machines�� Consider the machinemodel �� Assume that the machine is strongly BP transformable �De�nition� � ��qm � � ��qm � � �De�qm�� is nonsingular andA��A�� of Proposition�� hold� Under these conditions� there exists a dynamic output feedback controllerthat ensures global asymptotic torque tracking with internal stability� Furthermore�for all ��t �strictly positive bounded twice di�erentiable with known bounded �rstand second order derivatives the rotor �ux �r satis�es limt� j k�rk � ��t j � �

�

Proof� The control law is obtained from �� and �� #� setting �s � � To thisend� the de�nition of 'qed in Proposition �� $ is used� Notice that -qed is bounded� andcan be computed from the available measurements provided '� is known�

Convergence of '�qe � follows from the arguments of Section #� � Boundedness of'qed follows from ��!� and the boundedness of �rd and �� This establishes asymptotictorque tracking�

Electrical rotor �ux norm tracking is a consequence of the convergence of thecurrents to their desired values and ��t � k�rdk� since

�rd � �r � �De�qm�� 'qsd � 'qs & �De�qm�� 'qrd � 'qr

and De�qm is bounded�

�

B Fully actuated machines� ns � ne

For fully actuated machines Me � Ine � and as previously explained� � � can beobtained by a suitable selection of v for given 'qed and -qed� The main di�culty is to�nd 'qed such that �� is satis�ed� This is done by choosing the desired currentsfrom the BP transformed torque equation� since the matrices relating the transformedcurrents and the torque are no longer dependent on qm� which considerably simpli�esthe choice�


Proposition �� PBC for fully actuated machines�� Consider the machine model�� and �� Assume the machine is BP transformable �De�nition � � andA��A�� of Proposition �� hold� Let the desired currents and their derivativesbe de�ned as

'qed � e UqmP�� 'zed

-qed � U e UqmP�� 'qm 'zed & e UqmP��

� -zed

where 'zed is chosen to satisfy

� � � � 'z�edP�� UDe� P

�� 'zed &W�

� � P�� 'zed ��#!

with 'zed� -zed � Lne � Under these conditions� use of the dynamic output feedbackcontroller de�ned in �� with

v � De�qm-qed & Ce�qm� 'qm 'qed &Res�qm� 'qm 'qed ��##

will ensure global asymptotic torque tracking with internal stability� �

Proof� The expression for the torque in the transformed system is� according to��!�

� � � 'z�e P�� UDe� P

�� 'ze &W�

� � P�� 'ze

Setting '�ze � 'ze � 'zed and using ��#! gives

� � � '�z�e P

�� UDe� P

��

'�ze & '�z�e P

�� UDe� P

�� 'zed &W�� P��

�'�ze

Since '�ze � P �qm '�qe� and P �qm is a bounded transformation� it follows that

limt�

'�qe � � limt�

'�ze �

�limt� � �

It is clear that 'zed � Lne � 'qed � Lne � and -zed � Lne � 'qm � L � -qed � Lne � De�ningv as in ��## gives � � � and the arguments of Section #� hold� �

C Remarks to the controllers

Remark �� As can be seen from ��!� there is a need for the derivative of thetorque reference for the computation of -qed� This may in its turn lead to the need of-qm� if the speed 'qm is used in an outer loop controller� It will be explained in Section��! and Chapter �� how this need for acceleration measurement can be avoided byuse of linear �lters or nonlinear observers�

�� PBC for electric machines ��

Remark �� Notice that the assumption in Proposition �� $ of the � �� block ofDe being nonsingular implies that the number of actuated and nonactuated windingsmust be equal� This is the case for typical induction motors� where this matrix is anonsingular rotation matrix�

Remark �� Equation ��!� has a solution�� of the form

�rd�t ��t��

e U�� d�t��rd� � '�d�t �rr

��t��t� �d� �

This gives an interpretation of the desired �ux in terms of its rotation angle� whosespeed �the desired slip is related to the desired torque�

Remark �� During the derivation of the model and controller� it has been assumedthat a VSI has been used to generate the inputs to the actuated windings� If theinverter used is a CSI or a VSI with fast current control� it follows that the inputs tothe actuated windings will be the currents 'qs � 'qsd� where 'qsd is the vector of desiredcurrents for the actuated windings� as de�ned in the previous sections� Given theirgreat practical importance these socalled currentfed machines will be studied ingreat detail in Chapter ��

Remark �� The quadratic form in ��#! is in general not easy to solve for thecomponents of 'zed� Examples of solutions for certain machines are given in Section ��

Remark �� For !� machines the currents of the transformed system are usuallydenoted with subscripts d �direct�axis component� q �quadrature�axis componentand �zero�sequence component� For machines in which the symmetrical windingshas an isolated neutral� the zero sequence of the transformed currents is exactly zero�which de�nes a natural choice for desired value of this current�

Remark �� An interesting connection between the PBC described above and the�eld�oriented approach� that we brie�y review in Section �� may be established asfollows� The transformed torque equation is generally given as

� cf�d 'zq � �q 'zdg

where c is a constant� and �d� �q are d and q components of the transformed �uxvector� If �q 'zd can be made equal to zero and �d is constant� it will be possibleto control the torque by specifying 'zq� This is the basic idea of the �eld�orientedapproach� Notice that in this case the angle of the transformation is known� andthere is no need to estimate it�

��Recall that if the matrices A�t� andR tA�s�ds commute� then the di�erential equation �x�t� �

A�t�x�t�� x�� x t � �� has the solution x�t� � e

Rt

�A�s�dsx� see pp� �� in ��


PBC of underactuated electrical machines revis�

ited

In this section we revisit the PBC derived above for underactuated electrical machinesfrom the perspective of systems invertibility� a concept which is well understood inthe geometric formulation of nonlinear control theory �� $�� In particular� weshow that the BP transformability assumption needed for the realizability of thePBC controller is akin but not equivalent to an invertibility condition� For ease ofpresentation we will restrict ourselves here to the case when there are no permanentmagnets� hence ��qm � ��qm � V�qm � � Furthermore� we choose � to beconstant�

�� Realization of the PBC via BP transformability

For the purposes of this section it is convenient to express the electrical dynamics.e of the machine �� and the torque �� in terms of the �uxes de�ned as ��Hence� recalling �� of Remark ��! we can write �� and �� as

'�&ReD��e �qm� � Meu ��#�

��

��C��qm�


C��qm�� D��

e �qmW��qmD��e �qm

The PBC of Proposition �� can be written in the form

Meu � '�d &ReD��e �qm�d &

�K��qm� 'qm

�D��e �qm�� #�

d ��

��d C��qm�d ��#$

where �� d�

The �rst equation of the PBC ��#� is a �copy� of the motor dynamics ��#� withan additional damping injection term needed to get the strict passivity� The secondequation ��#$ is a constraint that �clamps� the controller dynamics� As explainedin Section #�!� the motivation for this constraint stems from the fact that the output

�� Realization of the PBC via BP transformability ��

to be controlled� that is � can be written in terms of the errors as

��

��C��qm�

��

��& �d

�C��qm��& �d

��

��d C��qm�d &

�

��C��qm��& �d

� � &�

��C��qm��& �d

where we have replaced ��#$ to get the last identity� Noting that C��qm is bounded�and recalling that �� we see from the expression above that � � provided �dis bounded�

The PBC ��#�� #$ is given in implicit form� to get an explicit realizationsome assumptions on the system .e are needed� a stage that we called �from currenttracking to torque tracking� in Section #�! above� To solve the problem we imposedthe condition of BP transformability of the motor �De�nition �� which is a condi�tion on De�qm and Rr that we use to reduce the expression for � to the form ��# �Applied to the machine itself we get

� � '��r R��r U��r

where U�� is skewsymmetric� This is a key property since it allows us to explicitlysolve this equation as

'�r �

k�rk�RrU�� r ��#�

�Recall that the BP transformability condition ensures� via ��! � that Rr and U��

commute� Applying these derivations to the controller equations ��#�� #$� weobtain the controller dynamics ��!�� which for the case '� � reduces to

'�rd ��RrU

�� rd

where we recall that � and � are the reference values for k�rk and � respectively�In this case� boundedness of �rd is guaranteed� because k�rd�tk � �� The calcula�tion of the controller equations proceeds then as follows� Once we have the explicitexpressions for �rd we can calculate 'qrd from the last ne� equations of ��#�� Fromhere� and �d � De�qm 'qed� we can calculate �sd and� upon di�erentiation� obtain '�dwhich we replace in the remaining controller equations ��#� to get u�

In summary what we needed to solve the controller equations for the machines wasan invertibility assumption� Indeed� the �output� equation � h�� qm was �rstexpressed as � h��r� '�r� and then we inverted this function to get '�r � h��r� �where hi�� i � �� ! are suitably de�ned functions�


Remark �� Periodic zero dynamics�� The equation ��#� shows that for thisclass of machines� if we choose as outputs and k�rk� the zero dynamics are periodic�See also Remark ��! � We will comment further on this property for the case ofinduction machines in the next chapter�

�� A geometric perspective

The invertibility condition above is akin to the condition of full rank of the decouplingmatrix in standard geometric control �� Actually we can approach the problem ofrealization of our PBC from that perspective� First� we need to complete an outputvector

y � h�� qm � �� k�rk� h�� qm� � � � � hns�� qm�� #�

with respect to which the system ��#� has a wellde�ned relative degree �r�� rns�Then� the system must be transformed to the normal form

'� � q�� Y �k�� qm & p�� Y �k�� qmu

y�ri�i � ai�� Y

�k�� qm & b�i �� Y�k�� qmu

where Y �k� denotes a vector containing as many derivatives of the output as needed�The relative degree condition ensures that the matrix

B�� Y �k�� qm��

�� b�� Y

�k�� qm��

b�ns�� Y�k�� qm

��

is� at least locally invertible� The PBC will then be de�ned as

'�d � q��d� Y�k�� qm & p��d� Y

�k�� qmu

u � B�� Y �k�� qm

!B"

�� y

�r��

y�rns�ns�

��

�� a�� d� Y

�k�� qm��

a�ns��d� Y�k�� qm

��#CA &

&�K��qm� 'qm

D��e �qm��

for which we have �xed the output and its derivatives to the reference values y�i� � y�i��

carried the inversion from the normal form� and added the damping injection�

The requirement of boundedness of the controller state �d will then be relatedwith the stability of the zero dynamics for the given reference outputs� Even thoughthis approach is more systematic� it requires the calculation of the normal form� astep which is usually far from obvious� Furthermore� it is not clear how to de�nethe additional outputs to obtain a square system� It is interesting to note that

� Examples ��

we have been able to avoid these computations invoking the practically reasonableassumption of BP transformability� One interesting application of this geometricderivation of PBC could be for the extension of these schemes to machines which arenot BP�transformable�

� Examples

In this section controllers are derived for some common fully actuated machines�Controller design for underactuated machines will be the issue of subsequent chapters�where the squirrel�cage induction motor will be studied�

Example �� Synchronous motors�� In the last years� synchronous motors� andin particular permanent magnet motors have become attractive alternatives to induc�tion motors in the low to medium power range �!$�� These machines are generallymore expensive than induction motors� but have higher e�ciency due to the fact thatthe rotor losses are negligible� This results in reduced size and cooling problems ascompared to induction motors� As low price high�energy permanent magnets becomeavailable� the market for these machines will increase even more�

The controller given by �� ## with currents satisfying ��#!� can be ap�plied to this type of motor as follows�

Using the transformation given in ��!��!$ with p� � �!� and the model givenin �� the torque can be expressed in new coordinates 'ze � � 'zd� 'zq� 'z��

� as

�!np f�Ld � Lq 'zd 'zq & �m 'zqg

where Ld � Lls &��LA & LB� Lq � Lls &

��LA � LB�

The desired currents are chosen as

'zed �

��

��np�m

��

from which it follows that 'zed� -zed � L� � whenever �� '� � L �

To satisfy �� taking the uncertainty of the resistances into account� K� ischosen as

K� � ��

W��qm 'qm & kI�� k �

The input is then given from ��## and ��


This approach can also be extended to synchronous reluctance motors�� with thesame inductance matrix as in �� In these machines there are no permanentmagnets or windings in the rotor� hence �m � and the torque is given as ��

�!np

�Ld � Lq 'zd 'zq

from which it follows that one of the desired currents should be constant� and theother proportional to desired torque�

Also� if the synchronous machine has a �eld winding on the rotor instead ofpermanent magnets� �m will be proportional to the current in the �eld winding� whichis usually chosen to be constant or varying according to a �eld weakening objective�The choice of the other desired currents could be done as previously explained� �

Example �� PM stepper motor�� As another example� it will be shown howto apply the proposed controller �� ## with currents satisfying ��#!� to aPM stepper motor�

With the model given in ��!��#� the transformation to the dq�frame is ��$�

P �qm � e �Uqm �

�cos�Nrqm sin�Nrqm� sin�Nrqm cos�Nrqm

�� U �

� �Nr

Nr

�where U satis�es ��!��!!� This transformation is orthogonal� and P��qm �P T �qm�

The torque expressed in new coordinates 'ze � � 'zd� 'zq�T is

� Km 'zq �KD sin�#Nrqm

and the desired currents can be chosen as

'zed �

�

�Kmf� &KD sin�#Nrqmg

�

Notice that �� is satis�ed for the choice K� � kI� � � where k � � and� � L implies 'zed � L�

� and boundedness of -qed follows from the boundedness of'� and 'qm� The input is then given from ��## and ��

It is worth to point out that the controller in �!!� can be obtained from thepassivity�based approach if it is applied to the full system� without dividing thesystem into electrical and mechanical parts�

As previously pointed out� the underlying assumption of BP�transformability isthe sinusoidally distribution of the MMF� It can be discussed whether this is a goodapproximation in the case of stepper motors� with concentrated windings� signi�cantair gap saliency and often hybrid rotor constructions� The BP transformation abovehas however been used in several papers� among them � �� and � ��

�

��This motor has been proposed as an alternative to other AC machines� see ��

� Conclusions ��

� Conclusions

�� Summary

In this chapter the output feedback global tracking problem for a generalized electricmachine model has been studied� A passivity�based method was used to design thecontroller in three steps� First� the dynamics of the machine was decomposed asthe feedback interconnection of two passive subsystems electrical and mechanical�Then� a nonlinear damping was injected to make the electrical subsystem strictlypassive� Finally� an energy�shaping controller was designed to make the currentsconverge exponentially to desired functions� such that the desired torque is generated�The main contribution is the establishment of physically interpretable conditions onthe model� such that the method can be successfully applied� To further relax theseconditions� it is believed that passivity ideas must be combined with the powerful newdynamic extension techniques for stabilization of nonlinear systems� Some researchalong these lines for the robotics problem has been reported in �# ��

The passivity�based approach gives control schemes which provide global stabilityresults for the closed�loop system� There is no need for observers since unmeasurablestates are not used� hence the robustness problems associated with observer�baseddesigns are avoided �e�g� numerical problems from open�loop integrations� unknownparameters� Further� PBC do not introduce singularities� and the need for specialprecautions to be taken at for example start�up is obviated� The performance of thescheme� as measured with the exponential convergence rate of desired currents �andconsequently outputs to their desired values� can be explicitly derived for each ma�chine using the results in Sections #�� and #� � It follows that the rate of convergenceis restricted by the convergence rate of the unactuated dynamics� i�e� the resistanceof the unactuated windings� This is a consequence of our inability to add dampinginto this dynamics� since the involved states are unmeasurable�

It must be pointed out that the energy properties of the system are invariantunder a change of coordinates� and this gives the possibility of controller implemen�tation in a general dq�frame� chosen from the objectives of minimizing computationalburden and increasing numerical robustness� This also allows for implementationswithout measurement of rotor position� if the stator �xed frame is chosen for modelrepresentation�

To establish a relationship of the controller in this work to existing schemes� itshould be noticed that this control input consists of a nonlinear damping term addedto the reference dynamics� Henceforth� it can be classi�ed as an indirect vector con�trol scheme� which is the most widely used implementation of FOC �especially wellsuited for operation close to zero speed �� In particular� for speed control of theinduction motor� it is shown in Section ��! that PBC exactly reduces to the indirect�eld�oriented controller under some simplifying assumptions� namely speed regula�


tion with use of a current�fed converter �or high�gain current control� for which theadditional problem of stator dynamics is not present� In practice� the assumptions ofconstant and known parameters will not hold� Resistances will for instance vary dueto temperature changes and the skin e�ect at high frequencies� and inductances willchange when magnetic saturation occurs� Some answer to this challenging questionswill be given� for the case of induction motors� in Chapters ��

�� Open issues

It is worthwhile to point out that no systematic comparison of all the control ap�proaches discussed in Section � has been done for motor control� Intuitively� theschemes seem to have much in common� but to the best of the authors� knowledge�no comparing analysis from the implementation of more than two or three schemeson the same experimental setup has yet been reported� Also� the schemes are notgeneral in the sense that estimation and control are not based on a compact and gen�eral model� Equations of dynamics are commonly specialized to a particular machinebefore any results are derived� A comparison of the above results in a general setting�would be a highly interesting and challenging task�

The problem of combined parameter and state estimation is still an unresolvedissue in general nonlinear control theory� and a lot of e�ort has been devoted to thisproblem of interlacing controller design with the design of adaptive observers� Asan example� the main driving force within the �eld of induction motor research is toprovide a satisfactory solution to the problem of rotor resistance adaptation� Adaptiveobservers have been designed� and the �rst results from interlacing these results withoutput feedback controller design� have been reported for the backstepping schemeand the adaptive feedback linearizing scheme� This problem is still open for the otherschemes�

Another issue to be solved in a general setting� is the problem of incorporatingadditional nonlinearities arising from inherently nonlinear magnetics and actuatorsaturation� Some results along this line using nonlinear control theory have beenreported in �!�� for a feedback linearizing controller and in �� for the PBC� but thisis still an active area of research�

The problem of digital implementation of controllers for nonlinear systems is yetanother problem� Usually nonlinear analysis starts with a continuous model� andstability results are established for the total continuous system� The controllers arehowever invariably implemented on digital processors� Today this is commonly doneby use of an emulation technique with some ad hoc discretization �e�g� ZOH � underan assumption of �su�ciently short� sampling period� This assumption can be recastin terms of a bound on sampling period relative time constants of the system in thelinear case� For nonlinear systems there is no such equivalent� and the performance is�sues of the resulting systems with nonlinear controllers which have been discretized by

�� Open issues ��

use of ad hoc methods� are yet not fully understood� As an example� it is not obviousthat exact cancelation of dynamics and stable zero dynamics can be achieved with adiscretized feedback linearizing controller� when directly applied to a continuous sys�tem� Promising �rst results towards the rigorous design of discrete�time controllersfor electric machines� have been been reported for the CSI induction motor in � ��see Section ��$ for a presentation of this scheme� and for the synchronous motorin ��#�� It is expected that the solution of this problem will lead to the design ofdiscrete nonlinear controllers at converter level� This means that the discrete natureof switched converters will be taken advantage of in the controller design� removingthe PWM block between controller and converter� and instead directly specifyingthe switching pattern by the discrete�time controller� Hysteresis controllers for directcontrol of torque and �ux� which specify the converter switching via lookup tables� arenow becoming alternatives to �eld�oriented controllers for industrial use �� Thisprinciple should be focused on also from the viewpoint of nonlinear control theory�and some interesting results have been reported in ��$�

Torque ripple due to current harmonics must be addressed to reduce mechanicalvibrations and to obtain higher power e�ciency� It is expected that the solution ofthe above points regarding nonlinear magnetics� unknown parameters and converterswitching will signi�cantly reduce this problem�

There have not been reported many results where both speed and �ux observershave been interlaced with controller designs� and rigorously analyzed using nonlineartheory� Observers for �ux and speed are generally designed separately in an ad hocway� even though a high quality estimate of speed is needed for good �ux estimates�and vice versa� This is a problem to be solved for both higher dynamic performanceand power e�ciency �! �� A natural extension of this problem� is to the design ofnonlinear controllers which can be implemented without rotational sensors� This is achallenging nonlinear estimation problem for which no rigorous solution has yet beenderived�

Chapter ��

Voltage�fed induction motors

The induction� motor� and especially the squirrel�cage induction motor� has tradition�ally been the workhorse of industry� due to its mechanical robustness and relativelylow cost� In a wide range of servo applications with high�performance requirementsit has now� due to advances in control theory and power electronics� replaced DCand synchronous drives� As a continuation of our studies on torquecontrol of thegeneralized machine in Chapter � we address here the problem of passivitybasedspeed�position control of this particularly important machine� The two phase squirrel�cage induction motor model� is �rst given in Section �� Various equivalent represen�tations� often encountered in the literature are also presented� The speed%positioncontrol problem is then formulated in Section �

In Section ! we add to the torquetracking PBC of Proposition �� an outerloop speedposition controller for generation of the desired torque signal� A nicefeature of this scheme is that it has the nestedloop con�guration that is standardin applications� Furthermore� we show in Chapter �� that� under some suitablemodeling assumptions� the PBC reduces to the wellknown FOC� Other extensionsof the scheme like adaptation and integral action� as well as an important remark forits practical implementation� are also presented�

We have in previous chapters discussed how di�erent PBCs can be designed�based on the choice of desired storage functions for mechanical �Chapter � andelectromechanical �Chapter systems� In Section # we illustrate this particularlynice feature for the induction motor� In this case the energyshaping stage is appliedto the full machine model� instead of just the electrical subsystem� using as a storagefunction the total energy of the motor� Although conceptually simpler� this controllerrequires� in contrast with the previous scheme� the implementation of an observer�

�The induction motor was invented by Tesla ca� �� and a �rst analysis of its dynamic behaviourwas presented some years later in ��

�This is the most commonly used model for control purposes� The derivation of the �� equationsfrom the ��model can be found in any standard textbook on electric machines� e�g� ��

!��

�� Ch� � � Voltage�fed induction motors

For both controllers we can prove global asymptotic speed%position tracking in anestedloop con�guration� While for the �rst PBC we give a complete proof� for thesake of brevity we present only the torque tracking version of the second PBC� InSection � we present experimental results for both PBCs�

As pointed out in Chapter � a classical and widely used control strategy forelectrical machines in general� and in particular for induction motors� is the principleof �eldoriented control �FOC� To underscore the connections and di�erences withour PBC� we present in some detail this scheme in Section �� We also discuss inthis section the theoretically interesting feedbacklinearization strategy� Connectionsbetween these three strategies are further elaborated for currentfed machines inSection �� !�

� Induction motor model

� Dynamic equations

Under the same assumptions about the physical construction of the machine as in Sec�tion � � the standard two phase �model� of an np pole pair squirrel�cage inductionmotor with uniform air�gap has ne � #� ns � nr � and electrical parameters

De�qm �

�LsI� Lsre

Jnpqm

Lsre�Jnpqm LrI�

�� qm � � ��qm �

Re �

�RsI� RrI�

�� Me �

�I�

�� J �

� ��

�� J �

eJnpqm �

�cos�npqm � sin�npqmsin�npqm cos�npqm

�� e�Jnpqm � �eJnpqm�

Ls� Lr� Lsr � are the stator� rotor and mutual inductance� Rs� Rr � are stator androtor resistances� I� is the identity matrix�

The dynamic equations are derived by direct application of the Euler�Lagrangeequations � �� as in Section � with the Lagrangian from ��#� This results in

De�qm-qe &W��qm 'qm 'qe &Re 'qe � Meu ��

Dm-qm &Rm 'qm � � 'qe� qm� L ��

� 'qe� qm ��

'q�e W��qm 'qe �� !

where

W��qm �dDe�qm

dqm�

� npLsrJ eJnpqm

�npLsrJ e�Jnpqm

�� #

�In this model the axes for the stator have a �xed position while those corresponding to the rotorare rotating at the rotor �electrical� angular speed�

�� Some control properties of the model ��

'qe�� 'q�s � 'q

�r �� 'qs�� 'qs�� 'qr�� 'qr��

� is the current vector with stator and rotor compo�nents� 'qm is the rotor angular velocity� Dm � is the rotor inertia� The control signalsu � �u�� u��

� are the stator voltages� L is the external load torque� and Rm � isthe mechanical viscous damping constant�

The �ux vector �� s � �

�r �� s�� s�� r�� r��

� is related to the current vector'qe via

� � De�qm 'qe ��

from which the second of these vector equations

�r � Lsre�Jnpqm 'qs & Lr 'qr ��

is of particular interest for use in the subsequent analysis�

Also� notice that due to the short circuited windings of the squirrel�cage rotor�the second of the equations in �� is given by

'�r &Rr 'qr � �� $

� Some control properties of the model

A Input�output properties

The model of the induction machine derived above is a particular case of the general�ized machine model �� of Chapter � It therefore has the passivity propertieswhich were established in Proposition �� of Section �!� In particular� the model�� may be rewritten in �Newton�s second law form� as

D�q-q� �z mass � acceleration

�

� �W��qm 'qm 'qe��'q�e W��qm 'qe

��R 'q &Mu& �� z

sum of forces

where D�q � diagfDe�qm� Dmg� R � diagfRe� Rmg� 'q � � 'q�e � 'qm�� M � �M�

e � ��

� � � ��L��The second term on the righthand side of the equation above� corresponds to

the dissipation forces� while the last two terms on the righthand side constitute theexternal forces� The cornerstone of the passivity�based design philosophy is to revealthe workless forces� in this case the �rst term on the righthand side� This is easilyestablished with the systems total energy

T �q� 'q ��

'q�D�q 'q


which has a rate of change �the systems work

'T � 'q��R 'q &Mu& �

Workless forces do not a�ect the systems energy balance� which results from theintegration of the equation above

T �t� T � � �z stored energy

� �Z t

�

'q�R 'qds� �z dissipated

&

Z t

�

'q��Mu& �ds� �z supplied

As a result of this fact� the e�ect of these forces can� roughly speaking� be disregardedin the stability analysis�

The energy balance equation also proves that the mapping �u��L�� 'q�s � 'qm��

is passive� with storage function T �q� 'q� Furthermore� as shown in Section �!�the motor model can be decomposed as the feedback interconnection of two passiveoperators with storage functions Te�qm� 'qe and Tm� 'qm� respectively� These passivityproperties� and their corresponding storage functions� will be the basis for the twoPBCs to be presented in this chapter�

B Geometric properties

We now exhibit the �invertibility� property of the induction motor model which waspresented for a general strictly BP transformable machine in Section �� We recallthat this property is essential for obtaining an explicit expression of the torquetracking PBC�

From �� ! and �� # we see that the torque can be written as

��

'q�e W��qm 'qe � npLsr 'q

�s J eJnpqm 'qr ��

where the fact that J and eJnpqm commute �J eJnpqm � eJnpqmJ � and the skew�symmetry of J �J � � �J � x�J x � � � x � IR� has been used� Now� solving�� for 'qr

'qr ��

Lr

��r � Lsre

�Jnpqm 'qs�

��

and substituting it into �� gives

� npLsr

Lr

'q�s J eJ npqm�r ��

Finally� �� can be solved for 'qs

'qs ��

LsreJnpqm ��r � Lr 'qr

�� Coordinate transformations ��

and then substituted into �� to give

� �np 'q�r J �r �npRr

'��r J �r ��

where 'qr � � �Rr

'�r from �� $ has been used� This is the key expression that we usedin Section �� to �invert� the systems dynamics� In this case ��#� takes the form

'�r �

k�rkRr

npJ �r

As pointed out in Remark ��!$� the equation above shows that the zero dynamicsof the motor with outputs and k�rk are periodic� This fact becomes clearer if weevaluate the angular speed of the rotor �ux vector relative the rotor �xed frame �theslip speed as

'� �d

dtarctan�

�r��r�

��

� & ��r��r�

�

'�r��r� � �r� '�r��r�

��

k�rk�'��r J �r

�Rr

npk�rk� ��

From this equation we that if and k�rk are �xed to constant values� the rotor �uxrotates at a constant speed� This expression also shows that torque can be controlledby controlling rotor �ux norm and slip speed�

�� Coordinate transformations

In order to highlight some aspects of the machine� the model is sometimes presentedin a particular set of coordinates�� On the other hand� the use of di�erent motorrepresentations has been a source of confusion� In this section we follow ��# � andintroduce a general transformation from which we can derive most of the modelsconsidered in the control literature�

The induction motor model� previously presented in its natural frames of refer�ence� will now be presented in a frame of reference rotating at an arbitrary speed a�t� In this model� the natural machine variables �current� voltages� �ux linkagesassociated with stator and rotor windings� are substituted with dq�variables� �direct

�The importance of coordinate changes was probably �rst underscored by Copernicus� whopointed out that the planetary motions are better understood from the sun�s perspective �� Achange of coordinates is� of course� also the underlying principle of FOC�

�There is no consensus in the electric drives community concerning the names of the di�erentrepresentations� here we follow the one adopted by ��


and quadrature variables associated with �ctitious windings� Although this modelcould be written in terms of new current variables� it is common practice to presentthe model in terms of �ctitious stator currents idq and rotor �ux linkages �dq

�dq��

��d�q

�� e�J �a�npqm��r� idq

��

�idiq

�� e�J a 'qs

�� !

udq��

�uduq

�� e�J au

where �a is the solution of '�a � a� �a� � � with a a function to be de�ned later�depending on each particular choice of reference frame�

Substituting the expression for 'qr from �� into �� $ and multiplying by LrRr

gives

Tr '�r & �r � Lsre�Jnpqm 'qs �� #

where Tr�� Lr

Rris the time constant of the rotor dynamics�

Noting that �r � eJ �a�npqm��dq� computing its derivate� substituting it into�� #� rearranging terms� multiplying from the left by e�J �a�npqm� and using idq �e�J a 'qs� �nally gives

Tr '�dq & Tr� a � np 'qmJ �dq & �dq � Lsridq ��

To express the upper two stator equations of �� in this new reference frame�it is started by expressing �� in terms of idq and �dq� This expression and therelation 'qs � eJ aidq are then substituted into the stator part of �� written asddt�s &Rs 'qs � u� or equivalently as

d

dt

�Ls 'qs & Lsre

Jnpqm 'qr�&Rs 'qs � u

Di�erentiating� multiplying from the left by e�J a� substituting for ddt�dq from ��

and rearranging terms� the stator equations can be written

d

dtidq & � aJ & �I��idq &

Lsr

�LsLr�np 'qmJ � �

TrI��dq �

�

�Lsudq ��

� � �� L�srLsLr

� is the total leakage factor of the motor� and � � RsLs�

& L�srLs�LrTr

�

To express the torque in terms of dq�variables� substitution of �r � eJ �a�npqm��dqand 'qs � eJ aidq into �� gives

� npLsr

Lri�dqJ �dq � np

Lsr

Lr��diq � �qid �� $

�� Coordinate transformations ��

De�nition of a Reference frame

Stator��xed frame �ab�framenp 'qm Rotor��xed frame

np 'qm & LsrTr

iq�d

Rotor��ux frame

or

np 'qm & ddtarctan

��r��r�

�Table � �� De�nitions of '�a � a�

Some of the most commonly used de�nitions for the reference angle �a of thisrotation� are given in Table � ��

A choice of reference frame which is of particular interest is the stator �xed frame�where rotor variables are associated with �ctitious stationary windings � �� Themodel is of special interest because it is widely used in the many implementations ofcontrollers based on backstepping ��# � and feedback linearization ��$ �� This model isdenoted the ab�model and can be derived from �� and �� by setting a � ��a � � Since this model is usually written in state space form instead of the secondorder Euler�Lagrange form in �� it is of interest to rewrite it as

'�dq � np 'qmJ�dq � �

Tr�dq &

Lsr

Tridq

d

dtidq � ��idq � Lsr


TrI��dq &

�

�Lsudq

This model has been used in many works in the control literature� where insteadof subscripts d and q� a and b are used� For ease of reference we write the equationsabove out in detail as

'�ia �

LsrRr

Ls�L�r

�a &npLsr

Ls�Lr'qm�b � �ia &

�

Ls�u� ��

'�ib �

LsrRr

Ls�L�r

�b � npLsr

Ls�Lr

'qm�a � �ib &�

Ls�u�

'�a � �Rr

Lr�a � np 'qm�b &

RrLsr

Lria

'�b � �Rr

Lr�b & np 'qm�a &

RrLsr

Lrib

-qm ��

Dm� � L

�npLsr

Lr��aib � �bia ��


where �ab � ��a� �b�� is the rotor �ux� iab � �ia� ib�

� � 'qs is the stator current� and

it has been assumed that Rm � � De�ning the state vector x�� 'qm� �

�ab� i

�ab�

�� themodel can be rewritten in the classical statespace form

'x � f�x &Gu ��

with an obvious de�nition of the vector �eld f�x and the constant matrix G withcolumns g�� g��

The state space ab�model of the induction motor was used in this work only forimplementing a simulation model of the induction motor� It is well suited for thispurpose� since it has no rotational transformations� It is also written down for thepurpose of comparison with the second order Euler�Lagrange model which has beenused in this work� since it is believed that this second order model is rather uncommonwithin the motor control literature� While the EL model has a structure suitable forPBC design� the model in �� is a set of mathematical equations which are wellsuited for controller design and analysis using tools from geometric control theory�

�� Remarks to the model

Remark �� Squirrel�cage rotor�� For analytical purposes it is common to sub�stitute the squirrel �singlecage rotor� which has a uniform conductor distribution�with an equivalent �ctitious rotor with the same number of phases as the stator� andsinusoidally distributed conductors� This implies that in the analysis� only the �rstorder harmonic of the rotor MMF is accounted for� Experimental results indicatethat analysis and controller designs based on this simpli�ed model will also be validfor the real machine� In cases of deep bar or doublecage rotors� care should howeverbe taken when modeling these with sinusoidally distributed windings � $��

Remark �� PBC with the dq�model�� It is interesting to remark that in our�rst works on PBC of induction machines� i�e� � #� �� we used the model �� $� We treated a as an additional input which we then later integrated to getthe change of coordinates �� !� In control terms� for a control system 'x � f�x� uwe introduced a change of coordinates z � 0�x� �a� with �a a signal to be de�ned�Then we designed the PBC based on the system

'z ��0

�xf &

�0

��a'�a � g�z� �a� u� a

with the �inputs� u and a� and added an integrator to recover the original controlsignals� This is an interesting idea which� to the best of our knowledge� has not beenexplored elsewhere�

Remark �� Flux control�� The rotor �ux norm needs to be controlled for sys�tem optimization �e�g� power e�ciency� torque maximization during changing oper�ating conditions and under inverter limits � ��

�� Remarks to the model ��

Remark �� Measured variables�� A common instrumentation of a standardhigh�performance industrial !� induction motor drive is the use of two current trans�ducers and one rotational transducer� Currents are often measured using Hall�e�ectsensors with magnetic compensation� and can give high precision measurements withhigh bandwidth �DC to � kHz and isolation from measured currents� With knowl�edge of the switching instants of the inverter� the currents in each of the phases canalso be computed from measurement of the DC�side current with a shunt resistor�This allows for current measurement without expensive sensors� Velocity measure�ment can be expensive� and estimation of rotational speed from position measurementwith a high resolution digital incremental encoder can give signi�cantly better resultsthan often noisy analog measurements with DC tachometers �� In some rare casesboth velocity and position transducers are used� but the most common approach isto use an encoder� and estimate speed by simple numerical di�erentiation� or by theuse of speed observers driven by reference� or estimated torque and updated fromdiscrete position measurements ��

Since rotational transducers and their associated digital or analogue circuits giveextra costs and reduce the mechanical robustness of the total system� there has beenan increasing interest in schemes without rotational sensors� In some of these cases�speed is estimated by exploiting the in�uence of the rotational voltages in the dynamicequations� These methods are parameter sensitive with typically low performance atspeeds close to zero� Recently� promising results from sensorless speed and even po�sition estimation have been obtained by modifying rotor slots �introducing magneticsaliencies� and injecting balanced high frequency voltage signals at the terminals�� By signal processing of measured voltages and currents in combination with aclosed�loop observer� sensorless control is achieved� This �eld is an area of active re�search� and some successful implementations have already been reported in the recentsurvey � #�� The rigorous solution of this nonlinear estimation problem is yet to bederived� and drives without rotational sensors su�er from a performance degradation�which hampers their use in high performance speed or position control applications�

Additional voltage transducers are also used� not only for some control schemeswithout rotational sensors� but also in experimental setups for parameter identi�ca�tion �� Signal �ltering is then needed� especially with PWM converters� In experi�mental laboratory setups and in some industrial applications �e�g� ships� torque andinput%output power are also sometimes measured using current and voltage trans�ducers on the DC�link� and strain gauge rosettes on the motor shaft�

Many of the nonlinear control schemes are derived under the assumption of thefull state being measured� This is rarely the case� since the rotor currents or the�uxes� are not directly available for measurement� Measurement of currents in thesquirrel�cage rotor is very di�cult� Flux sensors �Hall�sensors� extra sensing coilsrequire expensive modi�cations of standard motors� and are not robust to mechanicalvibrations and other conditions encountered in rough industrial environments� Also�due to space harmonics� it is very di�cult to get good �ux estimates by interpolating


measurements from a few point sensors� Hence� �ux measurement is impractical andagainst the bene�ts of the squirrel�cage motor ��!�� It is therefore only used inexperimental setups� or special purpose installations�

�� Concluding remarks

In this section the dynamic equations for an np pole�pair squirrel�cage inductionmotor with smooth air�gap have been presented� The second order model structurefollows naturally from the Euler�Lagrange approach for modeling� and is well suitedfor passivity�based analysis�

As pointed out in Section � �!� there are parameter variations due to heatingand magnetic saturation� and these have not been been speci�ed in the model givenin this chapter� It is well known that the rotor resistance Rr can vary signi�cantly�and ability to compensate for this variation� or at least to guarantee stability despitevariations� is of utmost importance for any control design�

For comparison purposes� the more commonly encountered stator �xed ab�modelhas also been derived� This model is used extensively for analysis with geometric tools�and in a large number of implementations of controllers based on other designs thanthe passivity�based� For implementation purposes� this choice of model is sometimespreferred since measured quantities and controls need not be rotated to other frames��#!�� It is however important to be aware of bandwidth considerations for controllersin such implementations� While currents will be constant in a dq�frame at stationaryconditions �constant �ux and speed� they will be oscillating with a speed dependentfrequency in the stationary frame� The bandwidth must consequently be higher forcurrent controllers if they are implemented in the stationary frame�

It is worthwhile to point out the generality of the model presented in this chapter�It could be interpreted as an equivalent model of the usual !� machine� or stemmingfrom a reduction of phases in the more general polyphase machine through transfor�mations like those presented in � �� Notice that depending on the transformationused to go from !� to �� there is a factor appearing in the torque equation� de�pending on the form �power invariant� non�power invariant� see � $�� or ��# � of thetransformation used�

� Problem formulation

We will now formulate the control problem to be solved in this chapter� For easeof presentation we solve the speed control problem� since the extension to positioncontrol is straightforward� see Remark � ��

�A change ��RNr � Rr � ��R

Nr is not unreasonable ��

� A nested�loop PBC ��

De�nition �� Speed and rotor �ux norm tracking problem�� Consider thesix�dimensional induction motor model �� with state vector � 'q�e � qm� 'qm�

��inputs the stator voltages u � IR�� regulated outputs speed 'qm and rotor �ux normk�rk� Assume�

A�� Stator currents 'qs� rotor speed 'qm and position qm are available for measure�ment�

A�� All motor parameters are exactly known�

A�� The load torque L�t is a known bounded function with known bounded�rst order derivate� such that jL�tj c� �� t � � ��

A�� The desired rotor speed 'qm��t is a bounded and twice di�erentiable functionwith known bounded �rst and second order derivatives� such thatj-qm��tj c� �� t � � ��

A�� The desired rotor �ux norm ��t is a strictly positive bounded twice di�er�entiable function with known bounded �rst and second order derivatives� such

that � �� 1� �� '� '� � � and -� -� � ��

Under these conditions� design a PBC that ensures global asymptotic speed and rotor�ux norm tracking� that is�

limt�

j 'qm � 'qm��tj � � limt�

j k�rk � ��tj �

with all internal signals uniformly bounded�

� A nested�loop PBC

In Chapter we adopted a feedback decomposition approach for the design of atorque tracking PBC� In this section we solve the speed�position tracking problemposed above by adding an outerloop controller to this torque tracking PBC� Thisleads to the nestedloop �i�q� cascaded scheme depicted in Fig� � �� where Cil isthe innerloop torque tracking PBC and Col is an outerloop speed controller� whichgenerates the desired torque� d� We will show in this section that Col may be taken

�This assumption is made for simplicity� the result can be extended for the case of unknownlinearly parameterized load torque as discussed in Remark ��

�In the case where the torque is explicitly given as an external reference signal� as in the torquetracking problem� we denote it �� If it can be considered as the output from a controller �whichtakes for instance the speed reference �qm� as an external input�� we use �d�


as an LTI system that asymptotically stabilizes the mechanical dynamics� The maintechnical obstacle for its design stems from the fact that Cil requires the knowledgeof 'd� and this in its turn implies measurement of -qm� To overcome this obstacle weproceed� as done in Section �� #�B� for the robotics problem� and replace -qm by itsapproximate di�erentiation� while preserving the global stabilization property�

'qm�d

u'qs

'qm

�

��

��

� �

�

Col Cil .e .m

Figure � �� Nested�loop control con�guration�

A very interesting property of the resulting scheme� which is further elaborated inChapter �� is that if the inverter can be modeled as a current source and the desiredspeed and rotor �ux norm are constant� the controller exactly reduces to the wellknown indirect FOC� hence providing a solid theoretical foundation to this popularcontrol strategy�

Even though the torquetracking PBC can be directly obtained from the deriva�tions in Section �#� we will present it in detail for three reasons� First� to underscorethe additional di�culties associated with the control of the mechanical subsystem�Second� we show that the damping injection term which was instrumental for thestrict passivation in Section �#�� can be obviated if we choose a di�erent storagefunction� It is however required if we use instead the usual desired electrical energy

Hed ��

'�q�e De�qm '�qe ��

where '�qe�� 'qe � 'qed� and 'qed is generated by the PBC�

Third� to provide further insight into the philosophy of PBC and highlight itsconnection with other control approaches� we present the design this time from the�asymptotic inversion� perspective taken in Section �� Comments to the relationwith the general results from Section �# will be given at appropriate points�

The rest of this section is organized as follows� First� in Section !�� we review thederivations of Section �� as specialized to the induction machine� The main resultof this section� a globally de�ned observer�less speed and rotor �ux norm trackingcontroller� is then given� followed by its proof� It is also shown how this controller

�� A systems �inversion� perspective of the torque tracking PBC ��

can be robusti�ed with an integral term in the stator currents� and how adaptationof stator parameters can be added� A negative result concerning rotor resistanceadaptation is then presented in Section !�� It is shown that there exists a fundamentalstructural obstacle in the closedloop equations for the application of the existingadaptive control theory based on passivity of the error system �or equivalently� ondecoupled Lyapunov equations�

Motivated by practical considerations� a dq�implementation of the scheme is de�rived in Section !�$� Examples of how to de�ne the desired rotor �ux norm forminimization of power losses and to mimic �eld weakening are also included� fol�lowed by simulation results in Section !�� For ease of comparison with the variousschemes� all experimental results� including the ones of the nestedloop PBC� havebeen grouped at the end of this chapter in Section ��

�� A systems �inversion� perspective of the torque trackingPBC

In this section we derive the torque tracking PBC from the perspective of systems�inversion� studied in Section �� As discussed in that section� this derivation iseasier to understand if the �ux vector is used in the calculations instead of the currentvector� Therefore� using �� we rewrite �� ! as

'�&ReD��e �qm� � Meu ��

�npRr

'��r J �r �� !

where� for ease of reference� we have repeated ��

The PBC of Proposition �� is a �copy� of the electrical dynamics of the motor�� with an additional damping injection term introduced to get strict passivityof the mapping v �� 'qs� see Section �#� In its implicit form� this yields

Meu � '�d &ReD��e �qm�d &

�K�� 'qm

�D��e �qm�� #

� �npRr

'��rdJ �rd ��

where �� d and �d �

��sd� �

�rd

�� Notice that the damping injection is inde�

pendent of qm�

An explicit realization is obtained as explained in Section �� by �inversion of��

'�rd ��

��t

�Rr

np�J & '��t��tI�

��rd� �rd� �

��

�


which can actually be solved as

�rd � e J �d��t

��

'�d �Rr

np��t�� d� � �� $

The description of the controller is completed by replacement of �rd and '�rd in thelast two equations of �� # to get �sd� After di�erentiation we get '�sd which can bereplaced in the �rst two equations of �� # to get�

u � '�sd &�I�

�RsD

��e �qm�d &

�K�� 'qm

�D��e �qm��

�From the expression above we can see a di�culty for the implementation of thenestedloop scheme of Fig� � �� Namely that the control law depends on '�sd� whichin its turn will depend on '�� Notice that the latter was assumed to be known inProposition �� $� On the other hand� the signal � will now be generated by anouterloop controller Col� which will generally depend on 'qm� We will see in theProposition � �� how to overcome this obstacle by the use of a linear �lter�

Let us now analyze the stability of the closedloop� The error equation for the�uxes is obtained from �� and �� # as

'��&

�Re &

�K�� 'qm

��D��e �qm��

Global convergence of '�qe �and consequently of�� to zero was established in Proposi�tion �� for a suitably de�ned damping injection term K��qm� 'qm� see �� usingthe storage function �� With this storage function it is possible to prove thatthe closedloop equations de�ne an output strictly passive mapping v �� '�qs� where vis some external signal added to the control input� It is important to stress the factthat '�qs is known since 'qs is measurable�

We will show now that it is also possible to prove convergence� even without thedamping injection� To this end� consider the storage function��

H� ��

��R��e ��

whose derivative satis�es

'H� � ��D��e �qm�� H�

An explicit state space description is given in Proposition ��Recall that �� De�qm��qe��This function was used in �� to give an �implicit observer� interpretation of the PBC con�

troller�

�� A systems �inversion� perspective of the torque tracking PBC ��

for some � � � Hence� �� exponentially fast� It is easy to see that the closedloopequations de�ne an output strictly passive mapping v �� s� with storage functionH�� Notice� however� that this time the output is not measurable� We will see inChapter �� that this feature of measurable signals is essential for handling the casewhen the load torque is due to some nonlinear dynamics that we capture in .m� like inrobots with AC drives� �It is also required for the implementation of hybrid PBC fordirecttorque control as presented in ��$�� Even though the analysis above clearlyleads to a simpler controller for the problem at hand� it is for its later extension tothe case with nonlinear load torques� that we present the controller with a dampinginjection term and the proof based on the storage function ��

To illustrate the second di�culty in the stability analysis of the nestedloop

scheme� let us turn our attention to the torque tracking error �� After

some simple operations from �� ! and �� we get

� �npRr

�'��r J ��r &

'��r J �rd & '�rdJ ��r

�

� np

�� '�q

�r J �r &

�

Rr

'�rdJ ��r

�

where we have used '�qr � � �Rr

'��r� and the de�nition of ��r to get the last identity� In thetorque control problem we assumed that the external reference � and its derivative'� were bounded� and convergence of � � was proved as follows� First� �rd and '�rdare bounded by construction� see �� Then� we have shown above that �� exp�� consequently also '�qr � and �r is bounded� From here we conclude that� � � In positionspeed control � and '� are not a priori bounded� since they willbe generated by Col� Therefore� Col must be chosen with care and a new argumentshould be invoked to complete the proof� Proposition � �� below shows that Col canbe taken as a linear �lter�

A Connection with system inversion

Before closing this section� we will view the PBC from the geometric perspectivediscussed in Section �� The purpose of the exercise is to show that if we complete ina suitable manner the �output� vector ��#�� then the standard inversion algorithmof geometric control� as applied to the reference signals� will give us the PBCcontroller above� This fact is important for at least two reasons� �rst because theinversion algorithm is a general systematic procedure� while the procedure used aboveis somewhat ad hoc� Second� it provides a clear connection with feedback linearizationwhere the same inversion algorithm is invoked� but now applied to the output signals�

To this end� we propose to complete the �output� vector for the system ��


as y � �� y�� where

y�� np��r

� I�

D��e �qm� �

npRr

��r '�r

Recall that can be written in the form �� !�

Following the inversion algorithm we evaluate the decoupling matrix by takingthe time derivative of y

'y �npRr

��r J��r

� �ReD

��e �qm

��u&m��

�� G��r� qmu&m��

where m�� is some function of �� It can be shown that

�ReD

��e �qm

��

�RrLsr

LsLr�eJnpqm

which is globally invertible� On the other hand��r J��r

��

�

k�rk�� J �r �r

Consequently� the decoupling matrixG��r� qm is nonsingular everywhere except whenk�rk � � This implies that the system �� with output y has relative degreef�� g� A feedback linearizing controller is chosen as

u � G��r� qm�� 'y� �m�� & v�

where v is an additional stabilizing controller� On the other hand� the control signalfor the PBC derived above can be obtained by �evaluating the inversion for thereference signals�� that is

u � G��rd� qm�� 'y� �m��d� & udi

where udi is the damping injection term and �rd is obtained from �� and �� $�

Roughly speaking� we can summarize the discussion above as follows�

While inputoutput linearization implements a right inverse of the system .�� qm�that is

uFL � .�� qm�y� & v�

the PBC implements a left inverse

uPBC � .��d� qmy� & udi

This interpretation is depicted in Fig� � � below� Notice that� except for the dampinginjection� the PBC is openloop in �� However� the loop is closed with qm�

�� Observer�less PBC for induction motors ��

�

��

� �

�

�

.

uFL

y�

FL

uPBC.

qm

y�

PBC

�.��

�.��y

y

Figure � � � Connection with system inversion�

�� Observer�less PBC for induction motors

The main result of this section can now be formulated�

Proposition �� Speed and rotor �ux norm tracking�� The nonlinear dynamicoutput feedback nested�loop controller

u � Ls-qsd & LsreJnpqm -qrd & npLsrJ eJnpqm 'qm 'qrd� �z

��sd

&Rs 'qsd �K�� 'qm '�qs ��

with

'qed �

�'qsd'qrd

��

��

Lsr

h�� & Lr ��

Rr�I� &

Lrnp��

dJieJnpqm�rd

��

�dnp��

J &��

Rr�I�

��rd

��

where

'�qe �

�'�qs'�qr

��

�'qs � 'qsd'qr � 'qrd

�

K�� 'qm��

n�pL�sr

#�'q�m & k�� Rr� k� �

d � Dm-qm� � z & L �� !

and controller state equations

'�rd �

�Rr

np��dJ &

'��I�

��rd� �rd� �

��

�� !�

'z � �az & b '�qm� z� � '�qm� �� !


with '�qm�� 'qm � 'qm� and a� b � � provides a solution to the speed and rotor �ux

norm tracking problem of De�nition �� That is� when placed in closed�loop with�� it ensures

limt� '�qm � � limt� j k�rk � ��tj �

for all initial conditions and with all internal signals uniformly bounded� �

Proof� First� notice that �� can be rewritten as

De�qm-qe & Ce�qm� 'qm 'qe &R�qm� 'qm 'qe � Meu �� !!

where

Ce�qm� 'qm��

� npLsrJ eJnpqm

�'qm

R�qm� 'qm��

�RsI�

�npLsrJ e�Jnpqm 'qm RrI�

�

By substitution of �� into �� !! and using �� $� it follows that the closedloop system is fully described by

De�qm-�qe & Ce�qm� 'qm '�qe & �R�qm� 'qm &K� 'qm� '�qe � �� !#

Dm-�qm � �z & � 'qe� qm� d �� !�

'z � �az & b '�qm �� !�

'�rd ��

Rrnp��

dJ �rd & ��rd

�� !$

where K� 'qm �� diagfK�� 'qmI�� g� For later convenience� �� !� and �� !� are

rewritten as �-�qm'z

��

� � �

Dm

b �a� �

'�qmz

�&

��

Dm

�� d

m'x � Ax& B� � d �� !�

The matrix A is Hurwitz for all positive values of a and b�

Under Assumptions A��A�� the system �� !#�� !$ is locally Lipschitz

in the state � '�q�e � '�qm� z� �

�rd�

� and continuous in t� This condition ensures that thereexists a time interval � � T where the solutions exist and are unique� First� k�rdk ��t� �t � � � T � and it is consequently bounded� Now� consider the quadraticfunction �� whose time derivative along the solutions of �� !# for all t � � � T �is given by

'Hed � � '�q�e �R�qm� 'qm &K� 'qm�sy '�qe �� !�

�� Observer�less PBC for induction motors ��

where the skew symmetry property

'De�qm � Ce�qm� 'qm & C�e �qm� 'qm

has been used� and

�R�qm� 'qm &K� 'qm�sy �

��Rs &K�� 'qm I�

��npLsrJ eJ npqm 'qm

��npLsrJ e�Jnpqm 'qm RrI�

�

is the symmetric part of R�qm� 'qm & K� 'qm� This matrix will in the rest of thischapter be denoted�� Res�qm� 'qm� The matrix is strictly positive de�nite� uniformlyin 'qm� namely

Res�qm� 'qm � �I� � �� #

where I� is the # # identity matrix� This can be proved by using standard resultsfrom matrix theory and the facts that Rr � and J e�Jnpqm � e�JnpqmJ � whichleads to �� # holding if and only if

Rs &K�� 'qm�n�pL

�sr

#�Rr � �'q�m � � �� #�

See Section #�! of Appendix D for a detailed derivation of this requirement� whichis ful�lled with the chosen de�nition of K�� 'qm�

Therefore� from the above and �� !� it follows that

'Hed � � '�q�e Res�qm� 'qm '�qe � inf

qm� �qm��Resk '�qek�

and it can be derived that for some constants me � and �e � independent of T

k '�qe�tk mek '�qe� ke��et� �t � � � T �� #

Notice that� unless 'qm escapes to in�nity in �nite time� limt� '�qe � � Thus� itmust be proved �rst that this is not the case by showing that the input � � d tothe linear �lter �� !�� !� is linearly bounded by the �lter state� To this end�notice that the desired torque d can be written as

d ��

'q�edW��qm 'qed

hence it follows that

� d ��

'�q�e W��qm '�qe & '�q

�e W��qm 'qed �� #!

��Notice that the symmetric matrix above is exactly the matrix Res�qm� �qm� de�ned in ��


and the following bound holds

j � dj npLsr

�k '�qek� & k '�qekk 'qedk�

�� ##

On the other hand� writing the desired currents in �� as

'qed �

��

Lsr

�Lrnp��

�Dm-qm� & LJ & �� & Lr ��Rr�

I�

�eJnpqm�rd

��

�np��

�Dm-qm� & LJ &��

Rr�I��rd

��

&

�� Lr

Lsrnp��J eJnpqm�rd

�np��

J �rd

�z

�� t & ��tz �� #�

and noting that

k��t�k �q� �L�srk Lrnp��

�Dm�qm� � �L�J � �� Lr ��Rr�

�I�k� � k �np��

�Dm�qm� � �L�J ��

Rr�I�k��k�rdk

� ��

k��t�k �q

�� LrLsrnp��

�� n�p

��

�k�rdk � ��

yields k 'qedk 1�� & 1��jzj� Replacing this bound� together with k '�qek mek '�qe� k��t � � � T in �� ##� it follows that

j � dj npLsr

�m�

ek '�qe� k� & mek '�qe� k1�� & npLsrmek '�qe� k1��jzj� �t � � � T

This last inequality proves� via Gronwall�s inequality� that x� and consequently'qm� can not grow faster than an exponential in the time interval � � T � Moreover�since all the constants in the above bound are independent of T � this argument canbe repeated to extend the time interval of existence of solutions to the whole realaxis�

Having proved that �� # holds as t � �� it must be proved that this impliesthat limt� x � � with 'qed bounded�

Inserting �� #! into �� !�� and using �� #� to express 'qed� results in thesystem

'x � Ax&

� �

Dm

'�q�e W��qm��t

� �'�qmz

�

&

��

�Dmf '�q�e W��qm '�qe & '�q

�e W��qm��tg

�m

'x � Ax&B�tx & c�t �� #�

�� Remarks to the controller ��

Calculation of norms gives

kB�tk �

DmnpLsr 1��mek '�qe� ke��et �� #$

kc�tk npLsr

Dmfmek '�qe� ke��et & 1��gmek '�qe� ke��et

From this �see Example �� in �� $� it can be concluded that the system 'x � �A &B�t�x is globally exponentially stable� and since kc�tk � as t � �� it followsthat x� � Further� from �� #� it can be established that 'qed is bounded�

The proof of asymptotic rotor �ux norm tracking follows from �see �� and�� $

�r � �rd � Lsre�Jnpqm� 'qs � 'qsd & Lr� 'qr � 'qrd

'�r � '�rd � �Rr� 'qr � 'qrd

and convergence of current errors to zero�

�

�� Remarks to the controller

Remark �� Relation with Proposition �� Notice that the innerloop PBCabove follow directly from Proposition �� with U�� npJ � In the case of a two�dimensional �ux vector� the initial condition on rotor �ux norm can easily be givenin terms of one of its components�

Remark �� Position control�� It is easy to see that choosing the desired torquein the controller above as

d � Dm-qm� � z � f �qm & L �� #�

yields global asymptotic position tracking for all positive values of a� b� f � In this case�the error equation �� !� has

x �

�� qm

'�qmz

�� A �

�

��

� fDm

� �Dm

b �a

�� B �

�

��

�Dm

��

The matrix A is Hurwitz for all positive values of a� b� f � and the proof of globalasymptotic rotor �ux norm and position tracking follows verbatim from the proof ofthe main result above�

Remark �� Adaptation of load torque�� We can extend the result in Propo�sition � �� to the case of unknown but linearly parameterized load

L � ��qm� 'qm


where � � IRq is a vector of unknown constant parameters� and ��qm� 'qm is a measur�able regressor� We replace the last equation of �� ! by the certainty equivalencelaw

d � Dm-qm� � z & (L

where (L � (��qm� 'qm� with (� an online estimate of � to be de�ned below� Noticethat the proof of convergence of '�qe is not a�ected by this change� with only equation�� !� being replaced by

'x � Ax& B� � d & �L

where �L�� (L � L� This driving term appears again in the last part of the proof in

�� #�� which should be replaced by

'x � Ax& B�L &B�tx & c�t

Thus we are confronted with the well�known problem of designing an estimator for afullstate measurable LTI system with exponentially decaying additive disturbances�The use of the estimator

'(� � ��x�PB��qm� 'qm� � �

with P a symmetric positive de�nite matrix satisfying A�P &PA � will solve thisproblem�

Remark �� Damping injection to .m�� To overcome the problem of mea�surement of acceleration it was assumed in �$�� that Rm � � and the speed controlstrategy

d � Dm-qm� &Rm 'qm� & L

was proposed� Two drawbacks of this scheme are that it is open loop in the speedtracking error� and that its convergence rate is limited by the mechanical time constantDm

Rm� De�ning the desired torque d as in �� ! with z from �� ! � allows for

e�ectively feeding back the speed tracking error without acceleration measurement��

Even though� the convergence rate of the mechanical subsystem is independent ofthe natural mechanical damping� it is however restricted by the convergence of theelectrical subsystem� This� in its turn� is limited by the rotor resistance� as seen from�� !��

Remark �� Damping injection to .e�� In Chapter we carried out� in asimple magnetic levitation system� a simulation study of the e�ect of damping injec�tion to .e in PBC� We showed that without damping injection the convergence ratewas inadmissibly slow� On the other hand� in Section � of this chapter we will present

��This result was �rst reported in ��

�� Integral action in stator currents ��

experimental evidence showing that the damping injection term introduces undesir�able high gains that excite the unmodeled dynamics� induces actuator saturation andamplify the noise� Also� in Chapter �� we will present simulation studies for robotswith AC drives where a similar undesirable behaviour is observed for a backsteppingbased design� whose stability analysis heavily relies on this kind of nonlinear dampinginjection terms� Further research is clearly needed to enhance our understanding ofthese nonlinear phenomena to be able to tradeo� between speed of response andhighgain injection into the control loop�

�� Integral action in stator currents

It is common in applications to add an integral loop around the stator current errorsto the input voltages� The experimental evidence presented in Section � shows thatthis indeed robusti�es the PBC by compensating for unmodeled dynamics� It isinteresting to note that the global tracking result above still holds for this case� asshown in the proposition below�

Proposition �� The result in Proposition �� is still valid if the integral term

uI � �KIs

Z t

�

'�qs dt� KIs �

is added to the control� with KIs a positive semide�nite matrix� That is� if we setu � uPBC & uI� where uPBC is de�ned in ��

Proof� To study the stability of this new system� a term ��

hR t

�'�qs dt

i�KIs

hR t

�'�qs dt

iis added to Hed� Computing the derivate of this new Hed with the new closed loopsystem �see �� !#�

De�qm-�qe & Ce�qm� 'qm '�qe & �R�qm� 'qm &K� 'qm� '�qe � � �� #�

with

� �

� �KIs

R t

�'�qs dt

�results in

'V� � � '�q�e Res

'�qe �KIs'�q�s

Z t

�

'�qs dt&d

dt

��

KIs

�Z t

�

'�qs dt

�� Z t

�

'�qs dt

��

� � '�q�e Res

'�qe

It follows that '�qe and the integral term will be bounded for a closed time interval�This is enough to complete the previous details proving that there is no �nite escape


time� and it follows that '�qe � L� � L�

� andR t

�'�qs dt � L�

� It is however not enoughto claim convergence of current errors to zero� For this� the additional requirementof -�qe � L�

will be su�cient� From �� #� it can be seen that this is the case if thespeed is bounded�

Notice from �� #$ that kB�tk � j �Dm

'�q�e W��qm��tj �

DmnpLsr 1��k '�qek� Now�

since '�qe � L�� it follows that Z

�

kB�tk� dt � �

and by use of standard results on stability of linear time�varying systems �see �� $��pp� $� �$� it follows that the origin is a globally exponentially stable equilibriumof the linear part 'x � �A&B�t� x of the system �� #�

'x � �A&B�t� x& c�t

Since '�qe � L� implies that the perturbation term c�t is bounded� it follows by use

of total stability arguments that x and consequently the speed 'qm will be bounded�With bounded speed 'qm it follows as previously explained that the current errorsconverge to zero� which again implies convergence of the speed tracking error to zerosince c�t� �

Thus� global asymptotic speed and rotor �ux tracking tracking with all internalstates bounded can be proved even when the nestedloop PBC is robusti�ed withintegral action in stator currents�

�

�� Adaptation of stator parameters

The controller in �� can also be modi�ed to compensate for unknown statorresistance and inductance� by using estimates (Ls� (Rs instead of real parameters inthe controller� This results in a closed loop system �assuming for simplicity there isno integral action with

De�qm-�qe & Ce�qm� 'qm '�qe & �R�qm� 'qm &K� 'qm� '�qe � �

�

� �(Ls � Ls

�-qsd &

�(Rs � Rs

�'qsd

�

To prove the same stability results as before using a Lyapunov approach� extra terms�Ls�

�(Ls � Ls

��

and�Rs�

�(Rs �Rs

��

with �Ls� �Rs � must be added to V�� Calcu�

�� A fundamental obstacle for rotor resistance adaptation ��

lation of 'V� then gives

'V� � � '�q�e Res '�qe & '�q

�s

n�(Ls � Ls

�-qsd &

�(Rs �Rs

�'qsd

o&d

dt

��Ls

�(Ls � Ls

��

&�Rs

�(Rs �Rs

��

Assuming that Ls and Rs are constant� and choosing parameter adaptation laws tobe

'(Ls � � �

�Ls'�q�s -qsd

'(Rs � � �

�Rs'�q�s 'qsd

results in

'V� � � '�q�e Res '�qe

The rest of the proof follows as explained in the previous section� This approachwill not be pursued here� since adaptation of only these parameters is of less interest�especially if integral action in stator currents is added�

�� A fundamental obstacle for rotor resistance adaptation

We have shown already in previous chapters that it is relatively easy to incorporateadaptation features to PBC� This stems from the following two facts� �rst� that PBC isbased on EL descriptions which preserve linearity �and minimality on the parameters�Second� that PBC enforces a passive operator in closedloop� which is a fundamentalproperty for adaptation� since as pointed out in Section ��#� � the standard �gradientor least squares estimators themselves de�ne also passive operators� Unfortunately�the error equations that result of the nestedloop PBC of induction motors exhibit afundamental obstacle for adaptation of the practically important parameter Rr� �SeeRemark �� for a discussion on parameter uncertainty� Namely� that the operatorrequired to be passive for a successful adaptation is not passive with respect to theclassical quadratic storage function� It is important to underline the quali�er �withrespect to�� because this does not rule out the possibility that it is passive for someother storage function�

Let us now proceed to establish this negative result� We assume that the onlyuncertain parameter is Rr and propose an adaptive implementation of the nestedloopPBC �� ! with a timevarying estimate (Rr replacing Rr� For simplicitylet us consider the case of � � const� In this case Rr appears only in �� !�� whichin the adaptive case takes the form

'�rd �(Rr

np��dJ �rd


Going through the calculations and de�ning the parameter error �Rr�� (Rr � Rr we

see that this error propagates to the current error equations �� !# as

De�qm-�qe & Ce�qm� 'qm '�qe &Res�qm� 'qm '�qe �

� 'qrd

��Rr ��

5From the derivations above we see that �� de�nes an output strictly passiveoperator 'qrd �Rr �� '�qr with storage function Hed� As �rst pointed out in � �� thisproperty allows us to design a globally stable adaptation law� in the case of full statemeasurement� as

'�Rr � �� 'q�rd '�qr ��

with the adaptation gain � � � The convergence proof can be easily established witha passivity argument as in Section ��#� � or with the Lyapunov function

V � Hed &�

��R�r

which gives 'V � � '�q�e Res�qm� 'qm '�qe ��k '�qek�� for some � � � Unfortunately� the

only computable current errors are '�qs� and �� is therefore not implementable�The question is then whether we can de�ne measurable dimensional vector functions0��0� so that� along the dynamics �� the operator 0�

�Rr �� 0� is strictly passive�

In this way� the update law '�Rr � ��0�� 0� would solve the problem�

To help us investigate this question� it is convenient to work in a coordinate framewhere the desired storage function �and for that matter all the electrical equations isindependent of qm� As shown in Section � this is the case if the ab�coordinates �� are used�� The electrical error equations can be expressed in these coordinates as

'ee � A� 'qmee &B0��d� �rd �Rr

0� � '�qs �

�I�

�ee

where ee�� '�q

�s �

��ab�� and we have de�ned

A� 'qm��

� �LsLsrLr�

K�� 'qmI� �LsLsrLr�

�� 'qmLsrRrLr

I� �� 'qm

�� B

��

�LsLr�

I��I�

�

with �� 'qm�� 'qmJ � Rr

LrI� and a regressor vector

0��d� �rd��

d��L�

sr

J �rd��See also �� for a derivation of the torque tracking PBC in this reference frame�

�� A dq�implementation ��

To investigate the passivity properties of the operator 0��d� �rd �Rr �� 0� wefollow the standard procedure in adaptive control and look for a constant matrixP � P� � so that W � �

�e�e Pee with P � P� � quali�es as a storage function�

In other words� we must satisfy the conditions of the KalmanYakubovichPopovlemma �see Section � in Appendix A

PA� 'qm & A�� 'qmP � ��

PB �

�I�

�� !

Notice that �� is satis�ed with the choice P �

�Lr�Ls

I�

I�

�� which is the one

corresponding to Hed in the ab coordinates�

Now� it is easy to see that the matrix A� 'qm can be factored as

A� 'qm �� B��qm� 'qm

Consequently� in view of the condition �� ! we have that �PA�� and thecondition �� cannot be satis�ed �with strict inequality� Therefore� we concludethat there does not exist a quadratic storage functionW � �

�e�e Pee to prove the strict

passivity of the operator 0��d� �rd �Rr �� 0��

�� A dq�implementation

For the VSI case� it is of interest to rotate the control in �� to a dq�frame� notonly for comparison with indirect FOC� but also for implementation purposes�

Using the results from Section ��!� the equivalent of �� can be derived in thereference frame of the desired rotor �ux�

As previously pointed out� �� consists of the desired stator dynamics anda nonlinear damping term� By using �� ! in terms of also desired quantities��

iddq � e�J a 'qsd� �ddq � e�J �a�npqm��rd� and comparing �� with �� it can be

��Superscript d will be used to denote desired quantities expressed in a dq�frame� e�g� desiredstator currents in a dq�frame as iddq�


seen that

u � Ls-qsd & LsreJnpqm -qrd & npLsrJ eJnpqm 'qm 'qrd &Rs 'qsd �K�� 'qm '�qs

mu � �Lse

J a�d

dtiddq & � aJ & �I��i

ddq &

Lsr


TrI��

ddq

��K�� 'qme

J a �idq � iddq�

�a �

Z t

�

a dt� �a� � np 'qm�

a � np 'qm & '�d

where '�d is de�ned by �� $� and iddq ��Lsr

�� & LrRr

'�� Lrnp�

d��ddq � ��

�� De�nitions of desired rotor �ux norm

In the following two examples of how the desired rotor �ux norm ��t can be designed�will be given� The �rst result is a direct adaptation of the results in � $$� to thepassivity�based controller for minimization of steady state losses� The other is anexample of how the well known �ux weakening approach can be mimicked�

Example �� Minimization of steady state losses�� To minimize power lossesin the motor� the function

Ploss � u� 'qs��zsupplied power

� 'qm��zmechanical output power

�� #

is considered�

The control u is �rst eliminated from Ploss by using �� and �� which gives

Ploss �

�Ls � L�

sr

Lr

�'q�s -qs &

�Rs &Rr

L�sr

L�r

�'q�s 'qs � LsrRr

Lr

'q�s e J npqm�r ��

Detailed derivations of this and the following expressions are given in Section � ofAppendix D�

It is assumed that the stator currents 'qs have converged to their desired values�given in �� and all occurrences of 'qs and -qs can be substituted by their referencevalues� This gives

Ploss �L�rLs � L�

srLr

L�srR

�r

-� '� &LsLr � L�

sr

Rr

-�� &LsRr & LrRs

L�srRr

'��

&RrLrLs & L�

rRs

L�srRr

'�� &

LrL�sr � L�

rLs

n�pL�sr

'�� d &

Rs

L�sr

��

&

�L�rLs � LrL

�sr

n�pL�sr

'dd &L�rRs &RrL

�sr

n�pL�sr

�d

��

��

�� De�nitions of desired rotor �ux norm ��

The minimization of the above criterion with respect to a general time�varying strictlypositive function� ��d� 'd� is a nontrivial dynamic optimization problem�

As a �rst approach� only the task of minimizing the above expression at stationaryconditions with constant torque is considered here� In this case ��t can also beconsidered as a constant� which gives

Ploss �Rs

L�sr

�� &

L�rRs &RrL

�sr

n�pL�sr

�

�� d �� $

By evaluating and setting �Ploss�� the only extremum is found to be

�opt� � �

sL�r

n�p&Rr

Rs

L�sr

n�pjdj

Evaluation of the second order partial derivative of Ploss with respect to �� gives

��Ploss��

� RsL

�r &RrL

�sr

n�pL�sr

�

� �d � � ��

This implies that the function �� $ is convex in �� and �

opt� � is a global mini�

mum�

It must be pointed out that the loss model is very simple� with no core losses�current%voltage limits or e�ects from nonlinear magnetics included� Despite this� themodel indicates that to minimize power losses� the �ux norm reference should beproportional to the square root of the desired torque� at least as long as the resultingreference is below the maximum allowable value� �

Example �� Flux weakening�� In this example it is explained how the con�troller can be used to mimic the well known �ux weakening approach for operationin the constant power region above nominal speed�

To get the desired smoothness properties of the norm reference� a linear secondorder �lter is introduced��

'x��t'x��t

��

� �

� �n � � n

� �x��tx��t

�&

� �n

�ref�t ��

��t � x��t� '��t � x��t

with �x�� x��

�ref� �

�� and n � �

The input signal to the �lter �� is given by

ref�t �

$N� � � 'qNm 'qm� 'qNmN�

�qNmj �qm�j � j 'qm�j � 'qNm

��Notice that the derivative of desired torque is also needed for dynamic optimization�


with N� the desired constant nominal value for the rotor �ux norm and 'qNm the basespeed�

With the given value for the damping constant �� it is ensured that the outputof the �lter� �� will always be positive� and singularity points in the controller areavoided�

The �ux reference has been de�ned in a feed�forward way from reference speed�instead of by using a memoryless feedback from actual speed� This is due to theadditional technicalities involved in stability analysis when an extra feedback loop isused�

A possible drawback of this approach is that speed control has to be tight toavoid unwanted saturation and low performance when operating at speeds close tonominal speed and during transients� This implies that the acceleration constraintof the system must be taken into consideration when de�ning the reference for speedtracking� �


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−500

0

500Speed reference

Time [s]

[rpm

]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

Time [s]

[Wb]

Flux norm reference

Figure � �!� References for speed and �ux norm�


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1

0

1

2Speed tracking error

Time [s]

[rpm

]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.05

0

0.05

Time [s]

[Wb]

Flux tracking error

Figure � �#� Tracking errors for speed and �ux norm�

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4

−2

0

2

4Currents in phases a, b

Time [s]

[A]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−40

−20

0

20

40

Time [s]

[V]

Voltages of phases a, b

Figure � �� !� Currents and voltages ia� ib� ua� ub�

To verify the qualitative behavior of the scheme in Proposition � �� a simulationstudy with a SimulinkTMimplementation of the induction motor ab�model was done�This is a sti� system with fast electrical dynamics� and relatively slow mechanicaldynamics� for which the choice of integration method has to be carefully considered�The Gear�method with small step�size was found to give satisfactory numerical accu�racy� The same model parameters �see Section �� as for the experimental setup were


used� and simulations were done under the assumptions of ideal conditions �zero loadtorque� known parameters� linear unsaturated ampli�er� speed and position measure�ments� References were generated by use of step and square wave functions whichwere �ltered using third order linear �lters of the form

h�p ��

� p��

& �

�

� p�

��& � p

��& �

The necessary �rst and second order derivatives were obtained from the state spacerealizations of the �lters�

Filtering of �ux and speed references was found to be of high importance foravoiding current and voltage saturations� and the values � � �� rad%swere chosen for speed and �ux norm reference �ltering� respectively�

The following parameters were used� a � � � b � ! � k� � ! � � � �#�Rr�Integral action was not used in the controller� and all initial conditions were set tozero�

References� tracking errors and voltages%currents are shown in Figs� � �!� ��As can be seen from these �gures� there is no interaction between �ux and speedcontrol� This is consistent with the previous analysis� The !� voltages� and currentsare well within limits of the system� The linear term k� in the gain K� was found tobe signi�cant for satisfactory behavior around zero speed� when the speed dependentterm is close to zero�

A PBC with total energy�shaping

Instrumental for the development of the nestedloop controller of the previous sectionwere the decomposition of the motor dynamics into the feedback interconnection ofpassive subsystems� and the possibility to make the mapping v �� '�qs output strictlypassive via output feedback� In this way we could apply the PBC methodology solelyto the electrical subsystem and obviate the need of an �explicit observer� In thissection we take a di�erent approach were we� instead of shaping only the energyof the electrical dynamics� design a PBC based on the total energy of the system�mechanical and electrical� The design philosophy is in principle simpler� however�the price to be paid is the need for an observer� This observer�based approach hasbeen pursued in �$$� #� �� An additional assumption that we require here is thatthe friction coe�cient Rm is not zero� This restriction is needed only to completethe stability proof and does not restrict the convergence rate of the scheme� which islimited� as in the nestedloop PBC �see Remark � �� by the openloop electricaldynamics� in this case via the observer rate� We have seen in the previous section

��Only two of the �� quantities are shown� since the third is a linear combination of the �rst two�See Section �� for the transformations from two�phase quantities�

�� Factorization of workless forces ��

how to select the reference torque � in a torque tracking scheme to achieve speedor position control� Consequently� we will concentrate here on the torque trackingproblem�

Similar to the trajectory tracking problems in robotics of Chapter �� for the designof a PBC the workless forces must be factored in a suitable way� More speci�cally�they must be linear in 'q� In the case of the induction machine an additional restrictionon this factorization is imposed by the lack of full state measurement� The outputfeedback torque tracking problem is solved with an interlaced controller�observer de�sign which follows the passivity�based approach� A proof of global tracking is givenunder the assumption of known motor parameters� As the previous nestedloop PBC�the new control law does not require measurement of rotor variables� is always wellde�ned and does not rely on �intrinsically nonrobust nonlinear dynamics cance�lation� To simplify the presentation we solve �rst the problem assuming full statemeasurement� and then we focus on the output feedback case�

�� Factorization of workless forces

To carry out the PBC design� a suitable linear factorization of the workless forcesdiscussed in Section � into a form�

W��qm 'qm 'qe��

�'q�e W��qm 'qe

�� C�q� 'q 'q

must be found� Speci�cally� C�q� 'q will be required to be such that�

i 'D�q � C�q� 'q & C��q� 'q�ii The third and fourth rows of C�q� 'q are independent of 'qe�

The �rst condition is equivalent to the one imposed in Chapter � for the roboticsproblem� It stems from the fact that� even though passivity ensures that

'q�� 'D�q� C�q� 'q� 'q �

for all factorizations C�q� 'q 'q� for the energy shaping stage we need the stronger con�dition

z�� 'D�q� C�q� 'q�z � � �z � IR�

which is ensured by the skewsymmetry property i above� The second condition isrelated with the underactuated nature of the machine� and will be clari�ed below�

Using �� # and the transposed of the last expression in �� it is clear thatthe objectives can be achieved with the choice

C�q� 'q ��

�� f�qm� 'qr�npLsrJ e�Jnpqm 'qm

�f��qm� 'qr

��


with

f�qm� 'qr�� npLsrJ eJnpqm 'qr ��

This factorization leads to the following compact model representation�

D�q-q & C�q� 'q 'q &R 'q � Mu& � ��


The following problem will be solved by the use of an observer�based PBC in thissection�

De�nition �� Torque and �ux tracking with unknown load�� Consider theinduction motor model �� with outputs torque and rotor �ux norm k�rk to becontrolled� Assume�

A�� The load torque L is an unknown constant�

A�� Stator currents 'qs� rotor speed 'qm and position qm are available for measure�ment�

A�� All motor parameters are exactly known� and the viscous mechanical damp�ing constant is nonzero� i�e� Rm � �

Let the desired torque ��t be a bounded and di�erentiable function with knownbounded �rst order derivative� and the desired rotor �ux norm be a strictly positivebounded and twice di�erentiable function ��t with known bounded �rst and secondorder derivatives� Under these conditions� design a control law that will ensure inter�nal stability and asymptotic torque and rotor �ux norm tracking� that is� the closedloop system must give

limt� j � ��tj � � limt� j k�rk � ��tj � ��

from all initial conditions and with all signals uniformly bounded�

�� Ideal case with full state feedback

For the sake of clarity of presentation� the problem will �rst be solved under thetemporary assumption of full state measurement �measurable rotor currents andknown load torque� This is referred to as the ideal case� It will then be explained inthe next section how the controller can be modi�ed to remove this assumption�

�� Ideal case with full state feedback ��

Following the PBC approach we want to shape the motor total energy T �q� 'q ��'q�D�q 'q to the incremental form Td�q� 'q � �

�'�q�D�q '�q where we de�ned the error

signals '�q�� 'q � 'qd �

h'�q�e � '�qm

i�� with 'qd the vector of desired currents and internal

desired rotor speed� 'qd�� 'q�ed� 'qmd�

�� The introduction of the internal speed reference'qmd is speci�c to this observer�based approach� and it should not be confused with anexternal speed reference� which we will denote with 'qm��

Equation �� can then be rewritten as

D�q-�q & C�q� 'q '�q & �R&K� '�q � �

where K is a positive semide�nite matrix �to be de�ned below that injects the re�quired damping in the output feedback case� It can be set to zero when the state ismeasurable� The right�hand side in the equation above is de�ned as

�� D�q-qd � C�q� 'q 'qd �R 'qd &K '�q &Mu& �

Setting this perturbation term to zero is part of the implicit de�nition of the PBC�and results in

Ls-qsd & LsreJnpqm -qrd & f�qm� 'qr 'qmd &Rs 'qsd � u ��

'�rd &Rr 'qrd � �� !

Dm-qmd � f��qm� 'qr 'qsd &Rm 'qmd � �L �� #

with

�rd�� Lsre

�Jnpqm 'qsd & Lr 'qrd ��

The second equation is the �clamped dynamics� condition that ensures the desiredtorque is delivered� This is de�ned by ��#$ and can consequently be solved like inthe previous controller� that is� by setting

�rd � �

�cos��dsin��d

�� eJ �d

��

�� '�d �

Rrnp��

� � �d� � ��

There are two changes to the previous observer�less controller� First� from �� we observe that for the implementation of the control signal �� the rotor currents'qr must be measured� Second� the �fth equation of � � de�nes an additionalcontroller dynamics �� #�

The above derivations are summarized in the following proposition�

Proposition �� PBC with full state measurement�� Consider the inductionmotor model �� in closed loop with �� where 'qsd� -qsd� 'qrd� -qrdare calculated from �� and �� using �� Then� for all initial conditionsequation �� holds with all signals uniformly bounded� �


�� Observer�based PBC for induction motors

The main result of this section� a nonlinear observer�based PBC� is presented in theproposition below�

Proposition �� Observer�based PBC for induction motors�� Consider theinduction motor model �� with outputs to be controlled torque and rotor �uxnorm k�rk� and Assumptions A��A�� Let the control law be de�ned as

u � Ls-qsd & LsreJnpqm -qrd & npLsrJ eJnpqm '(qr 'qmd &Rs 'qsd �K�� 'qmd '�qs �� $

where

'qrd � �eJ �d�

��Rr��

np�

�

'qsd ��

Lsr

eJ �npqm��d�

� & Lr

Rr'�

Lrnp�

�

�

and with controller dynamics

'�d �Rrnp��

� � �d� � ��

-qmd ��Dm

��npLsr

'(q�r J e�Jnpqm 'qsd � Rm 'qmd � (L &K�� 'qd '�qm

��

with 'qmd� � 'qm� � The gains K�� 'qmd and K�� 'qd are given as

K�� 'qmd��

n�pL�sr

#��'q�md & k�� k� � �� $

K�� 'qd��

n�pL�sr

#��

�'q��d & 'q��d

�& k��

Rr

� k� � �� $�

while the state estimator and load adaptation law are

De�qm-(qe &W��qm 'qm '(qe &Re'(qe � Meu� L�qm� 'qm 'ee �� $ '(L � ��L '�qm� ��L � �� $!

with 'ee�� '(qe � 'qe the observation error and

L�qm� 'qm �

�

npLsrJ e�Jnpqm

�'qm �� $#

Under these conditions� the closed loop system achieves global torque and rotor �uxnorm tracking with all signals uniformly bounded� �

�� Observer�based PBC for induction motors ��

Proof� Since the control law above uses the estimated instead of the real states�contrary to the ideal case� the error equation in this case takes the following form

D�q-�q & C�q� 'q '�q & �R &K� 'qd� '�q � S�qm� 'qd 'ee & �� $�

where K� 'qd � diagfK�� 'qmdI�� K�� 'qdg� �� L�

� � � � � � � �(L �L�

� and

S�qm� 'qd �

�� npLsrJ eJnpqm 'qmd

�npLsr 'q

�sdJ eJnpqm

�� IR��

On the other hand� from �� $ and �� the observation error 'ee satis�es thefollowing equation

De�qm-ee & �W��qm 'qm & L�qm� 'qm� 'ee &Re 'ee � �� $�

Now� consider the composite Lyapunov function candidate

V ��

'�q�D�q '�q &

�

'e�e De�qm 'ee &

�

��L� �L

whose derivative� taking into account the skew�symmetry of

'D�q� C�q� 'qand

'De�qm� �W��qm 'qm & L�qm� 'qm�

yields

'V � � '�q��R&K� 'qd� '�q & '�q

�S�qm� 'qd 'ee � 'e�e Re 'ee &�

��L�L '�L & '�q

��

Use of �� $! in the equation above and de�ning z�� '�q

�� 'e�e �

�� results in the followingquadratic function

'V � �z�Mz

with

M �

� R&K� 'qd ��S�qm� 'qd

��S��qm� 'qd Re

�Checking that �� $ and �� $� ensures strictly positive de�niteness of M �seeSection #� of Appendix D� i�e�

M � �I� � �� $$


it can be concluded that '�q � L� � 'ee � L�

and (L � L hold �i�e� boundedness andthat 'ee and '�q are also square integrable �i�e� belongs to Ln

� � n � #� �� Since 'qsd� 'qrdare bounded by construction� then 'qe is bounded� which together with 'ee in its turnimplies that '(qe is bounded� The fact that 'qmd is bounded follows from boundednessof '(qr and �� with Rm � � and implies 'qm bounded since '�qm is bounded� From�� $� and �� $� it now follows that -�q� -ee are bounded� Since '�q� 'ee are boundedsignals with bounded derivatives� they are also uniformly continuous� Together withsquare integrability� this implies convergence of current errors to zero� Hence� rotor�ux norm and torque tracking can be concluded� with all internal signals uniformlybounded� �

�� Remarks to the controller

Remark �� Observer structure�� Under the assumptions that u � L� and

'q � L� � global exponential convergence of the estimated currents to their real values

can be proved� using only the part of V quadratic in 'ee together with �� $�� Toget this result� a speed dependent term �� $# proportional to the observation errorin stator currents is used to update the estimates in �� $ � This is common toseveral reduced order observers for which global stability results exists� see � � �� Thechoice of L�qm� 'qm follows naturally from the Euler�Lagrange structure of the model�aiming at getting 'De�qm� �W��qm 'qm & L�qm� 'qm� skew�symmetric� Unfortunatelythe convergence rate of the estimation errors depends on the minimum resistance� i�e��Re�

Notice also that even global exponential convergence of current estimation errorsto zero is not enough to claim stability of the total system with estimated statesin the controller� Nonlinear damping terms must be introduced in the controllerequations to ensure global stability when estimated instead of real states are used�An interesting task would be to incorporate other globally valid and exponentiallyconvergent observers into this scheme� This would give the possibility of havinga convergence rate which does not depend on the resistances� and may lead to aperformance improvement�

Remark �� Comparison with observer�less case�� Comparing �� with�� $� it can be seen that the di�erence is in the third term on the right hand side�where '(qr 'qmd is used instead of 'qrd 'qm as in �� and in the nonlinear dampingterm� where the internal reference speed 'qmd is used instead of the real speed 'qm�This internal reference speed is de�ned by �� and it depends on the estimatedrotor currents and the desired stator currents� After estimates and real currents haveconverged to their desired values� this speed will indeed be the actual rotor speed�To extend the torque tracking objective above to classical speed%position trackingproblems� an outer loop is needed to de�ne the desired torque �� Since the deriva�tives of the desired currents are needed in the control �� $� '� must also be known�

�� A dq�implementation ��

The speed%position control loop in Section !� is an example of how the outer torquegenerating loop can be de�ned�

Notice that the observed rotor currents and the derivatives of the desired currentsare used only in the case of a voltage input �� $� and hence they are not neededif a CSI or a VSI with current control is used� In these cases the controls will be thestator currents 'qs � 'qsd� and the controller above exactly reduces to the controller ofProposition �� in Chapter ��

�� A dq�implementation

For the purpose of implementation� it is of interest to formulate the controller in anarbitrary rotating frame of reference� This can be done by using the results fromSection ��!� giving

�dq��

��d�q

�� e�J �a�npqm��r� idq

��

�idiq

�� e�J a 'qs

udq��

�uduq

�� e�J au� 'qdqr

�� e�J �a�npqm� 'qr

�a is the solution of '�a � a� �a� � � with a the angular speed of rotation for thereference frame relative to the stator �xed frame� The real rotor currents rotated toa dq�frame� have been denoted 'qdqr �

Using these de�nitions and following the same procedure as for the derivation of�� it follows that the observer in �� $ and �� $# can be rewritten

�Ls

nddt(idq & � aJ & �I��(idq &

Lsr�LsLr

�np 'qmJ � �TrI��(�dq

o� udq � e�J au

Tr'(�dq & Tr� a � np 'qmJ (�dq & (�dq � Lsr

(idq � TrnpLsr 'qmJ�(idq � idq

�� $�

Next� a relation between '(qdq

r and the other estimated and measured dq�quantities mustbe derived� Notice from �� $ and �� $# that

'(�r &Rr'(qr � �npLsr 'qmJ e�Jnpqm

�'(qs � 'qs

�Di�erentiation of (�r � eJ �a�npqm�(�dq and substitution of this expression in the equa�tion above� gives after a rearrangement of terms

'(qdq

r � e�J �a�npqm� '(qr � � �

Rr

�� a � np 'qmJ (�dq &

'(�dq & npLsrJ 'qm

�(idq � idq

� Substitution of the �rst two terms in the bracket with terms from �� $� results in

'(qdq

r � � �

Rr

�Lsr

Tr(idq � �

Tr(�dq

�� $�


To express the control �� $ in terms of dq�quantities� notice that if the desiredstator and rotor currents are written in terms of rotated desired quantities�� iddq� 'qdqrdas 'qsd � eJ aiddq and 'qrd � eJ �a�npqm� 'qdqrd� it follows that

-qsd �d

dt

�eJ aiddq

� eJ a

� aJ iddq &

d

dtiddq

��

-qrd �d

dt

heJ �a�npqm� 'qdqrd

i� eJ �a�npqm�

�� a � np 'qmJ 'qdqrd &

d

dt'qdqrd

From the above and '(qr � eJ �a�npqm� '(q

dq

r � it follows that

npLsrJ eJnpqm '(qr 'qmd � npLsrJ eJ a '(qdqr 'qmd

�npLsr'(q�r J e�Jnpqm 'qsd � �npLsr

�'(qdq

r

��J iddq ��

Substitution of the above expressions into �� $ gives the control in any referenceframe� Especially� with the choice

a � np 'qm & '�d

where '�d is de�ned in �� $ can be rewritten using �� as

u � eJ �npqm��d��Ls

�d

dtiddq & �np 'qm & '�dJ iddq

�& Lsr

�d

dt'qdqrd & '�dJ 'qdqrd

�&npLsrJ '(q

dq

r 'qmd &Rsiddq �K�� 'qmd

�idq � iddq

� where '(q

dq

r is given in �� $�� and 'qmd is de�ned by

-qmd ��

Dm

��npLsr

�'(qdq

r

��J iddq � Rm 'qmd � (L &K�� 'qd '�qm

��

'qmd� � 'qm�

The desired stator and rotor currents in the dq�frame are in this case given as �seeProposition � ��$

'qdqrd � e�J �d 'qrd � ��

��Rr��

np�

�

iddq ��

Lsr

�� & Lr

Rr'�

Lrnp�

�

�

The nonlinear gains K� and K� can be calculated directly from �� $ and �� $��noting that the squared norm of 'qsd is equal to the squared norm of iddq�

��Desired stator currents �qsd expressed in a dq�frame are denoted iddq �

�idd� i

dq

��



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1

0

1

2Speed tracking error

Time [s]

[rp

m]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.05

0

0.05

Time [s]

[W

b]

Flux tracking error

Figure � �� Tracking errors for �ux norm and speed� Observer�based controller�

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−5

0

5

10

15x 10−5 Estimation errors in rotor currents

Time [s]

[A

]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.05

0

0.05

0.1

Error of internal speed reference

Time [s]

[rp

m]

Figure � �$� Components of estimation error '(qr� 'qr and error between the real speed'qm and the internal speed 'qmd from �� Observer�based controller�


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−4

−2

0

2

4Currents in phases a, b

Time [s]

[A]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−40

−20

0

20

40

Time [s]

[V

]Voltages of phases a, b

Figure � �� !� Voltages and currents ua� ub� ia� ib� Observer�based controller�

Simulations with the same reference trajectories �see Fig� � �!� �lter values and underexactly the same conditions as in Section !�� were performed with the observer�basedcontroller in Proposition � ��$� The outer speed control loop was chosen as in �� ! to generate � � with �ltering of the speed error to allow for speed tracking withoutacceleration measurement� Control parameters were chosen as� a � � � b � ! �k� � ! � k� � � �� #�Rr� ��L � � All initial conditions were set to zero� Resultsfrom the simulations are shown in Figs� � ��

As it can be seen from Fig� � �� there is no interaction between speed and�ux norm control� as predicted from the analysis� Also for this controller �ltering ofreferences was found to be of high importance for avoidance of saturations in currentsand voltages� and the parameter k� in the additional damping term was important forimproving tracking performance for reference speeds close to zero� For small dampingRm� the parameter k� was also important for the quality of speed tracking�

Comparison with Figs� � �#�� from the observer�less case shows that thereis little di�erence between the two approaches under ideal conditions� This can beexplained as follows� After a short initial transient the internal speed 'qmd and therotor current estimates converge to their real values �and consequently also theirdesired values� and then the two controllers behave exactly similar�

�� Concluding remarks ��


In this section an observer�based solution to the output feedback torque and rotor�ux tracking problem for an induction motor model was proposed� The controlleris an outgrowth of the work in �$$� #� �� More speci�cally� it is an extension ofthe work presented in the last paper to include the important case of rotor �ux normtracking� In this approach the design of controller and observer is interlaced� andglobal stability results for the total system can be proved� provided that nonlineardamping terms are added to the controller to compensate for estimation errors�

It must be pointed out that the observer used in this approach is simple� withonly updating from current error terms in the rotor equations� It would in generalbe advantageous to have updating also in stator equations for making the observermore robust� The inclusion of error terms with adjustable gains will also give thepossibility of specifying a convergence rate less sensitive to resistance parameters�These are possible future extensions of this observer�based result�

However� the observer is only needed when a voltage input is used� since thecontroller proposed in this section reduces to the controller of Proposition �� seeChapter �� for a motor with a CSI or a VSI with current control� Consequently� therobustness results from �� see Section �� with respect to uncertainty in rotor�ux time constant Tr hold for this controller too in the case of current inputs�

Field�oriented control and feedback linearization

The concept of �eld�oriented control �FOC was introduced by the German researchersHasse and Blaschke in the beginning of the seventies� It was based on the spacevector description of the dynamic model� formulated in the �fties by the researchersLyon� Kovacs and Racz� for the analysis of transients in AC�machines� The ideaspresented in �� for vector control of a PWM inverter fed induction motor� showeda remarkable improvement in response as compared to previous scalar methods� whichwere based on steady state linear models� The ideas did not seem to be of generalinterest� and their realization was quite complex� Some years later a general theoryfor vector control of AC machinery� based on deep understanding of the physics ofthe systems and a nonlinear model� was presented in � #�� This theory is now thede facto standard for high�performance control of AC machinery� in good part dueto the strong in�uence by the pioneering work of Professor Leonhard� The methodconsists of a nonlinear change of coordinates together with a nonlinear asymptoticallydecoupling state feedback� Seen from the viewpoint of modern control theory� FOCwas one of the earliest implementations of ideas which are now considered to belongto the more general �eld of geometric control theory� The fact that it gives superiorperformance as compared to methods based on classical linear theory� has motivatedand paved the way for other applications of nonlinear control theory�


In the rest of this section the rationale behind rotor��ux�oriented vector control��and the arguments commonly used against this approach are explained� In particular�it is argued that better performance can be achieved with a controller that exactly �andnot just asymptotically linearizes and decouples the motor dynamics� A controllerthat achieves this objective is presented�

There are generally two types of FOCs� and in this presentation� we adopt thefollowing de�nitions of direct and indirect FOC�

De�nition �� Direct and indirect FOC�� Direct FOC refers to an imple�mentation where the full state of the system is measured �Hall�e�ect sensors� searchcoils� tapped stator windings � while indirect FOC refers to an implementation whereonly stator currents� position and speed are measured�

There is some inconsistency in the literature regarding these de�nitions� and some�times direct FOC �also called �ux feedback control refers to the case where �ux ismeasured or estimated� while approaches using the reference values are denoted indi�rect �or �ux feed�forward control� See� e�g� pp� � #� � in � $�� We prefer to takethe clear distinction of state or outputfeedback given above�

It must also be pointed out that there is no clear de�nition in the literature ofwhat exactly an estimator or a reference signal is� Our PBC has an implicit estimator�in the sense the �� with the reference �d interpreted as an estimate�

�� Rationale of �eld oriented control

The simplicity of controlling a DC motor is due to mechanical commutation� whichensures that the main �ux from the �eld winding in the stator is always orthogonal tothe magneto�motive force created by the current in the armature winding� It followsthat the torque� which is proportional to the vector cross product of �ux and current�will be proportional to armature current when �ux is kept constant� Hence� it caneasily be controlled to give high dynamic performance by use of two linear loops�controlling �ux and armature current� In AC machinery �ux and MMF distributionsrotate with di�erent speeds� resulting in varying relative angle� This motivates theuse of rotating reference frames to analyze the dynamics�

Looking at the torque equation for the rotated model of an induction motor in�� $�

� npLsr

Lr

��diq � �qid

�There are several implementations of FOC� depending on the frame of reference chosen for modelrepresentation� e�g� rotor�� magnetizing ux� and stator�oriented� The rotor�oriented implementationis what now commonly is called �eld�oriented control ��

�� Rationale of �eld�oriented control ��

it can be seen that if one of the components of �dq is held constant� while the otheris zero� torque can be controlled by controlling one of the rotated stator currents�analogous to the DC motor� To achieve this objective a reference frame in which therotor �ux vector is aligned with one of the axis� should �rst be chosen� Thus� theangle of the rotor �ux vector must be known�

Assuming that the d�axis of the reference frame has been aligned with the rotor�ux vector� it can be derived from �� using �� $� that the rotor �ux speed�relative to the rotor frame can be written in terms of dq�components of �ux andcurrents as

'� �Lsr

Tr

iq�d

Indeed� if it is assumed that j�dj � � � � and a is chosen to be the rotational speedof �ux relative to the �xed stator frame�

a � np 'qm &Lsr

Tr

iq�d

��

the second of the equations in �� gives

'�q � � �

Tr�q

Hence �q converges exponentially to zero�

The �rst equation in �� becomes

Tr '�d & �d � Lsrid & �t �� !

with �t and exponentially decaying term� Thus� �d can �in principle be controlled byid� For instance� if id �

�Lsr

with � � a constant� it follows that limt� �d � ��

Assuming that �d is held constant� while �q is still zero� it follows that

� npLsr

Lr�diq

and torque can be controlled by the q�component of the rotated stator currents� Thisis the essence of FOC�

The remaining problem is to control the rotated stator currents idq� This can eitherbe done by high�gain current control� or if this does not give satisfactory performance�by exact cancelation of parts of the dynamics in �� High�gain current controlleads� invoking a singularperturbation argument� to a simpler model� the socalledcurrentfed machine� This case is treated in greater detail in Chapter ��

Otherwise� with the nonlinear decoupling input

udq � �Ls

� aJ idq & Lsr

�LsLr

�np 'qmJ � �

TrI�

��dq & vdq

�� #


equation �� can be written as

d

dtidq � ��idq & vdq

The inputs vdq can now easily be de�ned to force id and iq to their desired values�Usually vdq consist of nested PI�loops as in ��!�

vd � H��p�idd � id

��

vq � H��p�idq � iq

��

where Hi�p� i � �� are PI�controllers� and the desired stator currents idd� idq could

be de�ned as

idd ��Lsr

�� $

idq �Lr

npLsr�d ��

with d generated by an outer speed PI�controller

d � H��p� 'qm� � 'qm ��

Instead of the feed�forward de�nition of idd� it could be de�ned using feedback as

idd � H��p�� d

with H��p a PI�controller�

Assuming rotor �ux is available for measurement� the voltage input for direct FOCmay then be implemented as

u � eJ audq

� eJ a�Ls

� aJ idq & Lsr

�LsLr

�np 'qmJ � �

TrI�

��dq & vdq

��

It is also possible to implement high�gain current control by neglecting all termsexcept vdq in the equation above�

Notice that to compute a from �� the norm of the rotor �ux must be strictlygreater than zero� This assumption does not hold at startup� giving a controllersingularity at this point� To avoid the unwanted controller blow�up for small valuesof rotor �ux norm measurements or estimates� some heuristics is added to the controlscheme to make it work� for instance exciting id before iq or setting �d used in thecontroller equal to a constant value� when the measured%estimated value is smallerthan a certain limit�

�� State estimation or reference values ��

�� State estimation or reference values

In the above derivations it has been assumed that all states including rotor �uxnorm and angle could be measured� In general this assumption does not hold� asexplained in Remark � �#� This problem has been a longstanding research topic� andgenerally there are two ways to solve it� The �rst one is to estimate the rotor �uxangle and amplitude� while the other is to use reference values for these two quantities�

As an example of the �rst method� rotor �ux can be estimated in open loop fromstator current measurements using the �rst equation of �� and its angle can befound by integrating �� with the estimated value of �d as

( a � np 'qm &Lsr

Tr

(iq(�d

��

where

(�d �Lsr

Trp& �(id ��

The estimated currents (idq are computed from measured currents using the estimated

angle (�a and the rotation de�ned in �� !� This simple estimation scheme has beenused for high�performance control of an induction motor in �! �� See � � � for othersolutions to the estimation problem�

To implement indirect FOC� the same feed�forward way of de�ning the desiredcurrents as in �� $ and �� is used� but now the rotor �ux speed in �� isalso computed using reference values

a � np 'qm &Lsr

Tr

idq�

and �a is found by integration as before�

The indirect approach has become the most popular implementation of FOC sinceit does not require �ux sensors or a �ux model� avoiding the need for estimation � $��Also� its performance at low rotor speed is generally better than for direct schemes� forwhich there are estimation problems present when the ohmic losses become dominantin the stator equation� and signal integration is problematic �� However� allof these schemes are highly sensitive to parameters� especially to changes in therotor time constant Tr� When this parameter is uncertain� asymptotic decouplingof �ux norm and torque is lost� resulting in second order dynamic �ux and torqueinteractions�

It is often argued that the indirect approach is much more sensitive to parameteruncertainty than other direct approaches� This issue has recently been addressedin �� where it is shown that the �rst requirement of the system� global stability�


is preserved for the indirect approach with as much as a ) error in rotor resis�tance estimate� This is a remarkably strong result� and to the best of the authors�knowledge� there is no such result for the direct approach� where controllers based ona �nonlinear separation principle� are used� See Section �� for a presentation ofthese robustness results�

Motivated by the successful use of the voltage decoupling terms in �� # incombination with estimated states� there have been several attempts to implementequivalents by using reference values instead of the real the states� see pp� �� in � $�� The �rst problem with these approaches is that the term �Ls

ddtiddq with

the derivative of the stator current references iddq� is either neglected� resulting inunwanted torque overshoot� or has to be computed in an ad hoc way by numericaldi�erentiation� The other problem is that these schemes are only suited for constantrotor �ux norm� The indirect scheme proposed in Proposition � �� see also Section!�$ is an alternative method which do not possess these problems� and which alsogives a theoretical explanation of why these reference value �decoupling� approachesactually work�

�� Shortcomings of FOC

There are mainly two arguments used against the various implementations of FOC�

i� It does not give full decoupling

The decoupling between �ux and torque �or speed control is only asymptotic��$ �� There is only decoupling when the �ux has converged to its constant value�giving also a decoupled and linear speed dynamics� This means that simultaneouslytracking of both �ux amplitude and torque%speed%position using FOC most likely willcause problems� especially if the �ux reference is not slowly varying� For instance� aspointed out in � �!�� operation in the �ux�weakening regime will excite the couplingbetween �ux and speed� This gives undesired speed �uctuations� and could possiblycause instability� The problem is due to the rationale behind FOC� which is to make itbehave like a DC motor� where torque is proportional to current when �ux is constant�However� this does not give a decoupling between speed%torque and �ux norm� whichare the outputs a tracking controller must be designed for� For the reasons above ithas been natural to operate the machine at maximum constant �ux level below ratedspeed� something which restricts the possibility of power e�cient operation�

The problem of full �dynamical decoupling of rotor �ux norm and torque �exactinput�output linearization� still under the assumption of full state feedback� wassolved in ��#!�� The basic idea for the dq�implementation �under the assumption ofideal �eld orientation� �q � is to choose �d and the torque

� npLsr

Lr�diq

�� Shortcomings of FOC ��

as controlled variables� instead of �d and iq� For a motor with current control loopsas previously explained� decoupling can be achieved directly by de�ning the currentreference for iq as

idq �Lr

npLsr

��d

�� !

with � an external torque reference�

Some additional insight into the decoupling problem can be gained if the newnonlinear decoupling terms in the q�direction are found by directly evaluating thedynamic equation for the torque� giving

' � npLsr

Lr

�'�diq & �d

d

dtiq

��

��

Tr& �

� &

Lsr

Tr

id�d � a

npLsr

Lr�did

� L�srn

�p

�LsL�r

'qm��d &

npLsr

�LrLs�duq

where �� !� iq �Lr

npLsr��d� and a as de�ned in �� have been used�

From the above it follows that the choice

uq ��LrLs

npLsr�d

�vq � Lsr

Tr

id�d & a

npLsr

Lr�did &

Lsrnp�LsLr

'qm��d

gives

' � ��

�

Tr& �

� & vq

and can be controlled with a linear inner PI�controller H�p

vq � H�p�d �

where d could be chosen as in �� The voltage decoupling term in the d�axis isstill given by the �rst of the equations in �� #�

With another choice of outputs� the decoupling problem is not as easy to solveas above� and tools adopted from di�erential geometry must be used for its solution�This approach will be explained in the next section�

ii� It is based on �a nonlinear separation principle�

It is well known that for linear time�invariant systems� the problem of stabilizabil�ity with only partial state measurement can be solved by use of an observer� at leastif the system is both stabilizable and detectable� The stability of the total systemis only dependent on the observer stability� and the stability of the feedback controlsystem when full state measurement is assumed� This is the so�called �deterministicseparation principle�


Motivated by the successful linear controller designs� this principle has been car�ried over to controller designs for nonlinear and non autonomous systems� giving a�nonlinear separation principle�� Generally no theoretical stability analysis are givenfor systems resulting from such designs� and the performance is only veri�ed throughsimulations or experiments� It can be showed that even for very simple nonlinearsystems� with an exponentially convergent state observer�� and known parameters�the approach of separate observer�controller design can lead to explosive instability interms of nite escape time� when estimated states are used as if they were real states�even if the controller with real states would ensure global exponential stability of thetotal system ��# �� Analysis of separate observer�controller design using lineariza�tion techniques leads to local stability results� even under the assumptions of knownconstant parameters� With only local results� it is very di�cult to predict what willhappen for certain choices of controller parameters and initial conditions under realoperation� and the designer is left with a trial and error approach to controller design�making tuning an often time consuming and di�cult task�

It must also be pointed out than when observers are used for implementation�asymptotic properties are added to the schemes� i�e� there is only exact decouplingafter the estimates have converged to their real values�

Even if some heuristics are implemented to avoid instability� estimation errorscan lead to severe e�ects on system performance and power e�ciency� especiallyduring high speed transients� This is why much of the research in nonlinear controlof induction motors has aimed at interlacing design of controller and observer� withadditional terms in the controller or the observer to counteract for estimation errors�and obtaining global stability results� These schemes have the bene�t of guaranteeingstability �under given assumptions� and hence giving a priori information aboutwhich modi�cations can reasonably be expected to work and which ones will probablynot� In the cases where exponential stability results are established for a nominalsystem� robustness result for bounded perturbations can also be established �� $��

It must be pointed out that even if the robustness and stability issues of the manyimplementations of FOC are not rigorously established� this scheme has throughyears of practical experience been developed to a level giving a performance whichis di�cult to compete with for other nonlinear approaches� at least for nonlineardesigns implemented directly from theoretical desk designs� Years of experimentalwork has resulted in modi�cations of FOC based on experience and intuition� andproblems of parameter uncertainty� �ux saturation and other unmodeled dynamicscan be compensated for by several ad hoc methods� Even if these methods are basedon �nonlinear certainty equivalence�� using estimated parameters as if they were realparameters in the controller� and proposed without theoretical justi�cation� the mostimportant aspect from an application oriented view is that they work and improveperformance� The rigorous analysis of the resulting schemes is left as challenging

�This is generally the best convergence that can be expected�

�� Feedback linearization ��

problems for the academics�

�� Feedback linearization

As a solution to the decoupling problem in standard FOC� an input�output lineariza�tion from voltage inputs to speed and square of rotor �ux norm in the case of thestator �xed ab�model� was derived in ��#!� �see also ��$ �� using the new powerfultools adopted from di�erential geometry�

The linearization procedure starts with �� and the de�nition of a set of newoutputs yi � IR

� �� IR� i � �� !� which are functions of the states x � � 'qm� ��ab� i

�ab�

�

y��x � 'qm

y��x � Lfy��x �npLsr

DmLr��aib � �bia� L

Dm

y��x � k�abk�

y��x � Lfy��x � � Rr

Lrk�abk� & RrLsr

Lr��abiab

y��x � arctan��b�a

In the equations above� Lfyi�x�� yi��x

�f�x is the directional derivative �orthe Lie derivative�� of the function yi�x along the vector �eld f�x� and satis�esLf �L

m��f yi�x � Lm

f yi�x�

The de�nition of new outputs is not de�ned for k�abk � �

Di�erentiation of the equations above under the assumption that the load torqueL is constant� results�� in

'y��x � y��x

'y��x � L�fy��x & Lg�Lfy��xu� & Lg�Lfy��xu�

'y��x � y��x

'y��x � L�fy��x & Lg�Lfy��xu� & Lg�Lfy��xu�

'y��x � Lfy��x

from which it can be seen that if the real inputs are given as�u�u�

�� B��x

� �L�fy��x & v�

�L�fy��x & v�

�� #

��Named after the famous Norwegian mathematician Marius Sophus Lie �� Notice that the terms Lg�yi�x�� Lg�yi�x�� i � �� are zero�


with auxiliary inputs v�� v�� and the decoupling matrix

B�x �

�Lg�Lfy��x Lg�Lfy��xLg�Lfy��x Lg�Lfy��x

�

�

� � npLsrDmLr�Ls

�bnpLsr

DmLr�Ls�a

�LsrRr�LsLr

�a�LsrRr�LsLr

�b

��

the �rst four equations of the resulting system will be

'y��x � y��x

'y��x � v�

'y��x � y��x

'y��x � v�

which are two second�order linear and totally decoupled systems for speed y��x andsquare of rotor �ux norm y��x� These two outputs can now be controlled indepen�dently with v� and v� outputs from simple linear controllers�

The �fth di�erential equation of the system represents the dynamics of the �uxangle

'y��x � np 'qm &Rr

np

�

k�abk� �Dm-qm & L

which is unobservable with the choice of outputs and new controls in this approach��$ ��

From �� it is clear that the above scheme has a singularity for zero rotor�ux norm� Also� for the implementation of this scheme� measurement of the rotor�ux is required� and with the use of observers� the points from the previous sectionregarding use of a �nonlinear separation principle� apply to this scheme as well�

It can also be seen from �� # that the need for exact cancelation of parts of thedynamics is essential for the implementation of this scheme� The terms �L�

fyi�x� i �

�� !� which depend on the full state vector x � � 'qm� ��ab� i

�ab�

�� must be cancelled withthe control signals u� and u��

It was also shown in ��#!� that the implementation complexity of this scheme isnot much greater than for FOC� Experimental results from various implementations�stator �xed frame of reference� decoupling control for square of rotor �ux norm andspeed or torque of this scheme have been presented in �� $#�� and �! � �dq�frameimplementation with high�gain current control�

In Section �� ! we carry out a detailed comparison between PBC� FOC andfeedback linearization for the case of currentfed machines�

� Experimental results ��

� Experimental results

�� Experimental setup

In this section the experimental setup used in the testing of di�erent induction motorcontrollers is described� The setup was built from scratch with basic ideas for im�plementation taken from � !�� but modi�ed to �t speci�c needs� and avoid reportedproblems with noise and signal transmission� Additional equipment for load torquegeneration and torque measurement were built from standard components� For thesoftware interface� an integrated system from dSPACE was chosen� This choice al�lowed for fast prototype implementation without extensive C or assembly languagecoding� The motor used in the experiments was a #�pole !� squirrel�cage inductionmotor with a voltage source switched converter and a standard PWM scheme� and thecontrollers were implemented on a DSP� A more detailed description of the inductionmotor part of the setup can be found in ��

A Hardware description

The controller was implemented on a DS�� controller board from dSPACE� Thisboard has a # MHz TI! C!� ! %! bit �oating�point DSP and a � MHz TI! P�#! %�� bit micro controller DSP� In addition there are # A%D�converters which are usedfor current �� bit resolution� ! �s conversion time and torque�� bit resolution�� s conversion time measurements� �� bit digital I%O of which � bits were usedfor pulse�width modulation and � bit for a converter enable signal� and two # bitencoder interfaces� of which one was used for position measurement� The microcontroller computed the switching signals for the symmetric carrier�based PWM ofthe three phases from reference values transferred to it from the main processor atthe end of each sampling interval� The three signals for the upper transistors in theconverter legs were converted to optical signals on an interface card before they weretransmitted through optical �bers to the converter� where complementary switchingsignals and blanking time of the converter � �s were generated in hardware usingan IXYS IXDP�! digital deadtime generator for !� PWM control�

A �� kW Lust FU !� voltage source converter with a DC�link voltage of UDC �! V was used� The converter could give maximum line currents of Imax � �� A�This converter was connected to a #�pole �np � Lust ASH�� I�!� # Wsquirrel�cage induction motor� as shown in Fig� � �� on page !�� The followingnominal two phase parameters of the motor were given in the data sheet� LN

s � �� mH� LN

sr � � �! mH� LNr � �$�� mH� RN

s � �� *� RNr � � *� Dm � �� kgcm��

'qNm � ! rpm� N � � Wb� N � �� Nm� Rm � � � Nms%rad� A Lust BC�

��The torque was measured only for illustration purposes� It was not used for feedback in any ofthe experiments presented in this section�


Filter

AT BUS

220 V AC

I/O CONNECTOR

Lust FU2235

INVERTER

DS1102 Controller Board

dSPACE

ADC

Electrical/OpticalInterfaceFilter Circuits

SWITCHING SIGNALS

FIBEROPTIC

TRANSMISSION

TI320P1425 MHz

TI320C3140 MHz

COUNTER

LOCK SIGNAL

ENCODER SIGNALS

BLDC

DRIVER

LEMCurrent

Transducers

Filter

220 V AC

SIMULINK

dSPACE RTI

MATLAB

PC 1PC 2AT BUS

I/O BOARD

LOAD SYSTEM DRIVE SYSTEM

INDUCTION MOTOR CONTROL SYSTEM

INDUCTION MOTOR

TRANSDUCER

TORQUEBLDC MOTOR

LOAD

RE

SOL

VE

R S

IGN

AL

AMPLIFIER

ibia

ME

ASU

RE

D T

OR

QU

EU U UA B C

TorqueCommand

Speed

Figure � �� Block diagram of experimental setup�

brake chopper was connected to the DC link for power dissipation�

To allow for optical transmission of the three switching signals and one enablesignal from the DSP board to the converter� the standard microprocessor board forvoltage%frequency control was removed from the inverter� and replaced by a speciallydesigned interface card� Over�current protection was implemented both in softwareand hardware�

Position was measured using an incremental encoder with # �� lines� and a quadru�ple counter was used� giving a position measurement resolution of ��

�� rad�

The currents were measured with LEM LA ��NP current transducers and �lteredwith �rst order analog anti�aliasing �lters� having a cut�o� frequency of �� kHz� before

�� Experimental setup ��

they were converted by the � bit A%D converters� An o�set correction of the currentmeasurements was done at startup� when the inverter switches were disabled and thecurrents in the motor windings were zero�

As a load for the induction motor� a current controlled BSM � A � brushlessDC�motor from Baldor with a BSC�� driver was used� This motor has a momentof inertia equal to ��! kgcm�� and is capable of producing a nominal torque of !� Nm� A separate PC with a PCL�$��B I%O board and external electronics was usedto control the load�

Torque was measured using a HBM T� � Nm strain gauge torque transducerwith an AE� � ampli�er� and the signal was �ltered with a �rst order low�pass �lter�This torque transducer has a moment of inertia equal to �� kgcm��

An overview of the total system is given in Fig� � ��

B Software description

Saturation

2

u_b

Reference voltagesfor PWM calculation

Numerator(z)Denominator(z)

Filter

Measuredtorque

Speed reference

Passivity−basedinduction motor controller

Controller

1

u_a

3

u_c

Speed reference generator

3

Measured signalfor current in

phase a

2

Increment ofrotor angle

1

Estimated speed

4

Measured signalfor current in

phase b

SIMULINK Block Diagram for C−Code Generation

5

Measured torquesignal

K

Scaling

K

Scaling

K

Scaling

Figure � �� Main block diagram for C�code generation from SimulinkTM�


i_q*

dot_i_q*

ddot_betadot_beta

3−>2 Phasereduction and

rotation to dq−frame

Compute u_dq

Compute uin dq−frame

Speed filtering,speed controller,calculation of i_q*

and dot_i_q*

3

delta_q_mCalculate

q_m

q_m

dot_i_d*

Calculate i_d*

u_d

Rotation tostator frame and

2−>3 phasetransformation

i_d*

Calculate rho_*

theta_a

rho_* n_p q_m

Calculaten_p q_m

u_q

i_q

i_d

Filtering of beta

beta

1

u_a

2

u_b

3

u_c

1Unfiltered

speed reference

beta_refSignal

generator

4

i_a

0.5*Rr

epsilon

num_speed(z)den_speed(z)

Filter

2Estimated

speed

−K−

n_p dot_q_m

5

i_b

Figure � �� Example of SimulinkTMblock diagram for controller�

The DSP board was installed in a � #��%��MHz PC with MatlabTM%SimulinkTM�and a RTI C code generator was used� which converted the controller graphicallydescribed in SIMULINK to C code which could be compiled and run on the DSP�For logging of signals and online tuning of parameters� the programs TRACE andCOCKPIT�� were used to communicate with the DSP board�

Computation of speed� position increment� current and torque measurements andthe communication between the main processor and the micro controller� was imple�mented in external C�code which was linked with the code generated from the blockdiagram� These routines also provided the mapping between software inputs%outputsspeci�ed in the block diagram� and hardware addresses on the board� The code forthe PWM running on the micro controller was written in assembly and down�loadedto the slave�processor during startup� In Fig� � �� the main block diagram usedfor code generation is shown� and in Fig� � �� an example of the diagram for thecontroller block is shown�

��MatlabTMand SimulinkTMare registered trademarks of The MathWorks Inc� RTI� COCKPITand TRACE are registered trademarks of dSPACE GmbH�


The controller was implemented with multirate computation� where the PWMcalculation was run relatively fast on the micro controller at a sampling period ofTPWM � � �s independent of control algorithm� with a slower computation of refer�ence voltages and speed control at a period of Tsampl seconds� These sampling periodsdepended on the implemented control algorithm�

C Controller discretization and speed estimation

In the derivations below a base sample period of T �s� for controller implementationis assumed� and y�k� k � Z� is used to denote a sample of the signal y�t at timet � kT � Discretization of controller equations were done using the ZOH �Zero�Order�Hold approximation of an integration� In this approach the discrete equivalents oftransfer functions for linear continuous systems are derived by using the formula

hZOH�z � �� z��Z�h�p

p

�where z�� is the delay operator� and Z denotes the z�transform� This gives ��$�

Linear �lter�y

u�p �

�

T�p& ��

y�k � e� TT� y�k � � & �� e

� TT� u�k � �

Integration�y

u�p �

�

p�

y�k � y�k � � & Tu�k � �

To avoid integral wind�up� conditional integration was implemented in all PI�controllers� In this approach� the integral term is held constant when the output ofthe controller exceeds an adjustable limit�

Since the induction motor used in this work had no speed transducer� speed hadto be estimated from discrete time position measurement� This was done using thesimple backward di�erence approximation ��

('qm �qm�k� qm�k � !

!T

This average speed is a rather rough �rst estimate with a resolution �assuming quadru�ple counter of

'qresm �

��Npos

!T


Npos is the encoder resolution in pulses per rotation� For T � ! �s and Npos �# �� the resolution is 'qresm � �#! rad%s � � #�� rpm � The backward di�erenceestimate of speed was smoothened using a discrete implementation of the linear �lter

h�p ��

!Tp& �

before it was used for control purposes�

D Phase transformations

To transform the measured !� line currents Ia and Ib to equivalent � phase currents'qs used in the controller calculations� the linear transformation

'qs �

�'qs�'qs�

��

r

!

� ��

p��

p!

� ��p�Ia

�p�Ib

�

was used� The !��voltages ua� ub� uc used for the PWM calculation were computedfrom the voltages u�� u� calculated by the controller as�

� uaubuc

��

r

!

��

��

p��

��p��

��

u�u�

�

Note that uc � ��ua & ub� hence only ua and ub are needed�

The form of the transformation used here is the so�called power�invariant form ofthe transformation� see � $��

E Pulse�width modulation

A standard symmetric carrier�based PWM was chosen for generation of the switchingsignals of the converter�

In this method the switching signals for the transistors in the three bridge legsof the converter �see Fig� � �� are generated to be symmetric to the middle ofthe switching interval� and the on�time for each upper switch ti

ON� i � fa� b� cg is

computed from

tiON

�

��

ui � �UDC�

��& ui

UDCTPWM �UDC

� ui UDC

�

TPWM ui �UDC�

where ui is the constant value of the reference at the beginning of the switchinginterval which starts at time t � kTPWM� k � Z��

�� Outline of experiments ��

To get the desired symmetry of the digital switching signals� the slave DSP gen�erates two signals for each upper switching signal� These are then passed throughan external XOR gate� Both inputs to the XOR gate are logical high at the be�ginning of each switching interval� t � kTPWM � The �rst input is set low at timet � kTPWM & TPWM� � tiON� � and the other at time t � kTPWM & TPWM� & tiON� �

The three TTL voltage outputs from the XOR gates were then converted to currentsignals suited for generating optical signals using external electronics� and transmittedto the converter� Complementary signals for switching of the lower transistors andblanking time were generated by an IXYS IXDP�! PWM controller in the converter�

R

RE

CT

IFIER

300 V

LoadingCircuit

STATOR WINDINGS

220 V

50 Hz

AC

INVERTER-MOTOR CONFIGURATION

C B A

u

uu

a

bc

Figure � �� Inverter�motor con�guration�

�� Outline of experiments

Several series of experiments were carried out with the equipment described aboveand di�erent controllers� and to limit the number of �gures� only plots from what wasconsidered as illustrative experiments are shown� Unless something else is explicitlystated� the load torque used in the experiments was only due to friction� However�the converter of the load was turned on during all experiments� Even if the refer�ence for the load torque�� was set to zero� this gave some additional high frequencyoscillations of small amplitude in the system� Also� to limit the number of plots�measured currents and reference voltages for PWM have only been included for a fewof the reported experiments� In those cases where saturation was experienced� thisis commented explicitly� The references were generated by linear �rst order �lteringof step%square�wave signals from real�time implementations of the SimulinkTMsignal

��The control of the load torque was feed�forward in currents �i�e� torque� for the brushless DCmotor�


generator� Higher order derivatives of references for position%speed and �ux ampli�tude ��t were obtained from state space representations of linear �lters� similar tothose reported in the simulation part of Section !�� For this reason� in the caseswhere both a desired quantity and an estimated or measured quantity are shown inthe same plot� the desired value can be distinguished from the other as the smoothestcurve�

It was aimed at showing that �ux norm and speed can be independently controlled�and for this reason either the speed� or �ux reference was held constant during eachexperiment� while the other reference was a varying function� The un�ltered time�varying references were chosen to be square�waves� with a maximum amplitude of the�ux reference equal to the nominal �ux level of the motor�

The outline of the rest of this section is as follows� In Section ��! some of theresults from an implementation of the observer�less scheme described in Section !� arereported� Section ��# contains experimental results with the observer�based controllerfrom Section #�#� In Section �� results from an implementation of the rotor��ux�oriented control scheme in Section �� are given for the purpose of comparison withthe passivity�based controllers� Finally� concluding remarks to the experimental workare given in Section ��

�� Observer�less control

The behavior of the controller presented in Section !� � was �rst investigated in aseries of experiments� For later convenience �� is rewritten here in terms of itsdi�erent components

u � Ls-qsd & LsreJnpqm -qrd & npLsrJ eJnpqm 'qm 'qrd &Rs 'qsd� �z

desired dynamics

� �n�pL

�sr

#�'q�m & k� '�qs� �z

nonlinear damping term

� KIs

Z t

�

'�qsdt� �z integral term

��

where a possible integral term in stator currents have been added�

Since �ux measurement was not implemented in the setup� a �ux observer had tobe run in parallel with the controller for the purpose of verifying �ux tracking� Forlater comparison with an implementation of FOC� the estimation scheme in �� on p� !�$ was chosen� To avoid the singularity in the rotor �ux speed esti�mation for zero �ux estimate� it was necessary to substitute (�d in the division �see�� by a small constant c � � � whenever (�d c�

To the desired torque de�ned in �� ! and �� ! �or �� #� for position con�

�� Observer�less control ��

trol� an integral term was added to compensate for unknown load torque� giving

d � Dm-qm� � z � f�qm � qm� & (L �� $

'z � �az & b� 'qm � 'qm�� a� b � � z� � 'qm� � 'qm�� '(L � ��Le� ��L �

where qm�� 'qm� are the rotor position and speed reference� and f � for speedtracking� The error term in the integral action was set to e � qm � qm� for positiontracking� and e � 'qm � 'qm� in the case of speed tracking�

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−5

0

5Speed regulation error

Time [s]

[rp

m]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.08

0.1

0.12

0.14

0.16

0.18

0.2

Estimated flux norm and its reference

Time [s]

[Wb

]

Figure � ��!� Speed regulation%�ux tracking without integral action in currents�

The controller was �rst tested without integral action in stator currents� and atypical response is given in Fig� � ��!� As can be seen from the �gure� speed regu�lation is satisfactory� except from some high�frequency ripple� due to a combinationof unknown parameters� PWM� load torque and unmodeled dynamics� There is how�ever an error in �ux amplitude� which can be explained as follows� In �� there isonly proportional action in stator currents� even if the gain is a nonlinear function ofspeed� There will always be some unmodeled dynamics in the system� in addition tothe introduced discretization e�ects� and parameter uncertainty� For this reason realcurrents deviates from their desired values� In the speed controller there is an outer

��It can be shown that discretization introduces coupling terms in the dynamic equations whichare proportional to sampling period and speed �� See also Section ��


loop integral action� which forces the speed error to zero� despite error in q�axis cur�rent� There is no such feedback in �ux control� which is feed�forward� Consequentlythe error in d�axis current gives an error in �ux tracking�

For the reasons above� the integral term in �� had to be used for satisfactoryperformance� in addition to the other terms� The following controller parameterswere used in the rest of the experiments reported here� � � ��Rr� k� � ! � KIs � �!� a � � � b � ! � ��L � !��

0.5 1 1.5 20

100

200

300

400

500

Estimated speed and its reference

Time [s]

[rp

m]

0.5 1 1.5 20.08

0.1

0.12

0.14

0.16

0.18

0.2


Time [s]

[Wb

]

0.5 1 1.5 20

100

200

300

400

500

Estimated speed and its reference

Time [s]

[rp

m]

0.5 1 1.5 20.08

0.1

0.12

0.14

0.16

0.18

0.2


Time [s]

[Wb

]

Figure � ��#� Speed regulation%�ux tracking with �upper two �gures� and without '�in the controller�

In Fig� � ��# the importance of the '� term in desired currents is shown� Thisfeed�forward term is signi�cant for high�performance �ux tracking�

As can be seen from Fig� � �� it was di�cult to get good low speed trackingperformance� This is due to the resolution of speed estimation together with friction�especially stiction in the load� Since the speed estimation gives the average speedbetween the sampling intervals� it is di�cult to detect the sign transition precisely�and this gives problems with compensation of stiction terms�

In Fig� � �� an example of load torque rejection is shown� after a step in loadtorque of approximately �� Nm� The controller compensates fast for the disturbances�and no steady state error is present�

�� Observer�less control ��

0 0.5 1 1.5 2

−10

−5

0

5

10

Est. speed and its ref.

Time [s]

[rpm

]

0 0.5 1 1.5 20.198

0.199

0.2

0.201

0.202

Est. flux norm and its ref.

Time [s]

[Wb]

0 0.5 1 1.5 2−2

−1

0

1

2Meas. line curr.

Time [s]

[A]

0 0.5 1 1.5 2−30

−20

−10

0

10

20

30Ref. for stator volt.

Time [s][V

]

Figure � �� Speed tracking%�ux regulation at low speed �� rpm�

0.5 1 1.5470

475

480

485

490

495

500


Time [s]

[rpm

]

0.5 1 1.50.195

0.2

0.205

0.21

0.215Est. flux norm and its ref.

Time [s]

[Wb]

0.5 0.55 0.6 0.65 0.7−150

−100

−50

0

50

100

150Ref. for stator volt. (Window)

Time [s]

[V]

0.5 1 1.5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Torque error

Time [s]

[Nm

]

Figure � �� Speed regulation with load torque disturbance� L � �� Nm for t � ��s� 'qm� � � rpm� Error between desired torque and measured torque is shown inlower right plot�


0 0.5 1 1.5 2−2

−1

0

1

2Position reference

[ra

d]

Time [s]0 0.5 1 1.5 2

−0.02

−0.01

0

0.01

0.02

[ra

d]

Position tracking error

Time [s]

1 1.05 1.1 1.15 1.2

−5

0

5

x 10−4 Position tracking error

Time [s]

[ra

d]

0 0.5 1 1.5 2

−4

−2

0

2

4

6

8

x 10−3

Time [s]

[W

b]

Flux norm error

Figure � ��$� Position control� Passivity�based controller� � � � Wb�

Fig� � ��$ shows an example of position tracking for �ltered steps in positionreference of �� rad� The maximum error is approximately �� The steady stateerror is only restricted by the resolution of the position measurement system� In areal implementation� the digital jittering would be eliminated by a dead�zone� Thecontroller parameters � � ��Rr� k� � ! � KIs � �!� a � � � b � �� L �$ � f � #� were used in this experiment�

It is well known that for small and medium size motors with relatively high sam�pling frequencies for control calculations and PWM� high�performance control can beachieved with only high�gain current control �� This was also experienced in thisexperimental work� were it was found that the desired dynamics in the controller �see�� had relatively low in�uence on the performance for the chosen sample periodof T � ! �s� when integral action was also used� However� for a sample periodtwice this value� the in�uence of the desired terms was signi�cant� as can be seenfrom Fig� � ��

It must be pointed out that in all the reported experiments there are interactionsbetween �ux and torque control� resulting in small �ux norm peaks during transients�For a real system with unmodeled dynamics and with parameters taken from thedata sheet� perfect control can hardly be expected� The peaks are not detrimentalfor system operation� and result in negligible speed transients� Also� for high speedoperation �more than rpm� the nonlinear damping term in �� became

�� Observer�based control ��

0.5 1 1.5

−100

−50

0

50

100


Time [s]

[rp

m]

0.5 1 1.50.197

0.198

0.199

0.2

0.201

0.202


Time [s]

[Wb

]

0.5 1 1.5

−100

−50

0

50

100


Time [s]

[rp

m]

0.5 1 1.50.197

0.198

0.199

0.2

0.201

0.202


Time [s]

[Wb

]

Figure � �� E�ect of desired dynamics in controller for high sampling period�Tsampl � � �s� Lower two plots are result from an experiment with only inte�

gral action and the nonlinear damping term in the controller�

large� and this resulted in ampli�cation of noise from measured currents� which againgave saturation in voltages� For this reason the nonlinear damping term had to bedisconnected for high speeds� With integral action in stator currents� the e�ect ofthis term was found to be negligible�

�� Observer�based control

The controller in Section #�# was implemented� with the desired torque as in �� $�Since this controller exactly reduces to the observer�less controller if stator currentsare controlled by high�gain� it was aimed at testing how well it worked without integralaction in stator currents� An extensive simulation study was carried out with itsdiscretized version� which work well under ideal conditions� The controller was thentested experimentally�

As can be seen from Fig� � �� except from some ripple� the speed regulationis quite good� but �ux regulation is far from good� This was general for all theexperiments� and can be explained with both missing integral action in stator currents�and only current error terms in the updating of the rotor quantities of the observer�


0 0.1 0.2 0.3 0.499

99.5

100

100.5

101Est. speed and its ref.

Time [s]

[rp

m]

0 0.1 0.2 0.3 0.40.15

0.2

0.25

0.3

Est. flux norm and its ref.

Time [s]

[Wb

]

0 0.1 0.2 0.3 0.4−1

−0.5

0

0.5

1Est. i_q and meas. i_q

Time [s]

[A]

0 0.1 0.2 0.3 0.40

1

2

3

4

5Est. i_d and meas. i_d

Time [s]

[A]

Figure � �� Speed and �ux regulation� Observer�based controller� Estimates ofelectrical quantities denoted by ��

The e�ect of the �rst point has been explained in the previous section� Since there isno updating in the stator terms of �� $ � the estimated stator currents drift o� fromthe measured values due to uncertainty in parameters� noise and other unmodeleddynamics� This again introduces errors in the estimated rotor currents� which are usedin the controller� It was possible to reconstruct a similar behavior under simulations�when the parameters used in the observer deviated from the real parameters�

�� Comparison with FOC

For the purpose of comparing the observer�less passivity�based controller with anotherscheme� the rotor��ux�oriented controller from Section � was also implemented� Morespeci�cally� the controller in �� was used� Estimated rotor �ux amplitude (�dand angle (�a were computed from �� and �� see p�!�$� The PI�controllersgiven in �� together with the current references from �� $� �� !�were used to give the voltage references vdq� For speed control the desired torque wasde�ned as in �� The controller parameters were the same as in the previoussection for the passivity�based controller� and the parameters K�P � ! � � K�I � �!�PI speed controller� K�P � �!� K�I � !�� PI current controllers were used in theimplementation of the FOC scheme�

�� Comparison with FOC ��

0.5 1 1.5 2

495

500

505

Est. speed and its ref. (PBC)

Time [s]

[rpm

]

0.5 1 1.5 20.08

0.1

0.12

0.14

0.16

0.18

0.2

Est. flux norm and its ref. (PBC)

Time [s]

[Wb]

0.5 1 1.5 2

495

500

505

Est. speed and its ref. (FOC)

Time [s]

[rpm

]

0.5 1 1.5 20.08

0.1

0.12

0.14

0.16

0.18

0.2

Est. flux norm and its ref. (FOC)

Time [s]

[Wb]

Figure � � � Comparison of the passivity�based controller �PBC with an implemen�tation of FOC� Speed regulation%�ux tracking�

Figs� � � � � � are representable for the comparison between the two schemes�In both cases the controller parameters were tuned such that currents saturated duringtransients� The FOC scheme generally gave slower responses� and somewhat highermaximum tracking errors� This scheme was also more di�cult to tune than thepassivity�based scheme� In the FOC controller implementation it was advantageousto use saturation limits corresponding to the systems constraints both in referencecurrents and voltages�

To investigate the robustness of the schemes� an arti�cial change in rotor resistancewas introduced by using a value di�erent from the nominal value in the controllers�Both controllers were then tuned to give a �best performance� in terms of transientsand overshoot� The FOC scheme was experienced to give a more oscillatory behaviorthan the passivity�based for di�erent values of Rr� Examples are shown in Figs�� ! for Rr � ��RN

r �

The execution time� was approximately �� s for the passivity�based scheme�with the �ux observer running in parallel� and �#� �s for the FOC scheme�

��Only the execution time was logged� and no signal generators were implemented�


0.5 1 1.5 2

−500

−400

−300

−200

−100

0

100

200

300

400

500

Speed reference

Time [s]

[rpm]

0.5 1 1.5 2

−100

−50

0

50

100

Speed tracking error (PBC)

Time [s]

[rpm

]

0.5 1 1.5 20.18

0.19

0.2

0.21

0.22

0.23Est. flux norm and its ref. (PBC)

Time [s]

[Wb]

0.5 1 1.5 2

−100

−50

0

50

100

Speed tracking error (FOC)

Time [s]

[rpm

]

0.5 1 1.5 20.18

0.19

0.2

0.21

0.22

0.23Est. flux norm and its ref. (FOC)

Time [s]

[Wb]

Figure � � �� Comparison with FOC� Speed tracking%�ux regulation�

0.5 1 1.5−500

0

500

Est. speed (PBC)

Time [s]

[rpm

]

0.5 1 1.50.15

0.2

0.25Est. flux norm and its ref. (PBC)

Time [s]

[Wb]

0.5 1 1.5−500

0

500

Est. speed (FOC)

Time [s]

[rpm

]

0.5 1 1.50.15

0.2

0.25Est. flux norm and its ref. (FOC)

Time [s]

[Wb]

Figure � � � Comparison with FOC� Speed tracking%�ux regulation� Rr � ��RNr �

�� Concluding remarks ��

0.5 1 1.5

496

498

500

502

504

506

508Est. speed and its ref.(PBC)

Time [s]

[rpm

]0.5 1 1.5

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Est. flux norm and its ref. (PBC)

Time [s]

[Wb]

0.5 1 1.5

496

498

500

502

504

506

508Est. speed and its ref. (FOC)

Time [s]

[rpm

]

0.5 1 1.50.08

0.1

0.12

0.14

0.16

0.18

0.2

Est. flux norm and its ref. (FOC)

Time [s][W

b]

Figure � � !� Comparison with FOC� Speed regulation%�ux tracking� Rr � ��RNr


The experimental testing of the proposed controllers can be summarized as follows�

The observer�less passivity�based controller was found to give the best dynamicperformance and robustness to unmodeled dynamics� as compared to a �direct FOCscheme and a passivity�based controller with observer� The observer�less controllerwas easy to tune� and could basically be down�loaded with parameters taken fromthe simulations� without the need for extensive tuning� Signi�cantly more tuning wasnecessary to make the FOC scheme give results comparable to the performance of thepassivity�based controller�

For the passivity�based controller� the feed�forward term from the derivative '�of the �ux reference improved �ux tracking signi�cantly� The e�ect of this simplemodi�cation of the passivity�based controller to allow for global �ux tracking is aninteresting result� especially when it is related to conventional implementations ofindirect FOC schemes� which only can handle �ux regulation�

It must be pointed out that high�gain current control was necessary for satisfac�tory performance of all the controllers which were tested� The nonlinear dampingterms which were introduced to prove stability� could not compensate for the unmod�eled dynamics of the motor and the converter� and integral action was needed� Thesedamping terms also have certain disadvantages with respect to current noise ampli��


cation� Together with the terms in the voltage controller stemming from the referencedynamics� they can be removed when integral action current control is implemented�at least when sampling frequency is high�

For the case of observer�based control along the direction of the result in Section#�#� it should be focused on including an observer which is more robust to unmod�eled dynamics into the scheme� This could for instance be a scheme which also hasupdating from stator current error terms in the stator equations� as in � � �� Anobserver�based controller is of interest for future extensions along the line of adaptiveobservers� and it is believed that it will be possible to include other observers withthe desired robustness properties into the scheme� provided they give exponentiallyconvergent estimates�

The aspects of discrete implementation and speed estimation also have to becarefully considered for performance improvements� In this experimental work stan�dard ad hoc schemes for controller discretization� speed estimation� and generationof switching signals have been used� This is a drawback of the implementation� sincethere is no theoretical justi�cation for such an approach�

Other experimental results from the application of the observer�less passivity�based controller to induction motors have recently been reported in ��! � �comparisonwith the scheme of �� showing similar promising results� See also Chapter �� forexperimental results with currentfed induction motors�

Chapter ��

Current�fed induction motors

�Experience does not err� it is only your judgment that errs inpromising itself results which are not caused by your experi�ments��

Leonardo da Vinci�

In Chapter � we proved that PBC of mechanical systems reduces� in regulationtasks with full state feedback� to the classical PD controller used in most roboticapplications� Furthermore� when velocities are not available for measurement thePBC methodology suggests to replace the velocities by their approximate derivatives�which is also a standard procedure in applications� This �downward compatibility� ofPBC with current engineering practice is a remarkable feature whose importance canhardly be overestimated� On one hand� it provides a solid systemtheoretic founda�tion to popular control strategies which enhances their understanding and paves theway for subsequent improvements� On the other hand� viewing the new controllersas �upgrades� of the existing ones� it facilitates the transfer of these developmentsto practitioners� In this chapter we will show that� under some simplifying assump�tions on the machine model� the PBC for electrical machines presented in previouschapters also has a �downward compatibility� property with the industry standard�eld�oriented controller �FOC�

FOC has already been introduced for voltagefed induction machines in the previ�ous chapter� It was pointed out that �eld orientation� in one of its many forms� is anestablished control method for high dynamic performance AC drives� In particular�for induction motors indirect FOC is a simple and highly reliable scheme which hasbecome the de facto industry standard� In spite of its widespread popularity� the sta�bility and robustness properties of FOC schemes are not theoretically well understood�An approximate analysis �based on steadystate behaviour� timescale assumptions�and linearizations� e�g�� !�� !� can be combined with the designer expertise tocommission the controller in simple applications� However� to meet large bandwidthrequirements� or other tight speci�cations� this ad�hoc commissioning stage may be

!��

�� Ch� �� Current�fed induction motors

timeconsuming and expensive� if at all possible� To simplify the o�line tuning ofFOC� and eventually come to terms with its achievable performance� a better the�oretical understanding of the dynamic behaviour of FOC is unquestionably needed�Such an analysis is unfortunately stymied by the fact that� as we have seen before� thedynamic behaviour of the closed loop is described by a complex nonlinear relation�ship� However� as we will see in this chapter� for currentfed machines it is possible tode�ne some suitable coordinate changes� that reveal useful passivity properties thatcan be pro�tably exploited to overcome this obstacle�

Realizing the practical importance of FOC� and motivated by the need to clarify itstheoretical underpinnings we analyze in this chapter the indirect FOC for currentfedinduction motors in detail� We start with a presentation of the model of the currentfed induction motor in Section �� and to explain the rationale of indirect FOC we�nd it convenient to recall in the next section �rst the simpler concept of direct FOC�which was presented in Section � of the previous chapter for the voltage�fed inductionmotor� We underscore also in this section the links of direct FOC and the theoreticallyinteresting idea of feedback linearization� which for currentfed machines are moretransparent� Then� we show in Section ! that the observerless PBC developed forinduction motors in Section ��!� exactly reduces to the well�known indirect FOC inspeed regulation applications with constant load torque� for currentfed machines� Acorollary of this result is a rigorous proof of the global asymptotic stability of FOC��rst reported in � �#��

In Section � we address the problem of robustness of FOC with respect to uncer�tainty in the rotor time constant� and we establish the stronger property of globalexponential stability� In Section � we give some simple rules for the o��line tuningof PI gains� which will ensure robust stability� Not surprisingly� instrumental for thisanalysis is a suitable decomposition of the system into a passive feedback intercon�nection�

A new discrete�time FOC for current�fed induction motors ��rst reported in � ��is presented in Section $� This controller ensures global asymptotic speed regulationand rotor �ux norm tracking provided a condition relating the sampling rate withthe controller parameters is satis�ed� In analogy with the continuous time controllerdesign� this condition disappears as the sampling rate goes to zero�

Many of the sections have experimental results included for illustration of thetheoretical results� In particular� we present in this Section # a detailed experimentalcomparison of FOC and a feedback linearizing scheme�

Concluding remarks to this chapter are given in Section ��

Remark �� Notation�� Throughout the chapter we will be interested in the ro�

�The importance of coordinate changes was probably �rst underscored by Copernicus who pointedout that the planetary motions are better understood from the sun�s perspective �� An ingeniouschange of coordinates is� of course� also the underlying principle of FOC�

Model of the current�fed induction motor ��

bustness with respect to parameter uncertainty of the di�erent controllers� When�ever we want to stress this point� we will use (�� to denote a xed estimate of aparameter �� Also� with the objective of simplifying the notation and the presenta�tion� in some sections where the value of certain parameters is of no relevance for thediscussion� they are set equal to unity�

� Model of the current�fed induction motor

In Chapter �� we have shown that the induction motor in the �xed stator frameis described by the state equations �� which we repeat here for ease ofreference

'�ia �

LsrRr

Ls�L�r

�a &npLsr

Ls�Lr'qm�b � �ia &

�

Ls�u�

'�ib �

LsrRr

Ls�L�r

�b � npLsr

Ls�Lr

'qm�a � �ib &�

Ls�u�

'�a � �Rr

Lr�a � np 'qm�b &

RrLsr

Lria

'�b � �Rr

Lr�b & np 'qm�a &

RrLsr

Lrib

-qm ��

Dm� � L

�npLsr

Lr��aib � �bia

where �ab � ��a� �b�� is the rotor �ux vector� iab � �ia� ib�

� � 'qs is the stator currentvector� and uab � �u�� u��

� is the vector of stator voltages� Rs� Rr �*� are stator androtor resistances� Ls� Lr �H� are the inductances of the stator and rotor windings and

Lsr �H� is the mutual inductance� � � �� L�srLsLr

� is the total leakage factor of the

motor� and � � RsLs�

& L�srLs�LrTr

� with the rotor time constant de�ned as Tr�� Lr�Rr

�s�� Dm �kgm�� is the rotor inertia and 'qm �rad%s� the rotor speed�

In many practical applications highgain current loops �sometimes with PI ac�tions of the form

u ��

��idab � iab

are used to force iab to track their corresponding references idab� where � is a smallpositive number� It is reasonable then to consider the singularly perturbed reducedmodel obtained by setting �� that is

'�ab � �RrLr�ab & np 'qmJ �ab & RrLsr

Lriab

Dm-qm � � L � npLsr

Lri�abJ �ab

��


with the skew�symmetric matrix

J �

� ��

�

The underlying assumption of this model is that the stator currents are exactly equalto their references� i�e� iab � idab� To further simplify the equations� we introduce the�globally de�ned change of coordinates

v � e�Jnpqmiab� �r � e�Jnpqm�ab ��

with the rotation matrix

e �Jnpqm �

�cos�npqm sin�npqm� sin�npqm cos�npqm

�� e �Jnpqm�� e �Jnpqm� � e Jnpqm

Hence v � �v�� v�� r � ��r�� r��

� are quantities expressed in a frame rotating withthe �electrical speed of the rotor� In the new state coordinates ��r � qm� 'qm�

�� andwith the new control inputs v� we have the following bilinear model

Tr '�r � ��r & LsrvDm-qm � � L � npLsr

Lrv�J �r

��!

Throughout the remaining of the chapter� we will assume that the behaviour of the�socalled currentfed induction motor is captured by the dynamical model ��!�As we will show below this apparently innocuous system can exhibit an amazinglycomplex behaviour and poses a signi�cant challenge for control system design�

Unless otherwise stated we will concentrate in the sequel on the problem of speedcontrol and assume that load torque L is constant� The modi�cations needed forposition control andor adaptation for a linearly parameterized L are analogous tothe ones explained for voltagefed machines� henceforth will be omitted�

Remark �� Validity of model�� In current�fed machines we assume that the�rotated stator currents v�t are equal to possibly discontinuous references calculatedby the controller� In real motors� stator currents must be continuous so that theycannot follow discontinuous references exactly� However� the stator currents can followthe references well if the stator inductances are su�ciently small� On the other hand�the assumption of perfect current control is based on su�ciently high ceiling voltageand switching frequency of the inverter� for the control bandwidth to be adequate�For instance� in the case of high power drives with thyristors switching at only a fewhundred Hz� perfect current control can not be assumed� and the interactions of thestator voltage equations must be taken into account for controller design�

� Field orientation and feedback linearization ��

Remark �� Relation with the asymptotic model�� It is important to un�derscore the fact that �r is a vector quantity� This model should not be confusedwith the machine model used in decoupling control� e�g�� $�$�� $�� $� of �!��which describes the asymptotic behaviour of the motor in closed�loop with an idealdirect FOC�

Remark �� Relation with the non�holonomic integrator�� The model co�incides �up to the presence of the term �r in the �rst equation of ��! and theload torque with the celebrated non�holonomic integrator of Brockett for which avast amount of research has been devoted in the last years� Interesting connections�extensions and simpli�cations between the timevarying controllers used for thissystem and the PBC presented in this text are explored in �$!��

� Field orientation and feedback linearization

� Direct �eld oriented control

De�ning the rotor �ux amplitude � k�rk� and the rotor �ux angle � � arctan��r��r��and introducing an additional input change of coordinates

idq��

�idiq

�� e�J �v

we can rewrite the model ��! in polar coordinates as

Tr ' � � & Lsrid'� � Rr

np�

� npLsrLr

iq

��#

Notice that is the output of a linear �lter with �input� id� while is simply theproduct of the second �control input� iq and � This two facts� together with thenatural time scale separation of the electrical and mechanical dynamics� motivatesthe classical direct FOC �see also Section �� where id is chosen to regulate toits reference value �� while iq is used to drive the torque to some desired referenced� as follows

idq ��

(Lsr

��

�Lrnp�

C�p '�qm� �z �d

��

� e�J ��npqm�iab�

��

where '�qm�� 'qm � 'qm�� 'qm� are the desired rotor �ux magnitude and rotor speed�

respectively� (�� denotes an estimate� C�p � ��KP &KI�p� p

�� d

dt� and KP � KI �

are tuning gains�


It is important to remark that for the implementation of the actual control signals�

iab ��

(Lsr

eJ ��npqm�

��

�Lrnp�

C�p '�qm� �z �d

��

the �ux angle � must be measured or estimated� The unavailability of this signal�coupled with the high sensitivity �with respect to the highly uncertain parameterTr of its estimate� are the main drawbacks of this approach� See Section �� foradditional comments�

� Indirect �eld�oriented control

v�

v�

�Dm

�p

L

'qm

�dd

�LsreJ �d

�r�

LsrLrRr

p��

LsrLrRr

p��

npLsrLr

� & Tr '�

�r�

�Rrnp��

�p

Lrnp�

d

'qm�KP � KI

p

�

'�

Figure �� Currentfed induction motor with indirect FOC� Motor model presentedin a frame of reference rotating with rotor �electrical speed�

To remove the need of the rotor �ux angle � from the direct FOC above we can thinkof replacing it by it�s �asymptotic certainty equivalent� estimate� that is the value towhich it will converge if and behave as desired� Applying this rationale to thesecond equation of ��#� and calling this signal �d� yields

'�d �(Rr

np��d ��

d � C�p '�qm � ��KP &KI�

p '�qm

where we have kept the same de�nition of d as above� We can now use this estimateinstead of the actual � in the direct FOC �� to get

idq ��

(Lsr

�� & (Tr '��Lr

np�C�p '�qm

� �� e�J ��d�npqm�iab

��$

�Recall that we assume current fed operation�

�� Observer�based feedback�linearizing control ��

where the term (Tr '� in the id�component of the currents is introduced to achieve �uxnorm tracking and is motivated by the �rst equation in ��#� The resulting controllaw� that is�

iab ��

(Lsr

eJ ��d�npqm�

�� & (Tr '��Lr

np�C�p '�qm

�

will be called in the sequel indirect FOC� This scheme may be found� for instance� in�� It is often presented in the literature without the �ux tracking term (Tr '��

The total closed�loop system consisting of the induction motor ��! and thecontroller ��$ is shown in Fig� ��

�� Observer based feedback linearizing control

The observer�based feedbacklinearizing controller �OBFL of �� may be viewedas a variation of direct FOC where� in order to achieve a linear system in closed loop�some terms are added to the control law to cancel the motor nonlinearities� A similarphilosophy was adopted in ��#!� for the case of the full motor dynamics� see also ��$ �where some adaptation terms are added to the basic scheme of ��#!�� Although theobjective of feedback linearization is quite luring� the resulting schemes su�er fromserious drawbacks� from both theoretical and practical viewpoints� First� they invari�ably require the explicit implementation of an observer� This� besides increasing thecomputational burden� makes the stability analysis extremely di�cult� For instance�it is well known that for nonlinear systems the certainty equivalence principle fails�Also� it widely recognized that at this stage� our understanding of nonlinear observersis quite rudimentary� It su�ces to say that in spite of many years of research� to thebest of our knowledge� a complete stability analysis for exact linearization schemes inthe full motor model case is conspicuous by its absence� However� it should be pointedout that for the simpli�ed model of current�fed machines this problem is elegantlysolved in �� Second� since these schemes are based on nonlinearity cancelations�it is expected that potential instability due to parameter mismatch will arise� Onesuch instability mechanism for the OBFL� which appears even in the state feedbackcase� is identi�ed in this work� and observed in the experiments�

For later comparison with direct FOC and to exhibit the instability mechanism�we will distinguish between the statefeedback and the output feedback cases in thepresentation of the OBFL�

�Called in that paper observer�based adaptive controller�

� Ch� �� Current�fed induction motors

A State feedback

First� we notice from the �rst equation of ��#� that � satis�es the di�erentialequation

Tr

d

dt� � �� & Lsrid ��

Hence� assuming that the state is measurable and that the parameters are known� wecan choose

id ��

Lsr

� � Trk�

��

with k� � a design parameter� to obtain a linear dynamics for � as

d

dt� � �k��

�

A feedback linearizing and decoupling control is completed with a suitable de�ni�tion of iq� for instance if L is assumed known�� we can choose

iab ��

(Lsr

eJ �npqm��

� � �Trk�

��

��

�Lrnp�L � (DmKP

'�qm

��

The connection between direct FOC and OBFL is easily established comparing �� with �� There are two essential di�erences� the utilization of the actual in�stead of � throughout� and the inclusion of the second and third terms in the �rstcomponent of the vector� These modi�cations are introduced to achieve the exactlinearization and decoupling� which are only asymptotic in direct FOC� This is ex�actly the opposite to the approach taken in PBC� where the actual rotor quantitiesare replaced by their desired values� Of course� the main contribution of �� residesin the ingenious construction of the observer to achieve global stability� as explainedbelow�

We observe that

eJ �npqm��

��a ��b�b �a

��

which will be used to explain the rationale of the OBFL below�

�Instead of the expression for iq above we could have chosen� as in FOC� a PI speed loop� that isiq �

��C�p��qm� This idea underlies the work of �� and has recently been pursued by �� Other

linearizing schemes are presented in �� see also Section ��

�� Observer�based feedback�linearizing control �

B Output feedback

In this case estimates of the rotor �ux and the load torque are provided by thenonlinear observer

'(�ab � � �

(Tr(�ab & np 'qmJ (�ab &

(Lsr

(Triab � np (Lsr

(Lr(Dm

�� '�qm & �� '�q�

mJ iab

'(L � �KI

(Dm

�� '�qm & �� '�q�

m

with �� design parameters� Notice that if we set these parameters to zero theobserver is just a copy of the rotor �ux dynamics ��

The control is a modi�ed version of �� and is given as

iab ��

(Lsr

D��(�ab� '�qm

�k(�abk� � k� �Tr

��k(�abk� � �

��Lrnp�(L � (DmKP

'�qm

�

where the elements of the matrix D�(�ab� '�qm are given by�

D��(�ab� '�qm�� (�a � np

(Rr(Dm

�� '�qm & �� '�q�

m(�b

D��(�ab� '�qm�� (�b &

np(Rr

(Dm

�� '�qm & �� '�q�

m(�a

D��(�ab� '�qm�� (�b

D��(�ab� '�qm�� (�a

Notice that we have added to the rotation matrix �� some additional terms�hence the control is not a certainty�equivalent version of �� It is interestingto note that� as shown in ��$!�� to ensure invertibility of the matrix D�(�ab� '�qm� itis enough to choose the initial conditions of the observer bounded away from zero�which imposes no constraint in a practical application�

C An instability mechanism of the OBFL

In the following we will show that� in the face of parameter uncertainty� the OBFLscheme may become unstable� We have decided to present here the state feedbackcase to underscore the fact that the instability is not due to the observation errortransient� but to the nonlinearity cancelation term� The same calculations can becarried out� leading to the same result� for the observer feedback case�

Proposition �� Consider the current�fed motor model �� in closed loop withthe OBFL� which in the state feedback case simpli�es to �Eq� �� in ��

iab �

��a �b��b �a

�� Lsr

�k�abk� � k� �Tr�

�/�Lr

np �Lsr�(L � (DmKP

'�qm

�


where �/��

� � and we have assumed k�abk �� In this case� the systembecomes unstable if the mutual inductance Lsr is underestimated and the gain k� issmall� More precisely� the �ux grows unbounded if

�LsrLsr

� �� (Trk��

��

�

Proof� The proof consists in showing that the term introduced in to cancel therotor �ux norm� i�e��

�Lsrk�abk�� induces an unstable behaviour on the �ux norm error

when �� holds�

The �rst term of the control is given as

id ��

(Lsr

� �(Trk�

�/�

That is� the estimated parameter version of �� Replacing the equation above in�� and doing some simple calculations we see that

'�/ � C��/ &

Tr�Lsr

(Lsr

� �� !

where

C��

Trf�� & Lsr

(Lsr

�� (Trk� g

This completes the proof� �

�� Remarks to OBFL and FOC

Remark �� High�gain control�� It is clear from �� that the instabilitymentioned above can be easily avoided by choosing k� su�ciently large� namely k� ��Tr

su�ces� This will have the additional bene�cial e�ect of reducing the steadystate

error in ��!� Experimental and simulation evidence have shown� however� thatlarge values of k� induce a resonant behaviour� This is illustrated in the simulation ofFig� �� The simulation was performed in SIMULINK using the motor parametersof Table �� see page !�� A step signal was applied for the speed reference� 'qm� � � rad%s� at t � s and we observed the speed and amplitude of the stator currentvector as k� was varied� Fig� �� a corresponds to the ideal case without parameteruncertainty� Figs� �� b�� f show the e�ect of varying k� as in the practicallyreasonable case when Lsr and Lr are underestimated ) and Rr is overestimated� )� Notice that instability is triggered when k� is small � �� as predicted by theproposition� but also when it is chosen to large �� Further studies are neededto understand the latter instability mechanism� but it is clear that the use of highgains to avoid instability is not a sensible approach�

�� Remarks to OBFL and FOC ��

0 0.5 1 1.5 20

50

100(b)

0 0.5 1 1.5 20

50

100(c)

0 0.5 1 1.5 20

50

100

(d)

0 0.5 1 1.5 20

50

100

(e)

0 0.5 1 1.5 20

50

100

[sec]

(f)

0 0.5 1 1.5 20

50

100

[rad

/s] ,

[A]

(a)

speed / reference

current modulus

Time [s] Time [s]

(a) (b)

(c) (d)

(e) (f)

[A],

[ra

d/s]

Figure �� Simulation illustrating instability of feedback linearizing control� �aIdeal case with k� � � � �b�f parameters perturbed� and k� � � � k� � �� k� � $�k� � � and k� � � �

Remark �� Modi�cation of FOC which gives robust �ux tracking�� We bringto the readers attention the �apparently not well known fact that a simple modi��cation to direct FOC allows us to achieve also �ux norm tracking� see e�g� �� This is an important feature that is sometimes invoked to claim superiority of exactlinearization schemes over direct FOC� To this end� in �� we choose

id ��

(Lsr

�� & (Tr '� � k� ��

with �� This gives from ��# the closed loop equation

Tr ' � C� & C�� & C�'� ��#

where C�� C� are some constants� and

C�� Lsr

(Lsr

k� �

Notice that ��#� in contrast to ��!� is always stable� hence avoiding the insta�bility mechanism of OBFL� When the parameters are exactly known we have

Tr'� � ��k� & ��

which ensures the desired tracking with arbitrary rate of convergence�


� Passivity�based control of current�fed machines

The main message that we should retain from Section � is that under assumptionsthat can be satis�ed in many practical applications� the stator currents can be takenas control inputs for the induction motor� In other words� in some applications theinverter can be modeled as an ideal current source� In this section we will prove that�under this condition� our PBC for voltagefed machines considerably simpli�es andactually reduces to the indirect FOC presented above�

�� PBC is downward compatible with FOC

In Section ��!� we derived an observerless PBC for speed and rotor �ux normtracking with the full voltagefed machine model� This controller has the followingfundamental downward compatibility property�

Proposition �� In the case of current�fed machines with constant speed referencethe PBC �� developed for the voltage�fed machine model exactly reducesto the indirect FOC ��

Proof� For currentfed machines the control signal is actually 'qsd� that is the �rstrow of �� and we do not need to calculate u as in �� Consequently 'd�which was required because of the presence of the term -qed� is no longer needed forthe implementation� Hence� we can remove the �ltered speed error and replace itdirectly by the speed error� In this way the controller reduces to

'qs � 'qsd ��

Lsr

�� & Tr

'��

�I� &

Lr

np��dJ

�eJnpqm�rd ��

'�rd �

�Rr

np��dJ &

'��

��rd� �rd� �

��

��

d � Dm-qm� � a '�qm & L ��$

where '�qm�� 'qm � 'qm� is the speed error and � is the timevarying �ux reference�

Now� notice that the controller states in �� can be exactly integrated as

�rd � �

�cos��dsin��d

�� eJ �d

��

�

where �d is the solution of

'�d �Rr

np��d� �d� � ��

�� Stability of indirect FOC for known parameters ��

By use of these expressions in �� we get

'qs ��

Lsr

eJ �npqm��d�

�� & Tr '�

Lrnp�

d

��

On the other hand� assuming that the desired speed is constant �-qm� � and re�placing the exact load torque cancelation by an integral action we get from ��$

d � ��a & KI

p '�qm� KI � ��

That is� a PI action around the speed error�

Let us now express the controller equations in the stator �xed coordinates intro�duced in Section �� To this end� recall that 'qs � 'qsd � iab� hence from ��

iab ��

LsreJ �npqm��d�

�� & Tr '�

Lrnp�

d

��

which is exactly the indirect FOC in ��$� �

�� Stability of indirect FOC for known parameters

The proposition below follows as a corollary of Proposition � �� To highlight someissues that will become important in the next sections we give a proof here only forspeed and �ux regulation� The proof for the tracking case follows verbatim from theproof in Section ��!� � when the stator currents are used as inputs�

Proposition �� Consider the current�fed induction motor model �� in closed�loop with the controller �� where L� 'qm� and � are constant�and the parameters Lsr� Lr and Rr are known� Under these conditions� the controllerensures global speed and rotor �ux norm regulation� That is�

limt�

j 'qm � 'qm�j � � limt�

j k�rk � �j �

holds for all initial conditions and with all signals uniformly bounded� �

Proof�

It follows from �� and �� that the PBC can be written as

v ��

Lsr��rd & Tr '�rd ��

De�ning ��r�� r � �rd and using ��! we see that '��r � � �

Tr��r� hence ��r converges

�exponentially fast to zero�


Finally� with some lengthy but straightforward calculations� we can get

Dm-�qm � Dm-qm �

npLsr

Lrv�J ��& �rd� L

�npLr

��rdJ �� L & �� &�

��rd��rd

� �KP �� &�

��rd�� '�qm �KI�� &

�

��rd��

�

p'�qm &

npLr

��rdJ ��r � L

Noting that k�rd�tk � �� we have that ��rd��r and ��rdJ ��r are exponentially de�

caying terms� Asymptotic stability of '�qm � follows invoking standard argumentsof LTV systems with exponentially convergent coe�cients� �

In Section � we will give an alternative proof of exponential stability based onconstruction of a strict �quadratic Lyapunov function�

Experimental comparison of PBC and feedback

linearization

Several experimental studies have been carried out to illustrate the theoretical resultsdescribed above� Some of these results are presented in this section� Other relatedworks� which have been reported within the industrial electronics community may befound in �#��

We present here an experimental comparison� between PBC and the OBFL of�� which was presented in Section �!� As shown in the previous section� thePBC controller is obtained as a particular case� for the approximate model used forcurrentfed machines� of the globally stable and globally de�ned output feedbackscheme of Section ��!� � which was developed for the full machine model� Oneimportant feature of this controller is that there is no need to reconstruct the state�i�e�� to estimate the rotor �ux� This translates into a considerable reduction in thecomputational complexity� a feature that� due to cost and numerical robustnessconsiderations� can hardly be overestimated in an application of this nature�

The experiments were carried out on a two�pole squirrel�cage current�fed induc�tion motor of �� kW and a microcomputer�based control system developed for anindustrial application by ��!�� In contrast with other experimental studies� wheresophisticated special purpose equipment is installed� we used in these experimentsstandard lowcost hardware �e�g�� Motorola �� microprocessor readily available

�Caveat emptor In a comparative experimental study it is di cult to distinguish the relativemerits �or demerits� of a technique from the talents �and prejudices� of the designer� The best wecan do to alleviate this di culty is to put at the disposal of the interested reader our experimentalfacility�


for a practical application� An exhaustive set of experimental results may be foundin ��!�� We have decided to present here only some representative curves on speedregulation� load torque disturbance rejection� and robustness to rotor resistance vari�ations�

The conclusions of our experimental comparison may be summarized as follows�

�� The high computational requirements of the OBFL forced us to double the sam�pling period achievable for PBC with obvious ensuing performance degradation�

� Even at a lower sampling frequency �with respect to the fastest achievable one�PBC systematically achieved better speed transient performance� faster loadtorque disturbance rejection� and enhanced robustness vis �a vis uncertainty inthe rotor resistance�

!� Commissioning of PBC was also simpler� because the performance of OBFL ismore sensitive to parameter uncertainty and its control e�ort was larger� Thelatter factor considerably limited the range of operation of the scheme�

Due to the extremely high computational requirements we were unable to test�with the installed equipment� backstepping�based designs� See �� for some experi�mental evidence of this scheme�

�� Experimental setup

D / AConverter

InverterController

MOSFETPWM

Inverter

PositionSensorPosition

SensorInterface

DCMotorRectifierE

E

Stator Current Feedback

Z8536

Z8530

Motorola

68000

InductionMotor

HostPC

Figure ��!� Experimental setup�

The experimental drive was assembled using commercially available products and cir�cuits made in the laboratory� In Fig� ��! we show the experimental setup consisting


of the induction motor� a DC motor to simulate load torque� the main card witha Motorola �� microprocessor� a timer and an input�output interface� the D%Aconverter� a position sensor with interface card� a PWM inverter with control card�current sensors� source of electric power� and a Macintosh host computer�

The parameters of the induction motor used in the experiments are given in Table��

Parameter Notation Value Unit

Rated power P �� kWStator resistance Rs $�� *Rotor resistance Rr �� *Mutural inductance Lsr ��! mHStator inductance Ls � � mHRotor inductance Lr � mHMoment of inertia Dm #�#� � � �� kgm�

Moment of inertia�with DC motor Dm #�� kgm�

Pole pairs np

Table �� Motor parameters�

A Hardware description

The main card has a � MHz Motorola �� bits microprocessor� a Zilog ��!�timer connected with the main processor for the management of the sampling time�and an Zilog ��! interface card used for the communication with the host computer�

The D%A converter card is inserted between the main card and the controllerfor the inverter� The resolution of this converter is � bits and it includes fourD%A converters �one for a reference voltage and the others for commands� Therotor position is transferred to the microprocessor from the position sensor� which isan incremental encoder of type GESINC�� with a resolution of lines �angularresolution !� o� � The power electronics which supplies the current to the motorconsist of a fast analogue currentloop controller card and a three�phase MOSFETinverter�

The current control is performed in a classical manner� Two analogue currentreference signals are transmitted to the control card� The currents in two of thephases are measured and compared with the reference signals� The regulation of thethird phase is carried out such that the mean value of the neutral point voltage isequal to half the value of the feeding voltage� This nonredundant control approach


leads to good performance and utilization of the converter and allows for consideringit as an ideal current source�

B Software description

The M�� can be programmed in C and assembly language� Assembly code wasused for the control routines which had to be executed in real time� The other routines�management of menus� data saving� etc� were written in C� For communication withthe processor� the application program Versaterm operated on a Macintosh was usedas an interface between the user and the microprocessor�

The real�time application requires on�line solution of di�erential equations� trigono�metric functions� square roots� integration� etc� A trigonometric function look�uptable was used by the controller� To enhance precision� given the short time availablefor calculations� we used the second order Runge�Kutta method for solution of thedi�erential equations�

Since the microprocessor can handle only �xed point numbers �of �� or ! bits�we introduced a scaling proportional to the size of the numbers used in the controller�To avoid as much as possible a phenomenon of truncation during the calculation�an optimal multiplier was chosen for the control variables� For the multiplication oftwo variables� we developed multiplication algorithms of �! bits�! bits� � and��bits�! bits� �� An appropriate algorithm was chosen according to the size ofeach �xed�pointed constant or variable� and care was taken to avoid over�ow whenthe result of the calculation exceeded ��

In the implementation of OBFL� the terms containing �� '�q�

m became zero in the�xed�point calculations even if we used the maximum multiplier � �� The reasonfor this was that �� had to be chosen very small to avoid oscillations� �� Therefore we neglected these terms in the experiments� In a simulation study ��!�� weestablished that the in�uence of this term is negligible �in currentfed applications�

Method Assembly codemultiplier �� and optimal multiplier optimal multiplier

multiplication algorithm ��bits ��bits� ��

PBC �� ms �� ms �� msOBFL � ms ��# ms �# ms

Table �� Computation time�

In order to determine the smallest possible sampling time� we made a study of

�It is worth underscoring that setting � � � does not change the stability properties of OBFL�

� Ch� �� Current�fed induction motors

computing time in C and assembly code for di�erent multipliers� The results aresummarized in Table �� Notice that the computing time of the assembly code forOBFL is two or three times larger than the corresponding time required for PBC�

To establish a good compromise between precision and speed we used the �

multiplier� with the multiplication algorithm of �! bits �! bits� �� To make afair comparison between the schemes� we carried out the experiments with a samplingtime of $ ms for both controllers�

�� Selection of �ux reference in experiments

The desired rotor �ux magnitude is a primary design and application parameterwhich had to be selected equal for the two controllers� In general� high�performanceservo applications require a low mean velocity� a continuous transient operation anda high peak torque with limited current amplitude ampli�ers� In such applications�the selection of the rotor �ux magnitude is usually based on the peak values of torqueor acceleration which can be achieved by a given limited current �� Anotherimportant constraint for a design optimization is that magnetic saturation e�ectsmust be considered in the selection of the �ux level�

There is a limit for the stator current which can be supplied by the inverter �thecurrent obtained with the maximal amplitude for the given gain of the inverter�This limit is � A� but for safety reasons a lower value� Imax � � A� was used�If the stator current magnitude is limited by the inverter to Imax� then the torquebecomes limited by the relative values of the �ux�producing current� id �

Lsr� and

the torque�producing current�

iq �Lr

Lsrnp

which are attainable at Imax� by the relation�

qi�d & i�q �

s�

Lsr� & �

Lr

Lsrnp� � Imax

with use of the constant �ux strategy� This limits the peak torque achievable with alimited current amplitude� In this case the optimal choice of rotor �ux can be foundby expressing the torque in terms of the stator current amplitude�

�L�srnpLr

jIj� sin� �

where

� � tan��iqid�

The maximum value of this expression corresponds to � � #� or id � iq ��

�� Speed tracking performance �

At Imax� the magnetizing current Imr can be selected as Imr � id �Lsr

� ��$A using q

i�d & i�q �

s�

Lsr� & �

Lr

Lsrnp� � Imax�

But the optimal choice of �ux depends on the current selected in the optimization�� Thus the torque%amp can be maximized at the rated current of the inverterI� � # A� In this case� the magnetizing current can be selected as Imr � �� A�

To evaluate this selection� we compared experimentally the speed tracking re�sponses for di�erent �ux levels at low and high speeds ��!�� As a consequence of thiscomparison� we used Imr � ! A� which gives � � ImrLsr � ��$� � � �� Wb� We alsolimited the desired torque for PBC to jdj � �� $# Nm by the relation

Imax �

s�

Lsr

� & �LrdLsrnp

��

�� Speed tracking performance

Time [s] Time [s]

Time [s] Time [s]

Des

ired

torq

ue [

Nm

]

Spee

d [r

pm]

Posi

tion

[pts

] , 1

turn

= 2

000

pts

Des

. cur

r. a

mpl

. [A

]

Figure ��#� Results for a periodic square wave with amplitude changing between rpm and � rpm� Sampling time ! ms� KP � � � � KI � � �


Fig� ��# shows the response after starting the operation from zero initial conditionsfor the PBC� with a periodic square wave of large amplitude and a sampling timeof ! ms� The rotor speed converged very fast after the transient changes of speedreference� The desired torque and the desired current reached their maximum valuesduring the transient� An integrator antiwindup technique ��!� was implementedto avoid saturations� We can see from the graph of desired torque� that it did notconverge to zero in steady state� because the integrator action was used to overcomefriction� The desired current amplitude converged to the selected magnetizing current�Imr � ! A plus the small torque drive current�

Spee

d [r

pm]

Spee

d [r

pm]

Spee

d [r

pm]

Spee

d [r

pm]

Time [s] Time [s]

Time [s] Time [s]

OBFLPBC

Figure �� a and �b show results with PBC for di�erent sampling times� �c and�d show results with OBFL for di�erent values of KP �

As discussed in the previous section� because of the heavy computational burdenof OBFL� we had to increase the sampling time to $ ms� Also� and perhaps as aconsequence of the increased sampling period� we were not able to tune properlythe controller gains when we used a large change in the set points which saturatedthe control signals� Therefore we used a low speed reference of � rpm for theperformance comparison of the two controllers� A larger regulation range for OBFLis possible with a slower reference� this at the cost of achievable bandwidth� Aspointed out before� the initial condition for the observer in the OBFL must be chosen

�� Robustness and disturbance attenuation ��

di�erent from zero� We used (�b� � ��$� � � �� equal to �� We also adjusted thegains of both controllers to have the most rapid response time without oscillations�which at the same time kept the current within the given amplitude limit of � A�

As can be seen from Fig� �� the response of PBC is faster than that of OBFLwith the same sampling time of $ ms� An increase of KP of OBFL for a more rapidresponse caused oscillations and stator current saturation� It is also shown in ��!��that an increase of �� and KI causes an overshoot in the speed tracking response�while a change of k� does not signi�cantly a�ect the speed response�

�� Robustness and disturbance attenuation

Time [s] Time [s]

Spee

d [r

pm]

Spee

d [r

pm]

PBC OBFL

Figure �� Variation of (Rr�� )� �a show results of PBC� �b show results ofOBFL�

Fig� �� shows the e�ects of �� ) errors in the rotor resistance estimate used inthe two controllers with a speed reference of � rpm and for the same sampling timeof $ ms� We can see that PBC is unquestionably more robust to the change of thisimportant parameter�

In Fig� ��$� we can see the response of the two controllers for load torque com�pensation� We applied an unknown load torque �about � � Nm at t � ! s usingthe DC motor which was connected to the shaft of the induction motor� The gainconstants of the controllers had to be changed because the moment of inertia waschanged to Dm � #�� kgm�� The two controllers compensated well for theload torque disturbance within s with an increase of about �� A in the currentamplitude� but the OBFL needed higher current amplitude than the PBC� both in


the initial transient �� A vs� �� A� and at stationary conditions� Notice that thespeed reference was reduced in this experiment to � rpm� which explains why thespeed tracking of OBFL was improved as compared to the previous experiment�

We also observed experimentally the instability mechanism of OBFL presented inSection �!� under the same test conditions as in the simulation shown in Fig� ��

Des

. Cur

r. A

mpl

. [A

]

Des

. Cur

r. A

mpl

. [A

]Sp

eed

[rpm

]

Spee

d [r

pm]

Time [s] Time [s]

Time [s]Time [s]

OBFLPBC

Figure ��$� �a and �b show speed and desired current amplitude for the PBC� �cand �d show the same quantities for the OBFL�

�� Conclusion

We have presented in this section an experimental comparison of two controllersbased on exact linearization and passivity ideas� It is our strong belief� and wehope the present study corroborates this point� that PBC outperforms schemes basedon exact linearization� mainly because the control actions required to impose thelinear dynamics typically will exceed the domain of validity of the models� This isclearly illustrated in the present experimental study where we were unable to achievegood performance of the OBFL for large and fast speed references� Of course� ourexperimental conclusions pertain only to the particular application at hand� Otherexperimental evidence of exact linearization controllers may be found in� e�g�� ! � ��To the best of our knowledge the present work is the �rst attempt to compare two

� Robust stability of PBC ��

di�erent novel nonlinear schemes on the same experimental facility� The recent paper�� presents a similar study with analogous conclusions�

We conclude this section with the following remarks�

� We have illustrated that� in spite of a widespread erroneous belief� linearization�at least in the form which is presented here does not allow us to imposearbitrary performances� even in the hypothetical case of no control constraints�In the present study the convergence rate of both schemes is limited by the rotortime constant� due to damping injection limitations in PBC� and to observerrate limits in OBFL�

� PBC will in general be simpler to implement than exact linearization controllers�This fundamental issue of computational complexity is� unfortunately� not fullyappreciated in the recent literature of nonlinear control� It certainly goes beyondthe arguments of availability of cheap and fast numerical processors� it pertainsinstead to poor numerical robustness of complicated arithmetic operations�

Robust stability of PBC

The stability proof presented in Section !� for indirect FOC �see also Section ��!� for a proof in the case of the full model critically depends on the cascade structure ofthe closedloop equations� which is unfortunately lost when the rotor time constantis not exactly known� It is well known that this parameter is subject to signi�cantchanges during machine operation� hence the question of robust stability with respectto parameter uncertainty arises naturally�

In this section we recall the results of �� and provide some answers to the questionof robust stability� First� we present a result that states that� under very weakconditions� all signals in the closedloop system are bounded� Then� we give necessaryand su�cient conditions for uniqueness of the equilibrium point of the closed loop�Interestingly enough� both conditions above allow for a ) error in the estimate ofthe rotor timeconstant� a requirement which is not hard to satisfy in applications�Then� we give conditions on the motor and controller parameters� and the speed androtor �ux norm reference values that ensure either global boundedness of all solutions�or �local or global� asymptotic stability or instability of the equilibrium� The basisfor these robustness results is the introduction of a new change of coordinates andthe construction of a quadratic Lyapunov function from which we can establish thestronger property of global exponential stability�

The closed loop system that we study in this section is described by a fourth ordernonlinear autonomous system that we repeat here for ease of reference� The motor


model is given by

Tr '�r � ��r & LsrvDm-qm � � L

� npLsrLr

v�J �r�� !

and the control inputs are generated by the nonlinear dynamic output feedback PBC

v ��

LsreJ �d

��Lrnp�

d

�� #

'�d �(Rr

np��d� �d� �

d � ��KP &KI

p '�qm� KI � KP � ��

The �ux norm and speed reference �� 'qm�� respectively� and the load torque L areassumed to be constant�

To simplify the expressions below� and without loss of generality for the purposesof this study� all motor parameters will be set to unity except for the rotor resistanceand the load torque� which are assumed to be constant but unknown� Setting allparameters except the rotor resistance to unity causes only small loss of generality�for two reasons� First� setting rotor moment of inertia and mutual inductance to unitychanges only the loop gain by a factor� which can be compensated for by a scaling ofthe velocity� Second� the unknown parameter of importance to indirect �eld�orientedcontrol is the rotor time constant� which is a function of both rotor resistance androtor inductance �which can change due to nonlinear magnetics� Therefore the e�ectof an unknown rotor time constant can be investigated by considering the e�ect of anunknown rotor resistance only� To generalize the uniqueness and stability conditionsderived in this section� Rr must be replaced by Rr�Lr and (Rr by (Rr�(Lr�

�� Global boundedness

��

�

�

�

b�t

d�G�p

e

Figure �� Inputoutput description of closedloop system�

�� Coordinate changes and uniqueness of equilibrium ��

By the use of an inputoutput formulation of the problem we established in �� thefollowing result� Since the proof is quite technical we have omitted it here� and referthe interested reader to Section � of Appendix D�

Proposition �� The current�fed machine �� in closed�loop with the PBC�� may be described as the feedback interconnection of an LTI systemwith a time�varying bounded gain as

d � G�pe

e � � � b�td

see Fig� �� where

G�p �pKP &KI

p� & �pKP &KIRr�Rr

with � an external bounded signal� and b�t � b �t & b��t� such that

jb �tj j(Rr � Rr

(Rr

j

b��t � L� �

�

The corollary below follows from the application of the L small gain theoremand the evaluation of the L� gain of G�p�

Corollary �� Assume the impulse response of G�p is positive and

(Rr � Rr

Then the feedback interconnection of G�p and the b�t�gain has �nite L gain� Thatis� the PBC is boundedinput boundedoutput stable�

�� Coordinate changes and uniqueness of equilibrium

To carry out the asymptotic stability analysis in the general case we �nd it conve�nient to work with a state space representation of the system� and introduce somecoordinate transformations� First� let us de�ne the coordinate transformation�

� �

��

��

��rdJ �r��rd�rd'�qm

��

�This assumption is made only to simplify the expressions� it can clearly be relaxed��Notice that this change of coordinates� although formally nonlinear because it involves a non�

linear function of the state variable �d� may be considered as linear viewing �rd�t� as a bounded

function of time�


where

�rd � eJ �d��

�This results in the following dynamic model

'� �

��

�Rr(Rr

��

�Rr

� (Rr��

�Rr

�KP �KP��

�KI

� ��

��&

��

Rr

��

KP L�L

�� $

Now� we shift the equilibrium to the origin� To this end� we de�ne the new

coordinates w�� 1� where 1� � IR� is an equilibrium of �� $� Below we will show

that� for all practical purposes� the equilibrium is unique� The transformed dynamicmodel becomes

'w �

��

�Rr(Rr

��w��

�Rr &�Rr��

1��

� (Rr��w��

�Rr � �Rr��

1��

�KP �KP

��1�� KP

��w��

�KI

� ��

1��w��

��w ��

The equilibria of the model in �� $ have the following property�

Proposition �� The equilibria of �� are independent of KP � KI� Further�the equilibrium is unique for all values of L if and only if � (Rr !Rr� �

Proof� The equilibria of �� $ are all solutions 1� � IR� to the equation

��

��

��

�Rr(Rr

��

�Rr

� (Rr��

�Rr

�KP �KP��

�KI

� ��

�� 1� &

��

Rr

��

KP L�L

��

For any equilibrium point we must have 1�� because the PI�controller integrates�� This simpli�es the equilibrium equations to�

�

��

�� Rr

(Rr��

�Rr

� (Rr��

�Rr

�KP �KP��

��

�� 1��

1��1��

��&

��

Rr��

KP L

��

From these equations� the following third order polynomial in 1�� is derived�

Rr(Rr

1�� (R�rL

1�� &Rr(Rr

�� 1�� R�

r��L � ��

�� Coordinate changes and uniqueness of equilibrium ��

If the equilibrium value of 1�� is known� then 1�� and 1�� can be calculated �as functionsof 1�� Henceforth� we will concentrate on the solution of �� In particular wewill investigate the conditions under which the function L � L�1�� is bijective� i�e��1�� is also a function of L�

The expression for L as a function of 1�� is

L �Rr

(Rr1�� &Rr

(Rr�� 1��

R�r

�� & (R�r1��

Clearly� L�1�� is continuous and surjective� Then it is a bijection if it is strictlymonotone� The derivative of L�1�� is

dLd1��

�Rr

(R�r1�� &

�!R�

r(Rr

�� Rr

(R�r

��1�� &R�

r(Rr

��

R�r

�� & (R�r1��

�� !

The denominator in this equation is always positive� Therefore� if the numerator isof constant sign� L�1�� is bijective� The numerator of ��! is a polynomial in 1��This polynomial is of constant sign if its discriminant is less than or equal to zero�that is� if

�R�r � � R�

r(R�r & (R�

r

The discriminant is a polynomial in (R�r which is less than or equal to zero for (R�

r ��R�

r � �R�r��

Also� if (Rr �p!Rr then all terms in the numerator of ��! are strictly positive�

Then� L is a monotone function of 1�� If� on the other hand� (Rr � !Rr then valuesfor 1�� can be found where d�L

d�� so that L as a function of 1�� is not monotone

anymore and therefore not bijective� This concludes the proof� �

As an example of the existence of multiple equilibria for certain ranges of L� theroots of �� will now be determined by application of the root locus technique tothe more suitable form

�� L(Rr

Rr

1�� &R�r�R�r��

1��1�� & ��


2

1.5

0.5

1

0

-0.5

-1

-2

-1.5

-1.5 0-0.5-1-2 21.510.5

Figure �� Root locus of the system equilibria for (Rr � !Rr�

2

1.5

0.5

1

0

-0.5

-1

-2

-1.5

-1.5 0-0.5-1-2 21.510.5

Figure �� Root locus of the system equilibria for (Rr � !Rr�

The uniqueness of equilibrium for (Rr � !Rr is evident from Fig� �� since there

�� Local asymptotic stability ��

are three coinciding real roots for one value of L� while for any other value of Lthere is only one real root� The non�uniqueness of the equilibria for (Rr � !Rr causesthe locus of Fig� �� to have three distinct real roots for a certain range of L� If(Rr � !Rr then the two poles go directly to the zeros without crossing the real axis�

Before closing this section it is interesting to �pull out� the nonlinear terms of�� as

'w �

��

�Rr(Rr

��

�Rr &�Rr��

1��

� (Rr��

�Rr � �Rr��

1��

�KP �KP

��1�� KP

��

�KI

� ��

1��

��w &

��

(Rrw�

� (Rrw�

�KPw�

w�

�� w�

��!�

Noting the presence in the right hand term of the scaling factor w�� recalling that

w� � d� 1d� and refering to Fig� �� with '� � on page !�� we see from ��!�that� roughly speaking� the closed loop system behaves �almost linearly� if the PIspeed loop is not too tight� That is� if w�

��is �small� and%or slowly time varying�

�� Local asymptotic stability

In this section we will study� via the rst Lyapunov method� the local asymptotic sta�bility of �� Towards this end� we see that the systems �rst order approximationis simply the �rst right hand term of ��!�� whose characteristic polynomial is givenby ��

s&Rr � (Rr��

Rr � �Rr��

1��

(Rr��

s&Rr�Rr��

1��

KPKP

��1�� s&KP

��

KI

��

1��

s

��

Given the complexity of the expression above� �recall that 1� is itself a nonlinearfunction of the motor parameters� we are unable at this point to make a generalstatement concerning the stability of the roots of this polynomial� Consequently� wewill only consider below some special cases� In particular we will show that� evenwith zero load torque� the equilibrium may become unstable�

A Case of known Rr

As shown in Section !� � when Rr � (Rr the equilibrium is GAS� However� to providesome tuning rules it is interesting to look at the behaviour of the roots of the linearized


system� for instance as a function of the load torque� To this end� we write thecharacteristic polynomial in a L�root locus form as

� & �LR

�r

��

s� &KP s&KI

�s&Rr��s� &KP s&KI�

The closed loop then has two poles at �xed positions determined by KP � KI and� asL increases� the double pole at s � �Rr moves along straight asymptotes in the lefthalf plane�

B Case of zero load torque

Even though this case will be studied with greater detail in Section �� where sometuning rules for the PI gains are derived� we present here this simple� but interesting�proposition�

Proposition �� Local stability for zero load torque�� Assume L � � Then�the system �� is locally asymptotically stable if either � (Rr Rr & KP orK�

P � KI � On the other hand� the equilibrium will be unstable if (Rr � Rr &KP anda large integral gain is used� �

Proof� When L � the characteristic equation reduces to

�s& Rr��s�s&KP &KI &KP �s& Rr� (Rr � Rrs

&�s&Rr� (Rr �RrKI �

The proof is completed noting that this equation has one root at s � �Rr� whilefrom the Routh�Hurwitz criterion we know that the other roots are on the open theleft hand plane if and only if

(RrRrKP & (RrK�P � � (Rr �Rr �KP KI

�

The proposition above shows that the system can be destabilized� in the sense ofhaving unstable equilibria� if the rotor resistance is overestimated� the proportionalgain is too small� and a large integral gain is used�

�� Global exponential stability

In this section we will investigate global exponential stability of the equilibrium usingLyapunov�s second method� Namely� we will construct Lyapunov functions of theform

V �w ��

w�Pw

�� Global exponential stability ��

where P is a positive de�nite symmetric constant matrix� To select P we �rst �ndpositive semide�nite matrices Pi that lead to expressions without cubic terms in thederivative of V � Second� linear combinations of these positive semi�de�nite matricesare constructed that lead to a negative�de�nite 'V �w� Finally� the positive de�nitenessof P is checked�

To illustrate the procedure we �rst construct a Lyapunov function for the casewhere Rr � (Rr� Then� we treat the case when L � � Rr �� (Rr� and derive a su��cient condition on (Rr for GAS� The general case is then illustrated with a numericalexample�

A Introducing positive semi�de�nite matrices to avoid cubic terms

To construct our Lyapunov�function candidate we consider for P linear combinationsof the following positive semi�de�nite matrices Pi� i � �� #�

P� �

��

� �

�� P� �

��

��Rr

��

�� (Rr

��

P� �

��

� KP

KP K�P

�� P� �

��

K�P KP

(Rr

KP(Rr (R�

r

��

The corresponding functions Vi�w ��w�Piw� i � �� # have derivatives

'V��w � w�P� 'w � �Rrw��

Rr�� (Rr

1��

w�w� � Rrw��

(Rr1��

��w�w�

'V��w � w�P� 'w � �Rr & (Rr

(Rr

w��

Rr

(Rr

w�w� & � (Rr &Rrw�w� &Rrw�w�

'V��w � w�P� 'w � �KIKPw�� KIw�w�

'V��w � w�P� 'w � �K�P �Rr & (Rrw

��

�KP

(Rr�Rr & (Rr &K�PRr

�w�w�

�KPKI(Rrw�w� �KPRr

(Rrw�� KI

(R�rw�w�

Since these derivatives have only quadratic terms� the derivative of V �w � w�Pwwill also have only quadratic terms if P is a linear combination of Pi� i � �� #�As a consequence� the global negative de�niteness of 'V �w can be easily checked�


B Lyapunov function for (Rr � Rr

For the nominal case Rr � (Rr� a Lyapunov function can be constructed that is validfor all L as follows� Realize that for the nominal case� the equilibrium is

1�� 1��

�1�� L

Consider the matrix

Pa � P� & P�

This choice of P results in the Lyapunov function candidate Va�w ��w�Pw with

derivative

'Va�w � � K�PRrw

�� R�

rKPw�� KIKPw

��

� � KPR

�r &K�

PRr

�w�w� �KPKIRrw�w� � �KI &R�

rKIw�w�

The cross�term in w�w� can be cancelled by adding a term in P� to Pa�

Pb � P� & P� &KI &R�

rKI

RrP�

which results in the candidate Lyapunov function Vb�w with derivative

'Vb�w � � �K�

PRr &KI &R�

rKI

Rr

�w�� R�

rKPw�� KIKPw

��

�� KPR

�r &K�

PRr &KI &R�

rKI

Rr

�w�w�

&� �KI &R�

rKI�KPKIRr

�w�w�

� �a�w�� a�w

�� a�w

�� b��w�w� & b��w�w�

This derivative can always be rendered negative de�nite by adding a component�z� & z�P� to the matrix Pb�

Pc � P� & P� &KI &R�

rKI

RrP� & �z� & z�P�

where the coe�cients z�� z� are chosen to compensate for the cross�terms as follows�

z� ��

Rr

b��a�

z� ��

Rr

b��a�

�� Global exponential stability ��

so that the derivative of the Lyapunov function Vc�w ��w�Pcw becomes

'Vc�w � �a�w��

�a�w�� a�w

�� b��w�w� & b��w�w�

�b��a�

w��

b��a�

w��

��b��a�

&b��a�

�w��

The function Vc�w is positive de�nite and its derivative is negative de�nite� thereforeit is a strict Lyapunov function for Rr � (Rr�

C Lyapunov functions for (Rr �� Rr� L �

For the case L � and Rr �� (Rr� the cross�term �Rr � (Rrw�w� appears in 'V��w�This constrains the construction of a Lyapunov�function� The following approach hasbeen used to derive su�cient conditions for GAS�

Add P� and P� to obtain a 'V �w with negative terms for w�� and w�

��

'V��w & 'V��w � �KIKPw�� KI & (R�

rKIw�w�

�K�P �Rr & (Rrw

�� KP � (Rr�Rr & (Rr &KPRrw�w�

�KPKI(Rrw�w� �Rr

(RrKPw��

Add an amount of P� such that the cross�term in w�w� is cancelled� which simpli�esthe quadratic term�

'V��w & 'V��w &KI�� & (R�

r

Rr

'V��w � �KIKPw�� Rr

(RrKPw��

��K�

P �Rr & (Rr &KI�� & (R�

r

Rr(Rr

�Rr & (Rr

�w��

��KP � (Rr�Rr & (Rr &KPRr &KI

� & (R�r

(Rr

�w�w�

��KPKI

(Rr �KI� & (R�

r

(Rr

�Rr & (Rr

�w�w�

� �a�w�� b��w�w� � b��w�w� � a�w

�� a�w

��

To make this expression negative de�nite� one can add a term zRrP� so that the


derivative of the total Lyapunov function becomes

'V �w � �a�w�� b��w�w� � b��w�w� � a�w

�� a�w

��

�zw�� z�w�w� � zw�

�

� �w�

��

z & a� �z� & b�� b�� z

�z� & b�� a� b�� a�

��w

� �w�Qw'V �w is thus negative if the matrix Q is positive� A necessary and su�cient conditionfor this matrix Q to be positive is that all its leading principal minors are positive�

z & a� � �� z & a� z

�� z & a� �z� & b��

z �z� & b�� a�

��

��z & a� z� & b�� b��

z z� & b�� a� b�� a�

��

The �rst two conditions are satis�ed� The third condition follows from the fourthwhich can be formulated as follows�

�z & a�a�a� � �z� & b��a� � b��a� �

The cross�factor � follows from (Rr and Rr� while z can be chosen� The value for zwhere the expression is optimal is calculated from setting the derivative of the aboveexpression to zero

a� � � �z� & b��

z ��

�

�a� �� b��

�Replacing z� the condition for positive de�niteness becomes�

a�a�#��

� a�b��

& a�a� � b��

This condition is always satis�ed for su�ciently small � as a consequence of thepositiveness of a�� so that there are values of (Rr� with (Rr �� Rr� where the systemis GAS� The boundary values of � for which 'V �w is still negative� are complicatedfunctions of all parameters in the model�

� O��line tuning of PBC ��

D Lyapunov functions for (Rr �� Rr and L ��

The previous approach can also be applied to the case where L �� but the resultingsu�cient conditions for GAS are not given here since they are rather complicated�Instead� a numerical example is given of a Lyapunov function that ensures GAS forthe particular parameter values KP � �� KI � �� Rr � � (Rr � � and allL�

Consider the candidate Lyapunov function

V �w � w��

P� & ��P� & P� & P�

�w

This function is positive de�nite� and its derivative is

'V �w � �w�Q�� w

where

�� Rr

�� (Rr

1�� Rr��

�� (Rr

1�� Rr��

and Q�� is a constant symmetric matrix whose o��diagonal coe�cients dependon �� and �� For V �w to be a Lyapunov function� 'V �w must be negative de�nite�and Q�� must therefore be positive de�nite� For the particular parameter valuesof the numerical example� this positive de�niteness can be proved using the propertythat �� and �� are bounded functions of 1��

��

�

(Rr

Rr

&(R�r1��

Rr

(Rr �Rr

R�r

�� & (R�r1��

�� (Rr

� (Rr � Rr��

R�r

�� & (R�r1��

1��

Remark �� We have gone with some detail in �byhand� calculations above onlyfor the sake of illustration of the ideas� It is clear that the procedure is much moree�cient� and can be made systematic� with modern symbolic computation tools�

� O��line tuning of PBC

It is well known that the performance of indirect FOC critically depends on thetuning of the gains of the PI velocity loop� a task which is rendered di�cult by the


high uncertainty in the rotor timeconstant� �Recall that we showed in Proposition�� that indirect FOC is identical to PBC in currentfed machines� In this sectionwe give some simple rules� �rst reported in �$�� for the o��line tuning of the PI gainswhich will ensure robust stability� As is wellknown� robust stability� as opposedto just stability� ensures better performance measures� We give then an algorithmthat� for each setting of the PI gains� evaluates the maximum error of the rotortimeconstant estimate for which global stability is guaranteed� In this way� withoutknowing the actual value of the rotor time constant� we can assess the performance ofall PI settings before closing the loop� Not surprisingly� instrumental for our analysisis a suitable decomposition of the system into a passive feedback interconnection�


It has been shown in Section !� that the system �� !�� is globally asymptot�ically stable if (Rr � Rr� Furthermore� in Section � we proved that stability is actuallyexponential and showed that the system remains stable under large variations of therotor resistance� The problem we address in this section is the selection of the pa�rameters KP and KI which will ensure� not just stability of the closedloop� but alsoa good transient performance in spite of the uncertainty in the rotor resistance Rr�To formulate mathematically this problem we make the following basic observation�

PI tunings which allow larger estimation errors in rotor resistance are more robust�hence their overall performance is better�

Thus the tuning problem can alternatively be formulated as�

De�nition �� PI tuning problem for induction motors�� Given the induc�tion motor and controller equations �� with controller parameters (Rr�KP � KI and �� Find a range of values of the motor resistance estimation error

�Rr�� (Rr � Rr

for which global stability of the closed�loop system is preserved� More precisely� we

want to �nd an interval �Rminr � Rmax

r � � IR such that� if Rminr Rr Rmax

r � thenthe system �� is globally stable��

Even though some answers to this question may be found in Section �� the pro�cedure relies on the generation of Lyapunov functions� hence its computationallyexhaustive and not very transparent to the user� In this section we give a very sim�ple numerical algorithm that� for each setting of the controller gains� generates the

It is clear that from the interval for Rr we can immediately obtain an interval for �Rr �whichwill contain zero� by simply subtracting !Rr�

�� Change of coordinates ��

required set of values for �Rr� The size of this interval� which we will call in the sequelthe performance interval� provides a robustness measure of the closedloop systemwhich guides the user in the choice of the PI gains�

Since we believe this result could be of interest for practitioners� and furthermorein the case we consider the expressions considerably simplify� we have decided towork out the details for the nonnormalized motor model� However� we still limit ourattention to uncertainty on Rr�

�� Change of coordinates

In order to solve the problem formulated above� we use the change of coordinatesintroduced in Section �� We will show below that this representation reveals somenew energy dissipation features of FOC which are instrumental for our theoreticaldevelopments�

Applying the transformation �� to the closedloop equations results in thefollowing nonlinear dynamic model

'� �

��

�RrLr

�Rrnp��

�� RrnpLsr

� �Rrnp��

�� RrLr

�npKPLsrLrDm

�KPLsrDm��

�� KInpLsrDmLr

LsrDm��

��

�� &

��

Rr��LrLsr

��&

��

KP

Dm� �Dm

�� L

where we have �pulled�out� the terms depending on L to underscore the fact thatit enters linearly in the equations� Henceforth� if we can prove exponential stabilityof the system with L � then the system will remain stable �in the sense of globalboundedness even when L �� This fact will be invoked in our subsequent analysis�

In Section � it is shown that the equilibria of this system are de�ned by verycomplex algebraic relationships� and the system can actually have multiple equilibria�On the other hand� when L � � the equilibrium is unique and is given by

�1�� 1�� 1�� 1��

��

Lsr

� � ��

Given this fact� and the stability consideration mentioned in the previous paragraph�we will treat the load torque as a disturbance and concentrate on the case L � �

Let us now shift the equilibrium to the origin by introducing the new coordinateszi � �i � 1�i� i � �� # to obtain

'z �

��

�RrLr

�Rrnp��

z��

npLsr

�(Rr �Rr

�

� �Rrnp��

z� �RrLr

�npKPLsrDmLr

�KPLsrDm��

�z� &

��Lsr

��KI

npLsrDmLr

LsrDm��

�z� &

��Lsr

�

�� z ��!


�� Local stability

In this section we present conditions that guarantee local stability of the closedloop�To this end we recall the indirect Lyapunov method which states that �under someconditions veri�ed here a nonlinear system is locally stable if and only if its linearapproximation is asymptotically stable� i�e� all the eigenvalues of the system matrixare in the open left hand plane� Thus we rewrite ��! �pulling out� its nonlinearterms as

'z �

��

�RrLr

�npLsr

�(Rr � Rr

�

�RrLr

�npKPLsrDmLr

�KP

Dm�KI

npLsrDmLr

�Dm

�� z &

��

�Rrnp��

z�z�

� �Rrnp��

z�z��KPLsr

Dm��z�z�

LsrDm��

z�z�

��

In compact notation we get 'z � ALz & F �z� The systems �rstorder approximationis simply 'z � ALz� whose characteristic polynomial is

det�sI � AL �

�s&

Rr

Lr

�g�s

with

g�s � s� &

�KP

Dm&Rr

Lr

�s� &

�KP

(Rr

DmLr&

KI

Dm

�s&

KI(Rr

DmLr

Thus� by applying the Routh�Hurwitz criterion� it can be shown that the conditionswhich must be satis�ed for local stability are

i� KP � �DmRrLr

�

ii� KI �

and

iii� c��

��KPDm

�RrLr

��KP

RrDmLr

�KIDm

�� KI

RrDmLr

�

KPDm

�RrLr

�

Noting that the �rst two conditions are trivially satis�ed with positive values forthe PI gains� the attention will be focused on the third one� It is easy to see that�because the denominator is always positive� this condition can be equivalently writtenin terms of the rotor resistance Rr as

iii� � Rr �KI

�RrKP

RrLr

�KI

� KPLrDm

��!!

�� Local stability ��

From the fact that

(Rr � KI(Rr

KP�Rr

Lr&KI

for all KP � KI � Lr� we can now state the following proposition�

Proposition �� Consider the model of the current�fed induction motor in closedloop with the indirect FOC �� If

Rr � (Rr ��!#

then the system is locally exponentially stable for L � � When L �� all trajectoriesenter �in �nite time a ball centered at the origin of radius jLj� �

The importance of this result is that stability is preserved� for all PI gains� providedwe underestimate the rotor resistance� However� our interest lies in obtaining tuningrules independent of Rr� hence we will study the case when

KI(Rr

KP�Rr

Lr&KI

KPLr

Dm

i�e�� when condition ��!! holds for all Rr� This inequality can be equivalentlywritten as

(RrK�P � KI�Dm

(Rr �KPLr

from which� after some easy manipulations� the following result can be obtained�

Proposition �� Consider the model of the current�fed induction motor in closedloop with the indirect FOC �� Then� the system �with L � islocally exponentially stable for all Rr if and only if one of the conditions below hold

Condition �

KP � Dm�Rr

Lrand KI � ��!�

Condition �

� KP � Dm�Rr

Lrand � KI K�

P�Rr

Dm�Rr�KPLr

��!�

hold�

�

It is interesting to note that the results presented in this section are independentof the parameter �� This will also be the case for the performance results givenbelow�


�� A performance evaluation method based on passivity

In this section we de�ne intervals of the rotor resistance for which global stability ispreserved as a function of the PI gains� As mentioned above the size of these intervalsgives a quantitative measure of the performance of the PI tuning� To this end� we�nd convenient to decompose the closedloop system ��! as the feedback inter�connection of two subsystems� One of the subsystems contains all the nonlinearitiesand turns out to be strictly passive� The second subsystem is LTI� Our motivationfor the introduction of this decomposition is twofold� First� as shown in Appendix A�the negative feedback interconnection of two passive subsystems is still passive� andif one of them is strictly passive then the closedloop system is stable� Second� forLTI systems there is a very simple analytic characterization of passivity in terms ofpositivity of the real part of its transfer function� Since this transfer function dependson the motor parameters and the PI gains� this positive realness test will provide uswith the desired resistance intervals�

We can rewrite ��! as the feedback interconnection of the following two sub�systems

G� � u� �� y�

�%�%

'z� � �RrLrz� &

��z�u�

u� � � �Rrnpz�

y� � z�z�

��!$

G� � u� �� y�

��

'� � A� & bu�u� � z�z�y� � c��

��!�

with

A �

�� Rr

Lr�

npLsr

�(Rr �Rr

�

�npLsrKP

DmLr�KP

Dm�KI

npLsrDmLr

�Dm

�� b �

�

��

��

�Rrnp

�KPLsrDmLsrDm

��

c �

�� Rr

np

��

�� z�z�z�

��

and the obvious interconnection structure

u� � �y�u� � y�

This decomposition is shown in Fig� �� where we have de�ned the transfer functionG��s � c��sI � A��b

G��s �Lr

(Rr

n�p��

Dm(Rrs

� &KPRrs&KIRr

DmLrs� & �KPLr &DmRrs� & �KP(Rr &KILrs&KI

(Rr

��!�

�� A performance evaluation method based on passivity ��

��

�s�Rr

Lr

G��s

� ��

�

�

�

�

z�

z� z�z�

�Rrnpz�

Figure �� Decomposition of the closedloop system�

The main features of this decomposition are� � We show below that the subsystemG�� although nonlinear� is passive� G� is an LTI relative degree one and minimumphase LTI system that can be made strictly positive real for suitable values of themotor and controller parameters�

We present now the main result of this section�

Proposition �� Consider the model of the current�fed induction motor in closedloop with the indirect FOC �� Assume that the conditions for localstability �� or �� or �� are satis�ed� and that the following inequalitieshold

f� � and f� � � pf�f� ��#

where

f�� D�

mRr(Rr &KPDmLr� (Rr �Rr

f�� K�

PRr(Rr �KIDm� (R

�r &R�

r

f�� K�

IRr(Rr

Then� the trivial equilibrium of the closed�loop system is GAS if L � � When L �� all trajectories enter �in �nite time a ball centered at the origin of radius jLj� �

To establish the proof we need the following lemma�


Lemma �� The subsystem G� � u� �� y� de�ned by �� is output strictlypassive� In particular� it satis�es the following inequality for all t �

�

��

Z t

�

u��y��d � �(Rr

np��

Z t

�

z��z��z��d

� Rr

Lr

Z t

�

z��d & �

with � � IR� �

Proof� Consider the function

V ��

z��

whose time derivative along the trajectories of G� is given by

'V � �Rr

Lrz��

(Rr

np��z�z�z�

The proof is completed by integrating the above expression over the interval � � t��

recalling that V � � and de�ning �� V � �

�

Lemma �� Consider the transfer function �� Then� the conditions of Propo�sition �� above �i� e�� stability and �� ensure that G��s is strictly positivereal� �

Proof� We will verify thatG��j satis�es condition �ii�� of the KalmanYakubovichPopov lemma� see Section � in Appendix A� First� notice that the real part of thetransfer function ��!� is given by

�fG��j g � kf�

� & f� � & f�

�a� � a� �� & ��a� � a� � ��#�

with

k �Lr

(Rr

n�p��

a� � KI(Rr

a� � KPLr &DmRr

a� � KP(Rr &KILr

a� � DmLr

�� A performance evaluation method based on passivity ��

The limit condition is clearly veri�ed with f� � � Now� from ��#� we see thatthe sign of this equation is only determined by the sign of its numerator� Moreover�viewing this polynomial as a function of the variable � two conditions must besatis�ed in order to guarantee the positivity of the transfer function� namely� Thecoe�cient f� must be positive and the polynomial

f� � f� � & f�

� & f� ��#

must not have real roots�

Finally� notice that ��# can be written as a polynomial of degree over thevariable x � �� The roots of this new polynomial are

1x�� f� �

pf �� #f�f� f�

It can be seen that if f �� #f�f�� then 1x�� are complex and therefore� the roots of theoriginal polynomial� 1 �� 1 �� are also complex� On the other hand� if f �� #f�f��the roots 1x�� are real of the form

1x�� f� f�

Hence� if f� � then 1 �� 1 � are complex� satisfying the condition for positivity�Finally� if f �� #f�f�� then 1x�� are again real but in this case with the followingstructure

1x� ��f� &

pf �� #f�f� f�

1x� ��f� �

pf �� #f�f� f�

It is easy to see that� if f� � then 1x� � and therefore 1 �� 1 � are complex� Inorder to get 1x� � � i�e� 1 �� 1 � also complex� the condition that must be satis�ed isf� �

pf �� #f�f� which can be equivalently written as � #f�f�� The proof of the

proposition is then completed by noting that the condition f� � � pf�f� satis�essimultaneously the three required conditions to guarantee positivity of the polynomial��# � �

We can now present the proof of Proposition ��

Proof� Consider the following function

V� � �(Rr

np

Z t

�

z�z�z�d � Rr

Lr

Z t

�

z��d � �� z �� Lemma ��

&�

��P� �


whose time derivative along the trajectories of ��!� is given by

'V� � �(Rr

npz�z�z� � Rr

Lrz�� &

�

��PA& A�P � & ��Pbz�z�

Invoking the strict positive realness of the transfer function ��!�� the above ex�pression can be written as

'V� � ��Q� & y�z�z�

� �Rr

Lrz��

�

��Q�

��kzk�

for some � � � This proves that the whole state z � L�� and furthermore that� � L � From ��!$ we conclude that 'z� � L� also� hence z� tends to zero� Finally�from ��!� and boundedness of z� we conclude that � � as well� �

Standard convex optimization techniques can be used to obtain from the inequal�

ities above the performance interval �Rminr � Rmax

r �� A very simple algorithm can�however� be derived as follows�

An algorithm for estimation of the performance interval

Step � Input data� Numerical values for the induction motor parameters� the rotorresistance estimate (Rr � and the controller gains KP � � KI � �

Step � Set Rr � (Rr�

Step � Check conditions for local stability ��!# or ��!� or ��!� and��# � If one of them is not satis�ed then Rmax

r �the maximum value thatguarantees global stability is found� Go to Step �� If both conditions holdproceed with the following step�

Step � Increment the current values Rr by a small number � � � and go to Step��

Step � Set Rr � (Rr�

Step � Decrement the current value of Rr by a small number � � �

Step � Check conditions for local stability ��!# or ��!� or ��!� and ��# �

If one of them is not satis�ed then Rminr �the minimum value that guarantees

global stability is found� and the seeking algorithm stops� If both conditionshold go to Step ��

�� Illustrative examples ��

�� Illustrative examples

In this section some numerical and experimental results are presented� The objectiveis twofold� First� to illustrate the sharpness of the local stabilityinstability boundarypredicted by the theory� Second� to validate our claim that the size of the performanceintervals indeed provide a measure of the transient performance behaviour�

-4

-2

0

2

4

6

8

10

Rr

02

4 K i

012345Kp

Figure �� Manifold of local stabilityinstability�

A Simulations

For clarity of presentation� we have in this section chosen an academic example wherewe set all parameters of the induction motor equal to unity except for Rr� which willbe changed throughout the experiments� Also� we have set (Rr � � *� In Fig� �� the stabilityinstability boundary predicted by condition ��!! is illustrated in theRr� KP � KI space� All values of the rotor resistance above this surface correspond tostable behaviours� while those below will yield an unstable closedloop system� Wehave simulated the system with KP � �� KI � �� for which the critical value forRr is Rr � #�� In Fig� ��! we show the time evolution of the speed for the stable�Rr � �� critical �Rr � #�� and unstable �Rr � # cases� with 'qm� � � rad%s and


'qm� � � �� rad%s� Notice that we chose the initial condition of the speed very closeto its reference to further underscore the stableunstable behaviour�

We then evaluated� using the algorithm of Section ��#� the performance interval

for this setting to get �Rminr � Rmax

r � � �� This is� of course� a very smallrobustness margin� thus we expect the performance to be below par� This is cor�roborated in the step response plot of Fig� ��#�a where the speed reference waschanged from 'qm� � � rad%s to 'qm� � � rad%s� To improve transient performancewe must retune the PI gains to enlarge the size of the performance interval� We set

then KP � �� for which we get the bigger interval �Rminr � Rmax

r � � � �#� �� Asexpected� the transient response� shown in Fig� ��#�b� is much better�

0 2 4 6 8 10 12 14 16 18 209.8

10

0 2 4 6 8 10 12 14 16 18 209.8

10

10.2

0 2 4 6 8 10 12 14 16 18 209.5

10

10.5

Time [s]

(a)

(b)

(c)

P

E

E

D

S

Figure ��!� Simulation showing the stabilityinstability boundary� Speed in �rad%s�versus time�

�� Illustrative examples ��

0 10 20 30 40 50 60 70 80 90 100−10

0

10

20

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

S

P

E

E

D

Time [s]

(a)

(b)

Figure ��#� Simulation showing the improvement of performance� Speed in �rad%s�versus time�

B Experimental results

We have tried our tuning rules on the experimental setup of the Laboratoire deG�enie Electrique de Paris� For the software interface an integrated system with anIntel � #�� MHz microprocessor and a DSP! C digital signal processor allowedus to facilitate the assembly language coding� The squirrelcage induction motorparameters can be found in Table ��!�

The motor is driven by a pulsewidth modulated �PWM inverter which has asampling period of $�� s� and MOSFET bridges with current feedback loopsas shown in Fig� �� The measurement of currents is done using Halle�ectsensors� which have good accuracy �the linearity error is about �) and isolates theacquisition system electrically from the inverter� This reduces the measurement noise�The rotor position is measured by a highresolution optical incremental encoder� witha sampling period of � � � �� s� and the rotor speed is estimated from the positionmeasurement�

Thus� from standard discretization considerations� the maximum sampling periodis the estimation time of the speed �� s�


Description Notation Value Unit

Nominal power P �� kWPower factor cos�� !Number of pole�pairs np Maximum speed 'qmax

m �# rpmMaximum stator voltage � VMaximum stator current � ANominal rotor �ux norm N ��# WbNominal stator resistance Rs � *Nominal rotor resistance Rr !�� *Stator inductance Ls #$ mHRotor inductance Lr #$ mHMutual inductance Lsr �## mHTotal leakage factor��r � �s � � �� Moment of inertia Dm � � kgm�

Table ��!� Motor parameters�

D CI. M.

POSITION SENSOR

LOADVOLTAGEINVERTER

P W MINTERFACECURRENT

INTERFACEPOSITION

D S P 3 2 CC A R D

DRIVESYSTEM

P C

CONTROLDIGITAL

SOURCE

Figure �� Block diagram of experimental setup�

First we illustrate the stabilityinstability boundary predicted by ��!!� To thisend� we used the controller gains KP � �� KI � $ and �xed the rotor resistanceestimate to (Rr � ! *� which is clearly very far from its nominal value� The resultis shown in Fig� �� where we have taken� as in the simulation example� the initialspeed very close to the reference� namely 'qm� � �� rpm� and 'qm� � ! rpm�respectively� It is clear that� for this particular setting� the bound ��!! is only oftheoretical interest�

We then �xed (Rr � !�� * and evaluated the performance interval for KP � ��

and various settings of KI � For KI � $ we have �Rminr � Rmax

r � � � �� !�� *� which

� Discrete�time implementation of PBC ��

is a very narrow range� Decreasing KI to KI � �� yields � �� *� and �nallyKI � � gives the bound � �� *� The corresponding step responses� depictedin Fig� ��$� corroborate our claim that transient performance is directly correlatedwith the size of the performance interval�

0 100 200 300 400 500 600 700 800 900 1000200

250

300

350

Time [pts] (1 point = 0.002 s)

Spee

d [rp

m]

Figure �� Experimental instability with KP � �� KI � $� (Rr � ! * and'qm� � �� rpm�

� Discrete�time implementation of PBC

The computation of an FOC scheme� which is a nonlinear dynamic feedback� is invari�ably carried out in discrete�time �with standard microprocessors or special purposedigital signal processors� Since both the machine model and the controller are non�linear� the stability analysis of a discretized implementation of the controller is farfrom obvious� Furthermore� it is not even clear how such a discretization should bedone� Some e�orts in this direction may be found in ��#�� In this section we present anew discrete�time FOC for current�fed induction motors ��rst reported in � �� whichensures global asymptotic speed regulation and rotor �ux norm tracking provided acondition relating the sampling rate with the controller parameters is satis�ed� Thiscondition disappears as the sampling period goes to zero� To ensure stability of thescheme we modify the the �ux norm reference� If the latter is �xed� a tradeo� mustbe established between the sampling period and the reference value�


0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

350

400

Time[pts] ( 1 point = 0.003sec)

Spe

ed [r

pm]

(a)

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

350

400(b)

Time[pts] (1 point = 0.003sec)

Spe

ed [r

pm]

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

350

400

Time[pts] (1 point = 0.003sec)]

Spe

ed [r

pm]

(c)

Figure ��$� Experimental performance improvement with KP � �� 'qm� � �� rpm and� �a KI � $� �bKI � �� cKI � � �

We show that the main� and far from obvious� e�ect of discretization is that thetasks of �ux and torque regulation have to be shared between the two control channels�as opposed to a continuoustime implementation where they can be decoupled� Aconsequence of this coupled control approach is an enhancement in rotor �ux trackingperformance� One additional feature of this scheme is that� compared with the �rstdi�erence approximation of the classical indirect FOC� the additional computational

�� The exact discrete�time model of the induction motor ��

burden is negligible� Establishing the stability for a discretized FOC without thismodi�cation� remains a challenging open problem�

The performance of this new discrete controller is illustrated with some experi�mental results�

Remark �� Discrete time notation�� For a given �constant sampling pe�riod T � IR�� s� we will use the notation x�k � x�kT � k � Z�� i�e�� x�k denotesthe samples of the continuous signal x�t at the time instants t � kT � Also� q willdenote the shift operator� that is q�ix�k � x�k � i� i � Z��

�� The exact discrete time model of the induction motor

We will be concerned here with the� by now wellknown� currentfed induction motormodel �� ! and PBC �� #��

Before proceeding with the derivations of the discretetime PBC it is importantto underscore that the tasks of �ux and torque regulation are decoupled between the�rst �ux controlling component and the second torque controlling component� It willbe shown below that to achieve a stable discretetime implementation� this principleof decoupling in control signals will have to be abandoned� This is the main� and farfrom obvious� modi�cation to FOC induced by the discretization� It is well knownthat discretization introduces coupling terms in the dynamic equations which arerelated to sampling period and speed � $�� This naturally leads to the question ofhow the discretetime control components should be coupled� to avoid a degradationof performance because of the discretization� An answer to this question is given thenext sections�

To obtain a stable discretetime FOC the key observation made in � �� is thatthe motor equations �� ! can be exactly discretized under the assumption of zeroorder hold control� In spite of the simplicity of the model� this is not an obvious fact�This is possible only due to the particular form of the bilinear term in the torqueequation� Namely� we can integrate �� ! to get

'qm�k & � � 'qm�k &npLsr

DmLr

v�k�J�Z �k��T

kT

�r�sds

�� T

Dm

L

On the other hand� since v�k will be constant in the sampling interval� we can alsointegrate the solution of the �rst equation in �� ! asZ �k��T

kT

�r�sds � �� aTr�r�k & �T � �� aTr�Lsrv�k� a�� e�

TTr

which allows us� using the skewsymmetry of J � to evaluate

v�k�J�Z �k��T

kT

�r�sds

�� aTrv�k

�J�r�k


and replace the expression above in the discretized speed equation to get

�r�k & � � a�r�k & �� aLsrv�k ��#!

'qm�k & � � 'qm�k &�� aTr

Dm�k� T

DmL ��##

�k �npLsr

Lrv�k�J �r�k ��#�

�� Analysis of discrete time PBC

It is well known that the closedloop system consisting of a continuous plant andthe ad hoc discretization of a continuous�time nonlinear controller is not necessarilystable� even when the original �continuoustime loop is stable� However� it is areasonable conjecture that discretizing the PBC �� #�� with a su�cientlysmall sampling time will yield a stable closed loop� Let us consider� for simplicity�the case of constant desired rotor �ux amplitude� i�e� '� � � In this case� an exactdiscrete�time implementation of the PBC is obtained as

v�k ��

Lsre J �d�k�

��

Lrnp�

d�k

��#�

�d�k & � � �d�k &TRr

np��d�k

d�k � C�q� 'qm�k� 'qm�� C�q � ��KP &KI

q � � ��#$

The stability of the system �� ! in closed loop with ��#��#$ remains to beestablished�

Notice that the discretization above preserves the decoupling between �ux andtorque regulation in the controller structure� Due to discretization e�ects� the decou�pling of torque and �ux control may however be lost in the closedloop system� In thenext section we will present a modi�ed FOC where the decoupling structure of thecontroller itself is not preserved� and it will be shown that in this way we can ensureGAS in the discretetime case� This new controller will� of course� give decoupledcontrol of �ux and torque in closedloop�

To motivate the new discrete�time algorithm� we will �rst highlight the di�cultiesin the stability analysis of �� !� ��#��#$ introduced by the discretization�To this end� it is convenient to recall �rst that the stability analysis of the continuoustime case� It heavily relied on the cascaded structure of the error system� namely onthe fact that ��r � exponentially� which together with

� �� &�

��rd��rd &

npLr

��rdJ ��r

�� A new discrete�time control algorithm ��

allowed to complete the stability proof�

Unfortunately� this kind of analysis cannot be applied to the discretetime con�troller ��#��#$� Essentially because� in contrast to �� we cannot expressv�k as a di�erence equation for the desired rotor �ux� which is now de�ned as

�rd�k � e�d�k�J��

��

�cos��d�ksin��d�k

�Hence� we are unable to recover the asymptotically stable error equation for the rotor�ux error�

�� A new discrete�time control algorithm

Several modi�cations to the PBC are required to obtain a globally stable discretetime scheme� First� to ensure the desired behaviour for the rotor �ux mentionedabove� we propose to choose

v�k ��

Lsr

��

�� a��rd�k & �� a�rd�k

��#�

with the desired rotor �ux de�ned as

�rd�k � e�d�k�J��k

�� k

�cos��d�ksin��d�k

��#�

As will be shown in the following analysis� it is essential for the stability analysis that��k is a time�varying �ux norm reference�

Inserting ��#� in ��#! leads to ��r�k&� � a��k� and consequently ��k� �Replacing this control law in ��#� we get after some simple calculations

�k �np

Lr�� a�rd�k & ��J�rd�k & ��k

where��k

��

npLr�� a

��rd�k & �� a�rd�k�J ��r�k

is an exponentially decaying sequence� We see from the last equation that to recoverthe property of �k� �kd we must ensure

npLr�� a

�rd�k & ��J �rd�k � d�k ��

To enforce this identity we �rst notice that �rd�k in ��#� satis�es the di�erenceequation

�rd�k & � ��k & �

��ke Tb�k�J�rd�k


with Tb�k�� d�k & �� d�k� which replaced in �� yields

npLr�� a

�rd�k & ��J �rd�k � npLr�� a

��k��k & � sin�Tb�k

Therefore� we choose

b�k ��

Tarcsin

�Lr�� a

np��k & ��kd�k

��

To ensure that this equation has a solution we must choose ��k such that

Lr

np��kjd�kj �

��k & �

�� a��

holds �k � Z�� Once we have proved that �k � d�k we can follow verbatim thearguments of the continuoustime case for the stability analysis of the mechanicalsystem� Stability limits for the PI gains can be computed by the insertion of thecompensator ��#$ in the di�erence equation ��## and the use of Jury�s stabilitycriterion on the resulting polynomial in q�

We can now establish the following result�

Proposition �� Discrete�time controller from �� Consider the mo�tor model �� with control inputs the �rotated stator currents v and measurableoutput the rotor speed 'qm� Let 'qm� denote the desired constant rotor speed� andassume load torque L is also constant� though unknown� De�ne the discrete�timecontroller as

v�k ��

Lsr

eJ �d�k��

��k��a cos�Tb�k� a

��a��kLr

��k�npd�k

��!

�d�k & � � �d�k & Tb�k

d�k � C�q� 'qm�k� 'qm�� C�q � ��KP &

KI

q � � ��#

with b�k as in �� sampling period T � � v�k � v�kT � 'qm�k � 'qm�kT �

a�� e�

TTr � The PI gains are chosen such that

� KI � KP � �Dm

��a�Tr ��

Let ��k � � which represents the reference value for the rotor �ux amplitudek�rd�kk� be chosen so as to satisfy��

�Notice that the upper bounds �� and �� become arbitrarily large as T � �� Hence�relaxing the restrictions on the choice of KP � KI and ��k �

�� Discussion of discrete�time controller ��

Under these conditions� for all initial conditions of the motor� we have that allinternal signals are bounded and

�i� limk�

j 'qm�k� 'qm�j �

�ii� limk�

j k�r�kk � ��kj �

�

�� Discussion of discrete time controller

We have introduced in the �rst component of ��! a term cos�b�k� Since b�kdepends� via �� on d�k we see that this term couples the �ux and torqueregulation objectives� It will be shown in the experiments below that the inclusion ofthis term enhances the rotor �ux tracking performance�

As pointed out above� the introduction of a timevarying reference for the �uxnorm is essential to guarantee global stability via a suitable choice of this reference�i�e�� If the �ux is kept constant we have to tradeo� between the samplingperiod and the reference speed�

From the de�nition of the control ��#� we see that v�k is bounded� and inparticular it satis�es the bound

kv�kk �

�� a��k & � & a��k

Since the bound depends only on the rotor time constant and the desired rotor �uxit might be used to study saturation e�ects�

We have considered here only the problem of speed regulation with constant loadtorques� It is clear from the proof that the extension to speed tracking or to positioncontrol can be solved simply by choosing a suitable compensator in ��#� Also� wecan easily treat the case of unknown varying load torque� e�g�� a linearly parameterizedfunction L � �� 'qm� qm

�� IRq� with a straightforward modi�cation to ��##and use of classical adaptive control techniques�

�� Experimental results

A Experimental setup

The experimental system consists of a squirrelcage induction motor with the ratedparameters given in Table ��#� a threephase inverter� an antialiasing �lter moduleand a dSPACE DS�� DSPboard �see also Section �� for the description of asimilar setup in a host PC�


Parameter Notation Value Unit

Nominal power P $� WNumber of pole pairs np Power factor cos� �$�Voltage U VNominal current iNab !�� ANominal speed 'qNm �# rpmRotor resistance Rr ��# *Rotor inductance Lr �� mHMutual inductance Lsr �� mHNominal torque N $�� NmNominal �ux N �� Wb

Table ��#� Motor parameters�

The measurements of the stator voltages and currents are made using Halle�ectsensors� These sensors have a good accuracy �the linearity error is about �) andisolate the acquisition system electrically from the inverter� This reduces the mea�surement noise� Aliasing errors due to the high frequency modulation of the inverterare avoided by the use of four Bessel lowpass �lters of order �ve� To obtain accuratematching of the �lters� we used switched capacitor �lters�

The rotor position is measured by a highresolution optical incremental encoderwith � lines per revolution� This resolution is increased by a factor of # with aquadruple counter on the DSP�card� The motor is driven by a pulsewidth modulatedinverter with MOSFET bridges and current feedback loops� Only two of the phasesare current controlled� and the third stator voltage is calculated to obtain a symmetricthreephase feeding�

For logging of signals and online tuning of parameters� the programs TRACEand COCKPIT were used to communicate with the DSPboard� The controller wasimplemented using multirate computation� with the calculations of the PWM runningat a sampling rate of �� s �� MHz� with a slower computation of referencecurrents and speed control at a rate of T in the interval �!#� �� ms � ��# Hz� kHz�

Speed was estimated from position using the backward di�erence approximationproposed in ��

'� � ��k� ��k � �

TThis backward di�erence approximation of speed was smoothened using a discretetime implementation of the �lter

��p� & ��

��p& �

� � � � � � � �$


In order to protect the equipment we also included a �# A current saturation�

To execute the program directly on the DSP�card� we used the Real�Time optionin SIMULINK� The speci�c parameter values of our system as well as the integrationmethod� in this case Runge�Kutta !� could be speci�ed directly by the use of thisgraphical interface� We used two di�erent sampling times in the control calculation�a slow one T � �� !#� ms for the determination of ��!��#� and a fast samplingtime T � �� $ � �s for the rotation to the stator �xed reference frame� It should bepointed out that it is essential for a good performance of the algorithm to carry�outthe latter calculation at a high sampling frequency�

0 1 2 3 4

−100

−50

0

50

100

New controller T=30 ms

Time [s]

Spe

ed [r

ad/s

]

0 1 2 3 40.1

0.15

0.2

0.25

0.3

0.35

0.4New controller T=30 ms

Time [s]

Est

. and

ref

. flu

x no

rm [W

b]

0 1 2 3 4

−100

−50

0

50

100

FOC controller T=30 ms

Time [s]

Spe

ed [r

ad/s

]

0 1 2 3 40.15

0.2

0.25

0.3

0.35

0.4 FOC controller T=30 ms

Time [s]

Est

. and

ref

. flu

x no

rm [W

b]

Figure �� Improvement of �ux tracking with new discrete controller�

B Experimental results

In the experiments it was aimed at the following performance evaluations�

� To compare the new algorithm with the discretized FOC for di�erent samplingperiods�

� To show the enhanced �ux tracking performance of this novel discrete approach�

� To illustrate the robustness to variations in the motor parameters and the tuninggains�


� To evaluate the performance of the algorithm at low speed conditions�

For the sake of brevity we show here only the plots showing the enhanced �uxtracking performance� The reader can consult � � � for more experimental results�

Interestingly enough� in these tasks the new controller systematically performedbetter than the standard FOC scheme� a feature that appeared more evident for slowsampling times� To illustrate this fact we carried�out an experiment where� to avoidcurrent saturation� for a speed increase of $ rad%s at t � �� s the desired �ux mustbe decreased from �!! Wb to � Wb� Fig� �� shows the responses of speed andthe estimated rotor �ux norm for a sampling period of T � ! ms� The latter wasobtained with a standard openloop estimator from stator currents as in �! �� see also�� and �� in Section ��

� Conclusions and further research

The currentfed model of the induction motor was given in Section � and the underly�ing assumption of this model explained� In Section the standard direct and indirectFOC schemes were presented� An observer�based feedbacklinearizing controller wasalso presented� and some of the drawbacks of this method were derived theoretically�and illustrated by simulations and experiments�

In Section ! we proved that the PBC for the voltagefed machine reduces toindirect FOC� This gives a passivity interpretation to FOC� whose importance canhardly be overestimated� since it provides a deep systemtheoretic foundation tothis popular strategy and paves the way for subsequent analysis� In Section � weestablished exponential stability �with a quadratic Lyapunov function of PBC androbustness when (Rr �� Rr of FOC�

Simple o��line rules for PI gain tuning �with guaranteed stability were presentedin Section �� Our contention� which is validated with simulations and experiments�is that with this rules we can improve the transient performance� The performanceenhancement is quantitatively measured with an indicator of robustness of the stabil�ity with respect to uncertainty in the rotor resistance� namely the largest allowableestimation error under which global stability is preserved� A very simple algorithmthat evaluates these ranges for each PI setting has been presented�

Finally� a globally stable discrete�time version of PBC was given in Section $ andexperimental result� showing enhanced �ux tracking� were given in Section $��

In spite of the remarkable stability robustness properties of PBC which we havepresented here� there is an obvious interest of considering adaptive controllers thatestimate Rr to improve performance� Experimental evidence that substantiates thisclaim may be found in ��!�� Providing a satisfactory solution to this fundamentalproblem is the main driving force of the �eld now� Important steps towards its

� Conclusions and further research ��

solution were given in ��$ �� Other results may be found in �� $�� $�� Also� itis interesting to see whether the performance limitation imposed by rotor dynamicscan be overcomed�

Chapter ��

Feedback interconnected systems

Robots with AC drives

Throughout the book we have stressed the fact that PBC is compatible with oneof the important viewpoints of systems theory that complicated systems are bestthought of as being interconnections of simpler subsystems� each one of them beingcharacterized by its dissipation properties� This aggregation procedure has threeimportant implications� First� it is consistent with the dominating approaches formodeling and simulation based on some kind of network representation and energy�ow� Second� it help us to think in terms of the structure of the system and torealize that sometimes the pattern of the interconnections is more important thanthe detailed behaviour of the components� Finally� it is indeed a designorientedmethodology which allows us to isolate the �free subsystems� sensors and actuators�

In Chapters and �� we have already shown how� via a decomposition of thesystem dynamics into its electrical and mechanical parts� we can exploit this featureof PBC to design practically useful controllers for electrical machines� We consideredin those chapters the case where the dynamics of the mechanical subsystem is essen�tially linear� Even though this crude model is suitable for a vast array of problems�there are many modern applications that require the incorporation of a more detailedmathematical model of the mechanical load to meet the performance requirements�A typical example� that we consider in this chapter�� is the problem of motion controlof robot manipulators actuated by AC drives� In this case a linear model cannotcapture the behaviour of the robot when it is moving fast and we have to look at thecomplete nonlinear coupled dynamics�

The approach that we develop in this chapter is applicable� not just to electrome�chanical systems� but to a very large class of feedback interconnected systems� For thisreason we consider �rst a more general feedback interconnection problem� and then

�The material reported in this chapter is based on work done in collaboration with Elena Panteleyand Paulo Aquino�

##�

�� Ch� � � Feedback interconnected systems� Robots with AC drives

derive as a particular case the example of robots with AC drives� Namely� we assumethe forward subsystem is an underactuated EL system and that� in the absence of thelatter� a stabilizing controller is known for the feedback subsystem� �In the case ofrobots with AC drives the subsystems are the motors and the robot� respectively� Theassumption of a known controller for the robot amounts to the standard assumptionof neglecting the motor dynamics� This scenario� which appears in many practicalapplications with the forward subsystem representing the actuator dynamics� leadsnaturally to the classical cascaded �nestedloop control scheme that we have encoun�tered already in previous chapters� We are then interested in establishing conditionsunder which we can design a passivitybased innerloop controller for the EL systemsuch that global tracking is achieved� These are expressed in terms of actuatorsensorcouplings� the �strength� of the subsystems interconnection� and the requirement oflinear dependence on the unmeasurable variables� Interestingly enough� this analysisdoes not invoke the standard timescale separation assumptions prevalent in cascadedschemes� but uses instead some �growth� conditions on the interconnections�

The innerloop controller is designed following the developments of Chapter �hence we will review it only brie�y� Further details may also be found in � �$�� Themain obstacle that we must overcome when the feedback �mechanical subsystem isnot LTI is that we cannot simply use an approximate di�erentiation �lter �as donein Chapter �� to avoid the need for the derivative of the torque reference� To solvethis problem we add a nonlinear observer�

After presenting our general result� we then use it in the design of an outputfeedback global position tracking controller for robot manipulators actuated by ACdrives� Similarly to the material in Chapter � the result applies to a fairly largeclass of AC drives� which includes as particular cases induction� synchronous andstepper motors� Instrumental for the observer design is the utilization of a new robotcontroller which is linear in the link velocities� We also present simulation resultswhich compare our controller with the one reported in �� $�� which was derived usingbackstepping ideas�

� Introduction

� Cascaded systems

We are in this chapter interested in the problem of controlling feedback intercon�nected systems of the form depicted in Fig� � �� We assume that both subsystems�.e and .m� are nonlinear with u the control input vector� ye� ym the measurableoutputs and an �unmeasurable coupling signal� The control objective is to makeym asymptotically track a desired �timevarying reference ym� with internal stability�

� Cascaded systems ��

u ye

ym

�

� �

�

.e

� �.m

Figure � �� Feedback interconnected system�

This type of con�guration appears in many practical applications� For instance�in Chapter where we consider the electrical machine� the subsystem .e containsthe electrical dynamics� and .m the mechanical part of the motor� In this case is a force �or torque of electrical origin� Another situation when this scheme arisesis when .e represents the actuator dynamics and .m the plant to be controlled� Inthese instances it is reasonable to assume that� if were a manipulated �controlvariable� then we dispose of a suitable controller for the subsystem .m� say Col� Thisscenario� which is adopted in Chapter is also followed here� and it leads naturally tothe cascaded �i�q�� nestedloop controller con�guration of Fig� � � � where Cil is aninnerloop controller to be designed such that tracks d �su�ciently fast� to avoidupsetting the stability of the outer loop� As already pointed out in Chapter � thistask is complicated by the dependence of .e on ym and the need to estimate 'd�

ym�d

uye

ym

�

��

��

� �

�

Col Cil .e .m

Figure � � � Cascaded �nestedloop control con�guration�

Cascaded controls are typically designed for linear systems invoking time�scale sep�aration assumptions� That is� if the innerloop is designed to have a large bandwidth�then it essentially behaves as a static gain for the outer loop� and stability bounds canthen be estimated using� for instance� singular perturbation techniques ��!�� Thisreasoning is most suitable for the case when .e represents the actuator dynamics�


which is typically faster than the plant itself� Besides the additional di�culty dueto the dependence of .e on ym� the extension of this technique to the nonlinear casedoes not seem obvious� Furthermore� the timescale separation assumption becomesquestionable in some modern technological applications with increasing performancerequirements�

In pure cascade systems� where ym is not fed back into .e� stabilizability of .e

is combined with a growth condition on the interconnection to insure stabilizabilityof the composition .m � .e� �see e�g� �� $� �!�� # � �$#� for a summary of the latestdevelopments� It is therefore expected that in our case� besides the growth conditionon the interconnection� we will require a �stronger� form of stabilizability �namely�uniform in ym for the subsystem .e�

Conditions for stability of the cascaded scheme can be easily derived under fairlygeneral conditions on the subsystems� For instance� assume .m is a �su�cientlysmooth statespace system� linear in the input � which is exponentially stabilizedby a �possibly nonlinear controller Col� whose output is linearly bounded by ym�Further� assume that Cil is such that for all ym we have the tracking performance

j � dj �t & �tjdj ��

where �t is used to denote exponentially decaying terms� Then� converse Lyapunovtheorems can be invoked to prove suitable convergence properties of the overallscheme� There are �at least two di�culties with this approach� First� the dependenceof .e on ym makes the de�nition of an innerloop controller Cil that ensures �� forall ym very di�cult� unless some coupling conditions between the two subsystems aresatis�ed� These conditions are expressed in terms of some restrictions on the func�tional dependence of on ym� Second� to ensure the desired tracking of the mappingsd �� and ym� �� ym� the controllers Cil and Col will usually require the knowledgeof higher order derivatives of d and ym� respectively� When both are combined thiswill translate� through the dynamics of .m� into the need of having available formeasurement� It will be shown in the following sections that this di�culty can beremoved by the use of an observer� provided some linearity assumptions on Col andCil are imposed�

In this chapter we illustrate how the ideas discussed above can be applied to thecase when .e is an underactuated EL system with partial state measurements� Theinnerloop controller will then be a PBC as the one discussed in Chapter � We willthen establish conditions on the actuatorsensor coupling in .e� the �strength� of thesubsystems interconnection� and the nature of the dependence on the unmeasurablevariables �i�e�� linearity� to ensure global tracking for the mapping ym� �� ym� Inessence� these three conditions are required to stabilize .e �uniformly in ym� toensure a growth requirement for the subsystems coupling �similar to the one requiredin � $�� and for the design of the observer� respectively�

�� Robots with AC drives ��

� Robots with AC drives

An important corollary of our main result is the design of a global position trackingcontroller for rigid robot manipulators actuated by AC drives when the only variablesavailable for measurement are the link positions and velocities� and the currents ofthe stator windings� Control of robot manipulators with AC drives is an interest�ing research topic� both from application oriented and theoretical viewpoints�� Theproblem was �rst postulated for the case of induction motors in ��!� where a locallyasymptotically stable scheme was presented� The scheme presented here extends theexisting results in several directions�

�� We prove global asymptotic stability of the closed loop� which is strictly strongerthan the local result of ��!� or the ultimate boundedness condition of �� $�� In�� which heavily borrows from � �� asymptotic stability is also established�

� We establish this result for the �fairly large class of AC machines treated inChapter � which includes induction� permanent magnet synchronous and step�per motors as particular cases�

!� In ��!� some of the terms coupling the robot and the motor dynamics are can�celled by the control to obtain a cascade connection of two subsystems� insteadof the feedback con�guration of Fig� � � � An important drawback of this ap�proach is� of course� the lack of robustness of the nonlinearity cancelation whichis clearly exhibited in ��! �� The solution presented here does not su�er fromthis drawback�

#� From the viewpoint of computational complexity� our controller is several or�ders of magnitude simpler than the schemes given in �� $�� see Section ��It should be mentioned� however� that in �� the more challenging adaptivecontrol problem is studied�

�� As usual in backsteppingbased schemes� the controllers in �� $� containhigher order terms which essentially act as high gains �� The latter willnot only degrade performance in the presence of noise� but also create seriousnumerical problems in a practical implementation� as discussed in Section ��

Instrumental for our development is the utilization of the robot controller of Propo�sition #�� in Chapter � �proposed in � �� which is linear in the link velocities� Thisfeature is essential for the design of an observer�

�We refer the reader to �� and �� for further motivation on the problem�


� General problem formulation

We consider feedback interconnected systems of the form depicted in Fig�� where.m is described by the state equation

.m �

�'ym� � ym�

'ym� � fm�ym & gm�ym��

with the full state ym � �y�m�� y�m��

� � IR�nm measurable� and � IRnm the unmeasur�able coupling signal� We will assume that fm�ym is locally Lipschitz in ym� and thatgm�ym is bounded� that is�

supym kgm�ymk � �� !

Consistent with our cascaded control approach� we will make the following as�sumption�

A�� For any given bounded reference signal ym�� y�m�� 'y�m��

� � IR�nm � withknown bounded derivative� we know a globally exponentially stabilizing staticstate feedback controller for �� which is linear in ym�� and linearly boundedin ym� That is� we know d of the form

d�ym�� K��ym� &K��ym�ym� �� #

kK��ym�k �kym�k& �

kK��ym�k �

such that� if � d� then ym � ym� is a globally exponentially stable equilibriumof ��

The subsystem .e is an EL system with generalized coordinates qe � IRne andLagrangian Le�qe� 'qe� ym�� Notice the dependence of Le�qe� 'qe� ym� on ym�� this willestablish the coupling between the two subsystems� The behaviour of .e is modeledby the EL equations of motion studied in Chapter �� that is

d

dt��Le

� 'qe�qe� 'qe� ym�� Le

�qe�qe� 'qe� ym� � Qe

where Qe � IRne are the external �dissipative and control forces� We will assumethat

Le�qe� 'qe� ym� ��

'q�e De�ym� 'qe

Qe � �Re 'qe &Meu

� General problem formulation ��

with� � dMe � De�ym� � De�ym�� dme � � Re � R�e � and� M�

e � �Ine�� We will further assume that De�ym� is globally Lipschitz in ym�� consequently

supym�kWi�ym�k � ��

for i � 1nm�� f�� nmg� with ��i the ith component of a vector� and

Wi�ym��

�De

��ym�i�ym�� i � 1nm

In summary� .e is described by

.e �

�De�ym�-qe &

ddt�De�ym� 'qe &Re 'qe � Meu

ye � M�e 'qe

��

Notice that� following ��$�� we have considered the natural outputs available for

measurement� For our further developments we �nd it convenient to de�ne 'qe��

�y�e � 'q�r ��

Finally� we will assume that � i�e�� the input to the subsystem .m� is given by

��Le

�ym��qe� 'qe� ym�

whose components can be alternatively written as

i ��'q�e Wi�ym� 'qe� i � 1nm �� $

The problem we consider in this chapter is formulated as follows

De�nition �� Global tracking problem�� Given the feedback system of Fig��described by �� with Assumption A�� Find conditions on thegeneralized inertia matrix� De�ym�� and the dissipation matrix� Re� of .e which willensure the existence of an inner�loop PBC of the form Cil � �ye� ym� d �� u suchthat

limt�

kym � ym�k �

with internal stability�

Remark �� It will become clear in the sequel that the results of this chapter are�mutatis mutandis� applicable to a broader class of systems� For instance� Le�qe� 'qe� ym�may contain potential energy terms� and the dissipation structure need not be lin�ear as assumed here� Also� the controller Col may be� in general� a dynamic outputfeedback�

�We concentrate here in the more interesting case of underactuated systems� the problem isconsiderably simpler in the fully actuated case� i�e�� when Me � Ine � See also Section ��


Remark �� Notice that we have considered an EL model for .e that does notdepend on the generalized coordinates qe� only on its derivatives� That is� thesecoordinates are cyclic �also called ignorable ��$�� Again� this is done for the sakeof clarity of presentation� Furthermore� this is the case of the electrical machinesconsidered in Section ��

Remark �� It is important to remark that� due to the dependence on ym�� thesubsystem .e �� does not de�ne a passive operator u �� ye�

� Assumptions

Before presenting the solution to the global tracking problem we discuss here theassumptions needed for its solvability�The innerloop PBC that we present here islargely inspired by the PBC derived in Chapter for the generalized AC machine�In this chapter we want to extend this result to a larger class of EL systems .e withthe additional complication that the feedback subsystem .m is not LTI anymore�It is useful then to revisit the design proposed for AC machines� particularly theassumptions required for the realizability of the controller� This analysis will helpus identify the class of EL systems for which our cascaded scheme will work� Wewill �rst concentrate in the torque control problem� and then consider the extensionto position control� Besides the invertibility requirement� in speed%position controlapplications we also have the di�culty that the calculation of the PBC needs 'd�which in its turn implies knowledge of acceleration� While this obstacle was removedin Section ��! with a linear �lter for the single machine case� a nonlinear observerwill be needed to handle the case of nonlinear .m� We recall that the innerloopmust be stabilized uniformly in ym� hence some severe restrictions on De�ym� areexpected� In particular� we have seen in the previous chapters that in order to beable to explicitly solve the controller equations we need an invertibility assumptionon the subsystem .e� To give a systemtheoretic �avor to this assumption we willpostulate it in terms of the solvability of a tracking subproblem� Then� we enunciatesome additional assumptions which are essentially related with the linearity requiredfor the solution of the observer problem�

�� Realizability of the controller

Motivated by the controller for electrical machines we propose an innerloop PBC Cilof the form

Meu � '�d &ReD��e �ym��d &

�K��ym

�D��e �ym��

�di ��

��d Ci�ym��d� i � 1nm ��

�� Realizability of the controller ��

where

Ci�ym�� D��

e �ym�Wi�ym�D��e �ym�� i � 1nm ��

and K��ym is a damping injection gain to be de�ned later�

To address the realizability issue of this controller we �rst notice that� due tothe structure of Me� the �rst ne� equations of the PBC �� can always be solvedfor u once �d and '�d are given� Hence we concentrate exclusively on the last ne� equations� which we write as

'�rd � A�ym��rd &B�ym��sd

where we have introduced the partition �d�� sd� �

�rd�

�� with �sd� �rd � IRne�� andde�ned �

B�ym� A�ym�

�� ReD

��e �ym� ��

with denoting some matrices �dependent on ym�� The realizability problem willbe solved if we can �nd a function �sd � g�ym�� rd� d such that the solutions of

'�rd � A�ym��rd &B�ym�g�ym�� rd� d

satisfy the constraint ��

Motivated by this observation� we will now de�ne an auxiliary problem� whosesolvability implies the realizability of the innerloop PBC�

A�� Consider the linear timevarying system�

'�rd � A�t�rd &B�t�sd ��

with A�t� B�t de�ned by �� Then� for arbitrary �possibly unboundedym��t� d�t� there exists a state feedback of the form

�sd � K��rd� ym� &K��rd� ym�d �� !

such that the closed loop system �� ! satis�es

k�dk � & �kdk �� #

and

limt�

�� d Ci�t�d � �di

��

with exponential rate of convergence� where Ci�t is given by ��

�Notice that we are treating A�ym�� B�ym�� as functions of time� �sd as an input and �rd asthe state�


Remark �� The following remarks are in order�

� We require �sd to be a�ne in d� as indicated in �� !� to be able to implementthe observer in the next stage�

� The requirement in �� # is needed for the proof of boundedness of �d� which aswe discussed in Section �� is a requirement to ensure that the implication �� d holds�

� Notice that the �clamping� condition �� must only be satis�ed asymptotically�

�� Other assumptions

We need three additional assumptions� First� that the nonactuated coordinates of.e are suitably damped� that is

A�� Re is of the form

Re��

�Rs Rr

�with Rr � �

Second� that the subsystems coupling is �weak� with respect to the unmeasurablesignals� Speci�cally� we need to ensure that does not contain terms which arequadratic in� 'qr� This is ensured by

A�� De�ym� is such that

Wi�ym��

� ne��ne��

�

The �nal assumption concerns the order of .m� which is restricted to avoid complextensor notation� and hence simplify the presentation� We will in Section � see how tohandle the case of robots with AC drives� where this assumption is not veri�ed�

A�� The subsystem .m is of order two� that is nm � ��

Remark �� AssumptionsA��A�� are imposed by the fact that .e is underactuated� They are not needed in the fully actuated case when Me � Ine�

�Recall that we have de�ned �qe�� y�e � �q

�r ��

� Problem solution ��

Problem solution

We are now in position to present the main result of this chapter�

Theorem �� Given the feedback system of Fig� �� described by �� Assume that an outer loop controller Col � �ym�� ym �� d� which satis�es theconditions of Assumption A�� is known� Under these conditions� there exists aninner�loop PBC� Cil � �ye� ym� d �� u such that

limt�

kym � ym�k �

with internal stability provided A��A�� are satis�ed� �

�� Proof of Theorem ��

The proof consists of two major steps� the design of the innerloop controller� andthe stability analysis of the overall cascaded con�guration� For the design of Cil�we takeo� from the PBC �� This control law is� unfortunately� non�implementable because it requires the measurement of 'ym� Thus� we propose animplementable scheme that uses an observer� It is at this point that we use theassumptions of linearity of Col on ym� stated in A�� and linearity of Cil withrespect to d as expressed in �� !� The design of the observer is the main technicalcontribution of the chapter�

A Inner�loop

A�� Non�implementable PBC

The innerloop PBC� implicitly de�ned by �� can �in principle be ex�plicitly solved invoking A�� More precisely� the assumption insures that� frommeasurements of ym� and d� we can solve online �� ! to get �d� '�sdfollows from di�erentiation of �� ! and knowledge of 'ym� and 'd� and can in itsturn be replaced in �� which can be solved for u�

Let us now prove the convergence of �� To this end� we derive the errorsystem

'��&ReD��e �ym��&

�K��ym

�D��e �ym��

and consider the desired energy function Led � ��TD��

e �ym�� whose derivative

�This is the same function as the one used in Section �� but it is now written in terms of �dinstead of �qed�


along trajectories of the error system gives

'Led � ��TD��e �ym�Res�ymD

��e �ym��

where

Res�ym�� Re &

�

d

dtDe�ym� &

�K��ym

�See �� in Chapter � Notice that� in view ofA�� d

dtDe�ym� � ym�W��ym�� and

from A�� we have that �W��ym� � � where ��ij denotes the �i� j�th submatrixof a block matrix� This� together with A�� ensures that when

K��ym � K�� ym � sup

ym

�y�m�

#�W��R

��r �W�

��

�

�W��ym�

��

we will have infym ��Res � � � � where �� is the minimum eigenvalue� Thisproves that 'Led ��Led� hence �� exponentially fast�

Writing down �� $ in terms of the errors �� as we did in Section �� invokingthe convergence proof above and the bound �� we get the bounds

k � dk �t & �tk�dk �t & �tkdk ��

where �t are some exponentially decaying functions and we have used �� # to getthe last bound�

Notice that A�� requires the controller equation �� to be satis�ed onlyasymptotically� but as will be shown later this will not a�ect the stability proof�

A�� Observer design

Unfortunately� this controller cannot be implemented in the cascade scheme of Fig�� because� as discussed above� u requires the knowledge of '�sd which demands 'd�This� in its turn� given that d is a function of ym �� #� would require 'ym� In Sec�tion ��! we proved that� in the case when .m is linear� we can use a linear �lter toovercome this problem� This solution is not feasible in the nonlinear case� for whichwe introduce a nonlinear observer and use the assumptions of linearity on ym� and d�

We will see now how we can construct an observer that will estimate 'd indirectly�To this end� let us denote with uN the �nonimplementable control that uses '�d�From �� we see that u depends linearly on '�d� The following chain of calculationsallows us to write uN as

uN � f��ye� ym� d & f��ye� ym� d '�d� f��ye� ym� d & f��ye� ym� d 'd� f��ye� ym� d & f�ye� ym� d 'ym�

� f�ye� ym� d & f��ye� ym� d

�� Proof of Theorem ��

where fi� i � �� are some suitably de�ned functions and we have used �� !�� # and �� to get the second� third and fourth equations� respectively� Finally�we observe that A�� ensures that in �� $ is linear in the unmeasurable part ofthe state of .e� i�e�� 'qr� consequently

uN � f��ye� ym� d & fo�ye� ym� d 'qr �� $

We propose now the control law

u � f��ye� ym� d & fo�ye� ym� dzr ��

where z�� z�s � z

�r �� IRne will be an estimate of 'qe generated as

De�ym� 'z &�

d

dt�De�ym� z &Res�ymz � Mev � L�ye� ym� d� z

withv�� u&K��ymye

and L�ye� ym� d� z an output injection to be de�ned below� The observer is motivatedfrom the fact that the system equations� after the damping injection� are of the form�see ��$ in Section �#

De�ym�-qe &�

d

dt�De�ym� 'qe &Res�ym 'qe � Mev

Hence the error equation results in

De�ym� '�z &�

d

dt�De�ym� �z &Res�ym�z � L�ye� ym� d� z

where �z�� 'qe � z� Consider now the quadratic function Vo

��

��z�De�ym��z� whose

derivative satis�es'Vo ��k�zk& �z�L�ye� ym� d� z

Now� �� can be written as u � uN � fo�ye� ym� d�zr� yielding the error equationfor .e

De�ym�-�qe &�

d

dt�De�ym� '�qe &Res�ym '�qe � �Mefo�ye� ym� d�zr

Thus the derivative of the desired energy function Led takes now the form

'Led ��k '�qek� � �ye � 'qed�fo�ye� ym� d�zr

The calculations above motivate the following choice for the output injection

L�ye� ym� d� z�� 1Mef

�o �ye� ym� d�ye � 'qed

with 1M�e � � � Ine�� which exactly cancels the cross term and leads to

'Vo & 'Led ��k '�qek� & k�zk�From here we conclude that '�qe � �and also �z � exponentially fast�


B Outer loop

Once we have established convergence of the innerloop we turn our attention to the

outer loop� First� adding and subtracting d� we rewrite �� in terms of �ym��

ym � ym� as

'�ym � Fm��ym &

�

gm�ym�

�� d ��

From A�� and the converse Lyapunov theorem �� of Chapter in ��!�� we havethat Fm��ym� the closed loop vector �eld obtained by setting � d� satis�es

��Vm��ym��ym

Fm�t� �ym ��k�ymk� ��

for some Lyapunov function Vm��ym which furthermore veri�es

k�Vm��ym

k �k�ymk ��

Combining these inequalities and using the various assumptions we get the bounds

'Vm ��k�ymk� & �k�ymkk � dk ��k�ymk� & �tk�ymk�� & kdk ��k�ymk� & �tk�ymk�� & k�ymk

��

where we have used �� and �� ! to get the �rst bound� �� forthe second bound and �� # �and boundedness of ym� for the last one� Asymptoticconvergence of �ym to zero follows immediately from the last inequality�

�

Remark �� The steps of the proof given above may be summarized as follows�First� we use the assumptions on actuatorsensor coupling� damping of the nonactuated dynamics� and decoupling between .e and .m �AssumptionsA�� A��and A�� respectively to design an innerloop controller which ensures that converges exponentially to d �uniformly in ym� This controller requires 'ym� henceits not implementable� At this point we invoke the assumptions on linearity of d andym� for the design of an observer� The proof is completed using A�� on exponentialstabilizability of .m and a converse Lyapunov theorem�

Remark �� The arguments used for the proof of stability of the outer loop aresimilar to the ones used in � $� to study stabilizability of cascaded systems of theform

'x � f�x� & g�x� �t�t

where 'x � f�x� is globally exponentially stable� Notice however that the key�growth� assumption in � $�� namely

kg�x� �tk ��tkxk

� Application to robots with AC drives ��

with �� a class K function� does not hold in our case� since �as seen from thederivations detailed in �� we dispose only of the weaker condition

kg�x� �tk �tkxk& �t

It should be pointed out that similar line of reasoning was used before in � ! ��

Application to robots with AC drives

In this section we apply our general Theorem � �$ to the problem of motion controlof robot manipulators with AC drives�

�� Model

We consider rigid robot manipulators where each joint is independently actuated by anAC motor� In this case the Lagrangian of the whole system satis�es the decompositionproperty of Proposition �� that is

L�q� 'q � Le�qe� 'qe� qm & Lm�qm� 'qm

where we have partitioned q�� q�e � q

�m�� then the system can be represented as in

Fig� � �� where both subsystems are passive� As discussed in Chapter �� robots areEL system whose Lagrangian is of the form

Lm� 'qm� qm ��

'q�mDm�qm 'qm � Vm�qm

with qm � ym� � IRnm the joint positions� � � dM � Dm�qm � D�m�qm � dm �

the inertia matrix� and Vm�qm the potential energy�

The dynamics of an nmdegrees of freedom rigid robot given in Chapter � can bewritten in the state space form �� with

fm�ym�� D��

m �ym��Cm�ym�� ym�ym� &Gm�ym��

gm�ym�� D��

m �ym�� !

See Chapter � for the de�nition of the terms and some relevant properties of themodel� We simply recall here that the Coriolis matrix satis�es

kCm�ym�� ym�k cMkym�k �� #

Also� the local Lipschitz condition on fm�ym� and �� ! are satis�ed� Notice thatwe have assumed that the link positions and velocities are available for measurement�


In Section �# the AC motors were classi�ed into underactuated and fully actuated�that is� machines where the voltages can be applied only to stator windings �e�g��induction motor� or to both stator and rotor windings �e�g�� synchronous motor with�eld windings� The latter class also includes machines� like the PM synchronous�PM stepper and variable reluctance motors� where the generalized coordinates consistonly of stator variables which are directly actuated by the stator voltages� Controlof underactuated machines is the most challenging problem� hence we will restrictour attention here to this class� The extension to the fully actuated machines isstraightforward� We therefore consider .e to be a block diagonal operator consistingof nm subsystems of the form �� with nei � #� that is

.ei �

�Dei��ym�i-qei &

ddt�Dei��ym�i 'qei &Rei 'qei � Meui

yei � M�e 'qei

��

where i � 1nm� 'qei � �y�ei� 'q�ri�� IR� are the stator and rotor currents� and the

inductance and resistance matrices are given as

Dei��ym�i �

�LsiI� Lsrie

J �ym��i

Lsrie�J �ym��i LriI�

�� Rei �

�RsiI� RriI�

��

with

J �

� ��

�� eJ �ym��i �

�cos��ym�i � sin��ym�isin��ym�i cos��ym�i

�

The overall inductance and resistance matrices of .e are de�ned as

De�ym�� block diagfDei��ym�ig� Re

�� block diagfReig

respectively� Notice that ne � #nm and that the rotor currents 'qri are not measurable�See Chapter �� for the de�nition of all the terms above�

Since nm �� A�� does not hold and we cannot apply directly Theorem � �$�However� we notice that since Dei��ym�i depends only on component i of ym�� thecomponents of the torque vector are given� for all i � 1nm� by

i ��

'q�eiWi��ym�i 'qei

with

Wi��ym�i ��Dei

��ym�i��ym�i �� $

This nice block diagonal structure simpli�es the �rst part of the design of Cil� allowingus to approach the problem as nm independent tasks� and easily adapt Theorem � �$�

�� Global tracking controller ��

�� Global tracking controller

Before stating our result we �nd convenient to repeat here Proposition #�� of Section�� which pertains to the outerloop controller Col for the robot subsystem� Thiscontroller was reported in � ��

Proposition �� Let

d � Dm�ym�-ym�� & Cm�ym�� 'ym��ym�r &Gm�ym��Kd'�ym� �Kp�ym� ��

where ym��t is a bounded� three times continuously di�erentiable reference� and

ym�r � ym� � c��ym�

� & k�ym�k ��

where the gains c� � � Kd � K�d � and Kp � K�

p � are chosen such that

c� � min

�� Kd

!��Dm & cM�

#��Kp

��Kd & ��Kd� q��Dm��Kp

��Dm

&'( �� !

with cM as in �� Under these conditions� every solution of �� de�termined by the control law � d satis�es

k�ym�tk� �e��tk�ym� k� for all t �

�

We can now state the main result of this section�

Proposition �� Consider the rigid robot dynamics �� with AC mo�tor drives �� where we assume that the link positions ym�� velocitiesym�� and stator currents ye of the motors are available for measurement� Then thereexists compensators Cil� Col of the cascaded control con�guration shown in Fig� ��which ensure global asymptotic tracking for all �bounded� three times continuouslydi�erentiable position references ym��

A Proof of Proposition ��

Due to the block diagonal structure of the system mentioned above� the �rst part ofthe proof until the de�nition of the observer� consists of verifying the conditionsof Theorem � �$� For the sake of clarity of presentation we will divide again thederivations into outer and innerloop controller� and observer design�

A�� Outer loop control


In this the following we will prove that the robot controller of � � satis�es assumptionA�� It is actually shown in � � that

Vm��ym��

�

��ym� & c��ym�

�Dm��ym� & c��ym� &�

�y�m�Kp�ym�

with c� � verifying

c�

�� &

s Vm�

��Kp

��

c� � c�

quali�es as a Lyapunov function� which clearly satis�es �� and ��

Besides exponential stabilizability� A�� imposes the crucial requirements thatd must be linear in ym� and linearly bounded by ym� This can be easily veri�ed from��

A�� Inner�loop control

The innerloop controller exactly coincides with the PBC for induction machinespresented in ��!� hence we go through it very brie�y� From �� we immediatelysee that� for each subsystem .ei� A�� and A�� are satis�ed� We will now provethat Assumption A�� is also satis�ed� With some lengthy� but straightforward�calculations we can show that in this case the matrices Ai and Bi��ym�i of �� are given by

Ai�� RriLsi

�iI�� Bi��ym�i

��

RriLsri

�ie�J �ym��i

where �i�� LsiLri�L�

sri� A state feedback that satis�es the conditions of AssumptionA�� is given in �� of Section ��!� It can be written in terms of the �uxes as

�sdi �

��idi & �

i�Lsi

Lsri�i�

�eJ �ym��i�rdi

with i� � a design parameter� �that de�nes the desired value for k�rk� Actually�in this case we can solve �� and obtain an explicit expression for �rdi as

�rdi � e�diJ�i�

�� !�

where

'�di �Rri

�i�di� �di� � �� !

�In this respect it is worth pointing out that� even though there are other controllers that ex�ponentially stabilize the robot� e�g�� to the best of our knowledge� only the one given abovesatis�es the additional linearity requirements of Assumption A��

�� Global tracking controller ��

It is easy to check that with the de�nitions above we have

�

��diCi�t�di � �di for all t �

which is clearly stronger than the asymptotic property required by AssumptionA��

The ideal control of Chapter ��!� which in this case is nonimplementable� canbe summarized as

uNi � vNi � �K�i��ym�i 'qsi

vNi � Lsi-qsdi & LsrieJ �ym��i -qrdi & LsriJ eJ �ym��i�ym�i 'qrdi &Rsi 'qsdi & �K�i 'qsdi

where

�K�i��ym�i �L�sri

#bi'y�m�iI�� bi � minfRsi� Rrig �� !!

has been chosen to satisfy the damping injection condition �� and the desiredcurrents are de�ned as

'qedi �

�� 'qsdi

'qrdi

��

��

�Lsri

eJ �ym��i �I� &�diLri�i�

J ��rdi

� �di�i�J�rdi

�� !#

The description of Cil is completed with �� !� and �� ! and the de�nition of diin �� A�� Observer design

As in the proof of Theorem � �$� the control law above is not implementablebecause of the dependence of vNi on -qedi� This dependence translates� via �� !#�into a dependence on 'di� which in its turn implies due to �� knowledge of thejoint acceleration -ym�� To overcome this problem� we must implement an observer�To this end� we follow the derivations of Section # and write uNi � f�i�ye� ym� d &f�i�ye� ym� d 'di� Up to this point� the block diagonal structure is still preserved�Now� looking at �� we see that� due to the presence of the Coriolis matrix�taking Kd diagonal� each component 'di depends on the whole vector 'ym�� Forthe subsequent substitutions we have to introduce� in an obvious manner� the vectornotation uN � f��ye� ym� d & f��ye� ym� d 'd� to �nally obtain� as in Section #� uN �

f��ye� ym� d & fo�ye� ym� d 'qr� with qr�� q�r�� q

�r�� q�rnm�� The actual control law

isu � f��ye� ym� d & fo�ye� ym� dzr

where� exploiting the block diagonal structure of .e� we propose the observer

D�ym 'z &

�d

dt�D�ym &R

�z �Mev � L


with

D�ym�� block diagfDei��ym�ig

R �� block diagfRei &

��K�i��ym�i

�g

Me�� block diagf 1Meg

and the output injection

L�ye� ym� d� '�qs �Mef�� ye� ym� d '�qs

Considering the quadratic function for the whole system

V ��

�z�D�ym�z &

�

'�q�e D�ym '�qe

we get exponential convergence of signals �z and '�qe as in the proof of Theorem � �$�

�

Remark �� Controller structure and removal of observer�� There are twomajor drawbacks of the scheme presented above� First� even though the block diag�onal nature of .e somehow propagates through the controller calculations� in theobserver stage this decentralized structure is unfortunately lost due to the motorcrosscouplings that appear in the evaluation of �� $� Second� the inclusion of theobserver considerably complicates the controller structure� increasing its �dynamicalorder and introducing via fo�ye� ym� d higher order terms� In order to remove theneed of the observer we have recently derived a robot controller independent of jointvelocity � �� More precisely� we can show that

� Dm�ym�-ym�� & Cm�ym�� ym��ym�� &Gm�ym��Kd�� ym��Kp�ym�

with'� � � �� & ��k� � ym�k& ��k�ym�k �� ym� & -ym��

in closed loop with �� ! ensures global uniform asymptotic stability �pro�vided the gains are suitably chosen� Since the observer is now obviated we obtain asimpler decentralized scheme�

Remark �� Global tracking with FOC�� An interesting corollary of Propo�sition � �� is that for currentfed induction machines �where the stator currents yeare now the control inputs the standard indirect �eldoriented control scheme� seeSection �� without any observer� can be combined with the robot controller of � �to yield a global tracking controller�


� Simulation results

To illustrate the performance of the controller in Proposition � �� and compare itwith a backsteppingbased scheme� we have performed in SIMNON simulations ofthe twolink robot with induction motors presented in Subsection X�B of �� $�� Thedynamic model of the robot and induction motors� as well as the initial states� aretaken from that paper� It is worth pointing out that in �� $� a timevarying �uxreference i� ��di in �� $� is needed to avoid singularities in the control calculations�This �somewhat arti�cial provision is not needed for our controller which is globallyde�ned� Consequently� we have used i� � Wb which is the average value �in thetime interval of interest of the reference given in �� $�� Our controller contains onlyfour tuning parameters� The coe�cients bi of the damping injection gains �� !!�the proportional Kp and derivative Kd gains of the robot controller �� and thegain c� �� Following the suggestion of Section �� we set bi � ��Rri� whereRri are the rotor resistances� After some straightforward tuning� we set up

Kp � � I�� Kd � I�

and chose c� � �� which satis�es �� ! � The resulting position tracking errors�motor torques� stator voltages and currents are depicted in Figs� � �!� �� Tofacilitate the comparison� the plots are given in the same scale as the corresponding�gures in �� $��

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

Time [s]

Pos

ition

err

ors

[rad

]

2

1

Figure � �!� Position errors ��ymi� i � ��


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−10

−5

0

5

10

15

20

25

30

Time [s]

Tor

ques

[Nm

]

2

11

Figure � �#� Motor torques i� i � ��

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−40

−20

0

20

40

Vol

tage

s [V

]

1

2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−40

−20

0

20

40

Time [s]

Vol

tage

s [V

]

1

2

Figure � �� Stator voltages u for the two motors��


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−30

−20

−10

0

10

20

30

Cur

rent

s [A

]1

2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−40

−20

0

20

40

Time [s]

Cur

rent

s [A

]

1

2

Figure � �� Stator currents 'qs for the two motors�

The following remarks are in order�

�� The position errors of our scheme converge to zero in approximately �� s� witha negligible overshoot in the �rst error �denoted by � in the �gures� This shouldbe contrasted with Fig� $ of �� $� where the errors converge to a neighborhoodof zero after � s and exhibits afterwards a sustained oscillation� We should recallthat the analysis in �� $� predicts only ultimate boundedness of the errors� Thesimulations show that inside the residual set the behaviour is oscillatory�

� Our simulations proved to be highly insensitive to the parameter c�� However�as thoroughly discussed in Section �� the gain of the nonlinear dampingterm bi is critical for large speeds� Our experimental experience has shown thatthis term ampli�es the noise and induces voltage saturations� This di�cultycan be alleviated by use of integral action in stator currents as explained inSection ��!� This provision can also be incorporated to the theoretical analysispresented here�

!� To compare the computational complexity of the proposed passivitybased con�troller �PBC with the backsteppingbased controller �BBC of �� $� and theobserverless controller OL�PBC of Remark � �� we provide in Table � �� thenumber of operations needed by each algorithm �per joint�


Numerical operationMethod Additions and subtractions Multiplications and divisions Total

PBC ��# $� !��BBC �!� !�� #��

OL�PBC $$ �� !�

Table � �� Comparison of computational requirements�

#� In spite of the slow convergence rate of the position errors in �� $�� there isa highly oscillatory initial transient in torque� voltages and currents in thesimulations of that paper� This seem to stem from the �ux observation transient�The transient is partially alleviated by the particular �slowly increasing choiceof position reference� it gets worse for other more realistic tasks� Our schemedoes not require �ux observers and� as seen from the �gures� the responses arevery smooth�

�� In trying to reproduce the simulations of �� $� we encountered some di�culties�Not only is the number of code lines two to three times larger than for ourcontroller� but a high sensitivity to numerical errors was also experienced� Weattribute these di�culties to the high gains introduced by the higher order termsinherited from the backstepping design� This problem is still present in a laterwork of the authors �� where the control signals contain terms of order seven�Since these terms are of the same nature as our damping injection� we expectsimilar noise problems to appear in a practical setting� This seems to be thecase in the experimental results of Section X�A in �� $��


In this chapter we have studied the problem of output feedback global tracking ofan EL system in feedback interconnection with a general nonlinear system� Thiscon�guration often appears in practical applications� In some instances it is possibleto introduce a control action that removes the feedback path from the interconnection�leading to a cascade system� From a practical viewpoint� this does not seem to beadvisable since cancelation of nonlinearities might induce serious robustness problems�See ��! � for a conclusive example�

For cascaded systems it is by now well known that stabilizability of the driving sys�tem combined with a growth condition on the interconnection insures stabilizabilityof the composition� In our feedback interconnected case� besides the growth conditionon the interconnection� we require a �stronger� form of stabilizability� namely that


it must be uniform in the feedback signal� Relying on these assumptions we haveinvestigated the feasibility of a cascaded control con�guration which is particularlywell suited for EL systems with �block diagonal� kinetic energy functions� For thesesystems� we have a natural decomposition into feedback interconnected passive sub�systems� Furthermore� in the controller design we can easily use energy shaping plusdamping injection ideas to design a PBC� Since we are interested in obtaining outputfeedback controllers we introduced a nonlinear observer� The design of the latter isthe main technical contribution of the chapter�

One of the main motivations of our research was the problem of global tracking ofrobots with AC drives� A solution� based on backstepping principles� may be found in�� $� �see also �� for some important extensions� We believe these results� thoughinteresting from the theoretical viewpoint� are of limited practical interest becauseof their high computational complexity and reliance on higher order terms which actas high gains� The �rst problem is a fundamental issue which� unfortunately� is notfully recognized in the recent literature on nonlinear control� It pertains� not justto the availability of fast and cheap �number crunchers�� but also to the numericalsensitivity of the calculations and its impact on the tuning procedure� see ��! �� Thedeleterious e�ects of injecting high gains into the control loop are well documented�

A fundamental building block for the cascade systems stabilization theory is theproof of � � � that global asymptotic stabilizability implies global input�to�state sta�bilizability with respect to input disturbances� Unfortunately� as shown in �� sucha result does not hold for output disturbances� which makes the problem of stabi�lization of feedback interconnected systems more challenging� We hope the materialpresented in this chapter will motivate the researchers to look into this practicallyimportant problem�

Chapter ��

Other applications and current

research

The objective of this �nal chapter is twofold� First� we point out to other applicationsof PBC that went beyond the scope of this book� Second� we collect speci�c problemsin PBC of EL systems on which we are currently working� They have not yet beenfully resolved� and thus put forth further avenues of study�

The application of passivity �or the more general concept of dissipativity for con�trol of large �exible structures has a very long history dating at least as far back asthe work of Opdenacker and Jonckheere in �� It is well known that for thisclass of systems� in the absence of gyroscopic e�ects� the structure transfer matrixis symmetric� Furthermore� the use of collocated sensor and actuators makes thestructure dissipative� The practical signi�cance of this feature is that� if placed infeedback with a passive controller� the closed loop will be robustly stable to dissipa�tive unmodeled system modes� This property is particularly important in mechanicalvibration problems since� in spite of their nice structural properties� their accuraterepresentation requires many �ideally in�nitely many eigenmodes� Two new prob�lems arise at this point� a convenient model reduction technique that preserves thedissipation structure of the system� Then� a systematic design methodology that en�sures the resulting controller is passive� In � �� the �rst problem is addressed viaLQG balancing techniques� while in �� the authors determine conditions ensuringthat an LTI system with an LQG compensator de�nes a dissipative loop� See also therecent book �� for a review of some more recent developments in the linear case�and some extensions to treat nonlinear multibody �exible space structures�

#�$

�� Ch� �!� Other applications and current research

� Other applications

In the applications of PBC to robot manipulators described in this book we restrictedourselves to unconstrained robots� that is without interaction with their environment�It can easily be shown that the passivity property of the robot is preserved even inthe later case� It has therefore been possible to extend many of the results reportedhere for practically important case of force and impedance control� See �� foran exhaustive list of references� Another robotics area where passivity has played aprominent role is in teleoperation� where the fundamental work �� opened the roadfor many additional extensions�

In the applications above� it is the ability of passivity to handle in�nite dimensionalsystems� that is suitably exploited� It is quite natural then that the concept hasalso found its way on vibration control of civil engineering structures with activedampers� see � !#� for a recent interesting survey� In � !#� a very original applicationof structural controller with spacewise distributed actuators is also reported�

Some applications to chemical engineering processes of PBC have been reportedin � #�� where the theory presented here for systems in EL form� is applied forsystems in extended Hamiltonian form ��!� A new line of research has been openedin � � � where an elegant mathematical framework has been developed for PBC ofthermodynamical systems�

The Department of Engineering Cybernetics of the Norwegian University of Sci�ence and Technology has also been very actively involved in applications of PBC� Inparticular the research on marine vessels is remarkable� see for instance �� whichcontains some full scale experimental results of a passivity�based observer which suc�cessfully �lters out the noise from the position measurements while attenuating per�turbations due to environmental disturbances� We invite the reader to see also ��where di�erent Lagrangian models of marine vessels are included�

We should also mention the work carried out on controlled Lagrangians at Prince�ton University and CalTech by Nahomi Leonhard and Jerrold Marsden� respec�tively� Applications of this theory to underwater vehicles may be found in theURL http��www�cds�caltech�edu��marsden�� This work� although much moretheoreticallyoriented� is intimately related to PBC as developed in this book�

The realm of application of PBC is clearly rich and diverse� and it is by no meanslimited to EL systems� The list of references above is� of course� not exhaustive� beingnecessarily constrained to the authors� knowledge� background and preferences�

� Current research ��

� Current research

In our research on PBC we currently address two main topics� the extension to otherengineering areas of application� and the assessment of the performance of PBC� Inthe �rst research area we have given particular attention to power electronics andpower systems applications� which we brie�y describe below� To tackle the problemof power systems stabilization we were confronted with the question of passivation ofsystems with constant forcing inputs� The state of the art of this more fundamentalresearch is also discussed below� The routes we have taken to study performanceissues are described at the end of the chapter�

� Power electronics

The research on power electronics is a natural continuation of the work reported herefor DCtoDC converters to tackle other types of devices� Motivated by some recentwork of Stankovi�c and coworkers � �� we have been looking at the applicationof PBC to the practically important series�resonant DCtoDC converters� Thedynamics of this system� although described by a simple third order model with wellde�ned static nonlinearities� exhibits a very rich behaviour that cannot be globallycharacterized� To get a better understanding of the system structure we have recentlycarriedout some research on existence and attractivity of periodic orbits in �� Inthis work we also provide estimation of the signals amplitude as we move away fromthe resonant case� One of the main stumbling blocks for the development of PBCs �orfor that matter any other kind of modelbased controllers for seriesresonant DCtoDC converters is the mathematical di�culty of dealing with strong discontinuities inthe systems model� To overcome this problem� � �� uses a socalled phasor dynamicmodel� which consists of a set of smooth di�erential equations that describe thedynamic behaviour of the Fourier coe�cients of the �rst harmonic approximation�We have been recently studying the utilization of this model for PBC� This has givenrise to the question of evaluation of the domain of validity of the model� as well assome issues concerning the �inversion� of the averaging process �used to derive themodel to obtain an explicit realization of the PBC�

Another power electronic application that we have studied recently is the ACtoDC converter� and more speci�cally the threephase voltage sourced reversiblerecti�ers� The outcome of all our research on power electronics is reported in the PhD thesis �$��

Our PBC methodology has been applied with some success in ��!� to the problemof nonlinear adaptive control of static condensers� These devices� which belong to theclass of �exible AC transmission systems� has gained wide popularity in the powersystems community for voltage stabilization and control of reactive power�


� Power systems

In the area of power systems we are currently interested in the problem of suppres�sion of low frequency oscillations� These oscillations appear in strongly interconnectednetworks because of changes in load and topology� and they may cause loss of synchro�nism and generator tripping� This problem is usually studied in the power systemsliterature adopting a sinusoidal quasisteady state approximation that ignores thedynamics of the network circuit elements� Further� the interaction between the gen�erator and the network is characterized via terminal voltage phasor and complexpower pairs� This sinusoidal characterization may prove inadequate in some studiesbasically due to the proliferation of switching controls in modern power systems�

A unifying framework to study this broad problem from the perspective of pas�sivity has recently been advanced in � �$� using linear models� As discussed abovethere is a clear motivation to try to extend this study to the nonlinear case� As a�rst step towards this end� we concentrate our attention in � �!� on the design ofpower systems stabilizers with the generator exciter being the actuation point� Thegenerator to be controlled is described by a standard EL model� with three forcingterms� The mechanical torque coming from the turbine� the terminal voltage of thenetwork and the �eld voltage� which is our control variable� In view of the signi��cant di�erences between the mechanical and the electrical time scales� the �rst signalcan be treated as a constant disturbance� The terminal voltage may be viewed asthe output of an operator� de�ned by the remaining part of the network� whichis in feedback interconnection with the generator� Our basic assumption is that thenetwork is always absorbing energy from the generator� whence the interconnectionsubsystem �as viewed from the generator is passive� The control objective is thento close a loop around the �eld voltage so as to passivate the generator system� Asa preliminary result� we characterize� in terms of a simple linear matrix inequality� aclass of linear statefeedback controllers which achieve this objective�

Related research was recently reported in �� where a PBC� in the form of LgVor Jurdjevi�cQuinn control � !$�� was utilized to enlarge the domain of attraction forthe particular case of a singlegenerator with in�nite bus connection�

�� Generation of storage functions for forced EL systems

The passivation objective described above is done with respect to the standard errordynamics storage function� That is� we take as usual Hd � �q�D�q�q� with �q someerror signal� Although in some other applications� like the ones reported in this book�this choice proved suitable�� in the power systems stabilization problem the �energyshaping ability� of the controller is seriously crippled with the use of this storage

�Although� frankly speaking� di cult to rationalize from any physical viewpoint� this choiceturned out to be mathematically convenient�

�� Performance ��

function� This stems from the fact that the main instability source in the problemis the existence of an external disturbance� the mechanical load� which althoughconstant� shifts the equilibrium away from the origin� This problem triggered in��$$� our interest towards the closely related fundamental question of generation ofLyapunov functions for extended Hamiltonian systems �of the form ��! with externaldisturbances�

Motivated by the energy balance equation of passive systems

'H � 1u�y � x�Rxwhere H�R� 1u� x are the total energy of the system� the dissipation matrix� the con�stant external disturbance� and the systems state� respectively� it seemed natural topostulate

H�x� 1u

Z t

�

y�d

as a candidate Lyapunov function� Notice that the term 1u�R t

�y�d is precisely

the energy externally supplied to the system� Hence the new function is exactly thedi�erence between the energy of the system and the supplied energy�

To check whether this function can be used as a Lyapunov function� the �rst basicquestion is� of course� if we can write 1u�

R t

�y�d as a function of the state x�t�

Unfortunately� this is not always possible� However� we identify some cases for whichwe can construct a suitable new �output�� Interestingly enough� for linear systems theresulting Lyapunov function is the incremental energy� thus our derivations provide aphysical explanation of it� An easily veri�able necessary and su�cient condition forthe applicability of the technique in the general nonlinear case is also given�

This new storage function has recently been utilized as a basis for the generationof a new family of PBC for DCtoDC converters in �$��

�� Performance

As we have shown throughout this book� the issue of stabilization of a large classof EL systems is essentially settled� The next natural step is to assess their perfor�mance� This is a fundamental question for which an answer is needed to convince thepractitioners of the advantages of our theoretical developments�

The �rst performance indicator to be considered is disturbance attenuation� �theAchilles�s heel of highgain designs� If we consider the class of disturbances with �niteenergy� this translates into the evaluation of the L��induced norm of the closed�loopoperator mapping the disturbances to the output of interest� Clearly� this indicatormay be very conservative� particularly for nonlinear systems� where the possibility offrequency�mixing� stymies the utilization of LTI �lters to discriminate the more viabledisturbances� In spite of this potential limitation� we carried out in � !!� a comparison�


from a disturbance attenuation perspective� of PBC and feedback linearizing control ofa boost DC�to�DC converter and a rigid robot manipulator� For the former we provedthat for both controllers there exists a lower bound to the achievable attenuation levelwhich is independent of the design parameters� Furthermore� for the PBC we obtainedan upper bound for the disturbance attenuation� which is ensured provided we sacri�cethe convergence rate� It came as a rather nice surprise that this study gave us someclues on the basic tradeo� between robust stability and convergence for PBC� For thecase of rigid robots we showed that both approaches yield arbitrarily good disturbanceattenuation without compromising the convergence rate� See Proposition #�#�

Performance analysis has typically been recasted in the literature in terms orrobustness of stability with respect to unmodeled e�ects�� PBC are indeed robustlystable� because of their inverse optimality properties and the �strong� form of stabilitythat we can typically establish for these designs� e�g�� GAS for mechanical systems�and even exponential stability for electrical machines� But this does not mean thatthey will give good performance� for instance fast and smooth transient responses�We have seen already in Sections ��# and �� that the inability to add damping insome underactuated systems puts a hard bound on the achievable bandwidth� Also�the convergence rate of the PBCs developed for electrical machines is limited by thetime constant of the unactuated electrical subsystem��

In the authors� opinion the main motivation to consider nonlinear models of phys�ical systems �at least the class considered in this book is that they faithfully capturethe actual behaviour of the system� The formulation of performance evaluation interms of unmodeled e�ects seems then quite contrived� This is particularly distressingwhen the class of unmodeled e�ects is reduced to linear dynamics� This brings usback again into the role of modeling in control systems design� To elaborate on thispoint let us consider the problem of friction in mechanical systems studied in Chapter�� Given the dissipative nature of this phenomenon� it can certainly be claimed thatPBC will be robustly stable with respect to friction� But friction� if it is not explicitlytaken into account in the controller design� will make the system behave below par�Therefore� the question of performance for this problem has to be addressed througha careful study of models that capture the frictions e�ect� and not by �robustifying�the design�

We close this chapter with some scattered thoughts on the question of perfor�mance�

� Adaptation� There is no need to elaborate on the interest of adaptation since�after optimal control� this is the most performanceoriented technique in controltheory� It is also unquestionably needed in many applications� For instance�

�We underscore the word �stability� because some far fetched claims have been made regardingrobust performance� particularly in the H� control literature�

�It is fair to say that� as pointed out in Section �� the same bound applies� this time to theobserver convergence rate� for feedback linearizing schemes�

�� Performance ��

PBC of robot manipulators� which is certainly robustly stable to uncertaintyin the payload� because the variations cannot destroy the physical property ofpositivity of the inertia matrix� or the existence of minima in the potentialwell� However� adding adaptation features to PBC we can actually identify theparameters of the inertia matrix and relocate the minimum of the potentialenergy at their desired value�

� Analysis tools� It is clear that the tools we use for stability analysis are notsuited to evaluate performance� let alone design highly�performant controllers�Lyapunov analysis gives us� at best� estimates on convergence rates� Inputoutput analysis provides us additionally with estimates on the gains of theoperators mapping the various signals� These estimates are very hard to obtain�and are usually very conservative�

� Gain scheduling via �atness� It has been mentioned already that �atnesscharacterizes a class of systems for which trajectory planning is trivial� Manyphysical systems are �at for that matter� all the ones we considered in thisbook with� furthermore� a physically meaningful �at output� Now� in PBCsome controller parameters depend on the reference signals� For instance� thegain � of the PBC for �exible joint robots studied in Section �� #�B�� dependson the link reference qp�� which turns out to be a �at output for that system�In its current form the virtual robot simply places itself at the �nal positionand then pulls up the robot� hence it is not surprising that the performance isnot very satisfactory as witnessed by the simulations presented there� A moresensible approach would be to plan a smooth trajectory for the virtual robot�this translates into the de�nition of a timevarying gain � which depends onthe �at output� Current investigations are under way in this direction�

� Performance as optimal energy transfer� It is useful to view control asa mean to regulate the energy transfer in a physical system� For instance� wehave shown in Section �� that the action of the switch in the boost converteris simply to modify the dissipation structure to permit the transfer of magneticenergy in the inductance to electric energy in the capacitor� The same analogycan be made for mechanical systems� where PBC adds dashpots and springsto enforce a certain pattern of energy transfer� Adopting this perspective� itis reasonable to expect that a sensible �performance theory� should be closelylinked to energy considerations� This point of view is adopted in the interestingpaper �� where it is claimed that the workless forces� which are disregardedin standard PBC� can be actually �shaped�� We should point out that theadditional terms needed towards this end are similar to the ones introduced inbackstepping design� This has been illustrated in the example of the levitatedball in Chapter �

�� Ch� � Other applications and current research

We would like to �nish this book with a wise warning from Michael Faraday� whoseremarkable scienti�c and personal life sets forth an unvaluable example for the newgenerations�

�By adherence to a favorite theory� many errors have at timesbeen introduced into general science which have required muchlabor for their removal�� To guard against this requires a largeproportion of mental humility� submission and independence��

M� Faraday�

Appendix A

Dissipativity and passivity

Dissipativity is a fundamental property of physical systems closely related with therather intuitive phenomena of loss or dissipation of energy� It was introduced byWillems as a generalization of the wellknow property of passivity in the seminalpaper � �� Typical examples of dissipative systems are electrical circuits� in whichpart of the electrical and magnetic energy is dissipated as heat in the resistors� Asimilar role is played by friction in mechanical systems� To mathematically de�nethe property of dissipativity we must introduce two functions� the supply rate� thatis the rate at which energy �ows into the system� and the storage function� whichmeasures the amount of energy that is stored inside the system� These functions arerelated via the dissipation inequality� which states that along the time trajectoriesof a dissipative system the supply rate is not less than its increase in storage� Thisexpresses the fact that a dissipative system cannot store more energy than is suppliedto it from the outside� with the di�erence being the dissipated energy�

In its more general formulation the notion of dissipativity does not require thede�nition of inputs and outputs of the dynamical system� For the purposes of thisbook we �nd� however� useful to make this distinction� and call them u � IRm andy � IRm� respectively� We will further restrict ourselves to a particular class ofdissipative systems� namely passive systems� for which the supply rate function issimply u�y�

In the remaining of the appendix we recall some basic properties and classicalresults of passive systems� Most of the technical lemmas� theorems and propositionsare borrowed from �� $ �� For another recent text covering more modern conceptssuch as feedback passivity see � !$�� Before proceeding with the precise mathematicalde�nitions let us illustrate the basic concept with a simple example�

#$�

�� A� Dissipativity and passivity

� Circuit example

+

(a) (b)

RLC

-

v�t i�t

L

R

i�t

C

v�t

Figure A�� RLC network

Consider the LTI RLCcircuit of Fig� A�� The dynamic behaviour of the circuit canbe easily obtained applying Kircho��s law as

v � Ri &�

C

Z t

�

i�d & Ldi

dt

Now� multiplying by i we get

iv � Ri� &�

Ci

Z t

�

i�d & Lidi

dt

or equivalently

d

dt

!BB" �

C�

Z t

�

i�d�� z V

&L

i��zT

#CCA � vi� Ri� �A��

where we used the functions V and T to denote the electric energy stored in thecapacitor and the magnetic energy of the inductance� respectively� Integrating �A��from to t we get the energy balance equation

H�t��zavailable

� H� � �z initial

&

Z t

�

v�i�d� �z supplied

�Z t

�

Ri��d� �z dissipated

A� � L� and L�e spaces ��

where we have introduced H �� V&T the total energy of the circuit� In this example

the supply rate function vi is the power delivered from the external source to thenetwork� while the storage function H is the total energy of the system� Accordingto the de�nition above the RLC circuit is clearly dissipative�

If in the RLCcircuit above we adopt the convention of viewing v as input and ias output although there is no particular preference for either one of them we havethat the system . � u �� y is passive�

� L and L e spaces

We consider the set 6 of all measurable realvalued ndimensional functions of timef�t � IR� � IRn� We de�ne the set

L�� fx � 6 j kfk��

�

Z

�

kf�tk�dt ��g

with k � k the standard Euclidean norm� This set forms a normed vector space overthe �eld of real numbers with norm k � k�� We introduce now the extended space L�e

as

L�e�� fx � 6 j kfk��T �

�

Z T

�

kf�tk�dt �� TgClearly L� � L�e� because the extended space contains signals whose L� norm maygrow to in�nity� but only at in�nity�

We also de�ne the inner product and the truncated inner product of two functionsu and y as

hu j yi ��

Z

�

u�t�y�tdt

hu j yiT ��

Z T

�

u�t�y�tdt

� Passivity and nite�gain stability

Even though the concepts of passivity and inputoutput stability are developed inde�pendently of the de�nition of the system state� see e�g� �� for the sake of simplicitywe will restrict ourselves here to systems of the form

. �

�'x � f�x� u� x� � x� � IRn

y � h�x� u�A�

with state x � IRn� input u � IRm and output y � IRm� In this way� �A� de�nes acausal dynamic operator . � L�e � L�e � u �� y�


We have the following de�nitions�

De�nition A�� Dissipativity�� . is dissipative with respect to the supply w�u� y �IRm IRm � IR if and only if there exists a storage function H � IRn � IR�� suchthat

H�x�T H�x� &

Z T

�

w�u�t� y�tdt �A�!

for all u� all T � and all x� � IRn�

De�nition A�� Passivity�� . is passive if it is dissipative with supply rate w�u� y �u�y� It is input strictly passive �ISP if it is dissipative with supply rate w�u� y �u�y � �i kuk�� where �i � � Finally� . is output strictly passive �OSP if it isdissipative with supply rate w�u� y � u�y � �o kyk�� where �o � �

De�nition A�� L� stability�� . is said to be L� stable� if there exists a positiveconstant � such that for every initial condition x�� there exists a �nite constant �x�such that

kyk�T � kuk�T & �x��

The following corollary is obvious from the de�nitions�

Corollary A�� L��stability and dissipativity�� A state space system . is L��stable if it is dissipative with supply rate w�u� y � �

��kuk� � kyk�� for some � � �

The proposition below follows immediately from the de�nitions and an argumentof completion of the squares�

Proposition A�� OSP � L��stability�� If . � u �� y is OSP then it is L��stable��

Proof� The proof follows straight forward observing that OSP implies the existenceof �o � and � IR such that

�o kyk��T hu j yiT �

Hence

�o kyk��T hu j yiT � &�

�p�ou�

p�oy

��T

also holds� and therefore�o kyk��T

�

�okuk��T � �

�

�This type of stability is sometimes referred as strong �or stability with �nite gain� to distinguishit from the strictly weaker property of " � L� � L��

A� � Feedback systems ��

Feedback systems

We present in this section some well known results about the feedback system depictedin Fig� A� � Each of the subsystems .i� i � �� is a statespace system of the form�A� � We further assume that the interconnection is wellposed� that is the operator

u �� y� with u�� u�� u� and y

�� y�� y�� is causal and maps L�e signals into L�e

signals�

u�

y�

e�.�

.�e�

y�

u�&

&

&

�

Figure A� � Feedback interconnection of passive systems

We start with the basic property of invariance� under feedback interconnection� ofpassivity�

Proposition A�� Invariance of passivity�� Consider the input�output system de�picted in Fig� A�� If .� and .� are both passive then . � u �� y is also passive� Iffurthermore they are OSP then . � u �� y is also OSP� �

The main theorem concerning stability of the feedback system is given below�

Theorem A�� Passivity theorem� Consider the input�output system depicted inFig� A�� Suppose there exists constants �i�� o�� i�� o�� such that

he� j y�iT � �i�ke�k��T & �o�ky�k��T & �

andhe� j y�iT � �i�ke�k��T & �o�ky�k��T & �

for all e�� e� � L�e and all T � � Then� . � u �� y is L��stable provided

�i� & �o� � � �o� & �i� �

�

Several interesting criteria can be obtained as special cases of this theorem� Forinstance� that input �or output strict passivity of the subsystems ensures L�stability�Notice though that the theorem does not require that both operators be passive� sincethe excess of passivity of one of them can compensate for the lack of it on the otherone�


Internal stability and passivity

It is rather clear that input�output stable systems are also internally stable� i�e� stablein the sense of Lyapunov� if some observability properties are satis�ed �#�� !�� Toformalize this relationship we need the following de�nition�

De�nition A� �Zero�state observability and detectability� A state�space sys�tem 'x � f�x� x � IRn is zero�state observable from the output y � h�x� if for allinitial conditions x� � IRn we have �y�t � � x�t � � It is zero�state detectableif �y�t � � limt� x�t � �

To simplify the presentation we will specialize in this section to systems a�ne inu� that is

.a �

�'x � f�x & g�xu� x� � x� � IRn

y � h�x

with g�x and nm matrix�

The main result of this section is due to Hill and Moylan �� !�� We present herea version given in � $ ��

Proposition A� �Proposition �� of �� Suppose the system .a is OSP withpositive semide�nite storage function H � �

�i If .a is zero�state observable then H�x � for all x ��

�ii If H�x � for all x �� H� � and .a is zero�state detectable� thenx � is a locally asymptotically stable equilibrium of 'x � f�x� Furthermore� if His radially unbounded� the stability is global� �

For the feedback system of Fig� A� we have the following statespace version ofthe passivity theorem�

Proposition A�� Proposition �� of ��

�i Suppose .� and .� are passive with storage functions which have strict localminima in x� � 1x� and x� � 1x�� respectively� Then �1x�� 1x� is a stable equilib�rium of the feedback system� without external inputs� i�e�� u� � u� � �

�ii Suppose that .� and .� are output strictly passive and zero�state detectable�and the corresponding storage functions are proper� have a global and uniqueminimum in x� � � respectively x� � � Then � � is a globally asymptoticallystable equilibrium of the feedback system without external inputs�

�

A� � The Kalman�Yakubovich�Popov lemma ��

� The Kalman�Yakubovich�Popov lemma

Lemma A�� The Kalman�Yakubovich�Popov lemma�� Consider a stable LTIsystem with minimal state space representation

'x � Ax &Bu� x� � x�

y � Cx

where x � IRn and u� y � IRm� and the corresponding transfer matrix H�s � C�sI �A��B�

The following statements are equivalent

�i� H�s is positive real� that is� all poles are on the open left�hand plane and thoseon the j axis are simple with Hermitian positive de�nite residues� and � �� which is not a pole of H�s we have

H�j &H��j � � � � IR�

�ii� There exists matrices P � P� � and Q � Q� � such that

A�P & PA � �QPB � C� �A�#

�iii� The operator H � u �� y is passive with storage function V �x � ��x�Px�

Also� the statements below are equivalent�

�i�� H�s is strictly positive real� meaning that H�s � � is positive real for some� � � that is� all poles are on the closed left�hand plane

H�j &H��j � � � � IR�

andlim��

��H�j &H��j � � �

�ii�� There exists matrices P � P� � and Q � Q� � such that �A�� holds�

�iii�� The operator H � u �� y is passive and furthermore� for all t � � the followingidentity holds

� ujy �t��

kx�Qxk��t & V �x�t� V �x�


�

A proof of this lemma can be found in �� $� and the references therein� We closethis appendix with the following basic result�

Lemma A�� Let y � G�pu� where G�p is an n m strictly proper�exponentially stable transfer function and p � d

dt� Then u � Ln

� implies that y �Ln� �Ln

� 'y � Ln� � y�t is continuous� and y�t� as t�� If in addition� u�t�

as t�� then 'y�t� � �

Appendix B

Derivation of the Euler�Lagrange

equations

In this appendix the EL equations of motion are derived from an integral princi�ple� This fundamental equations are used throughout the text in this book for theformulation of the dynamic equations of motion of the physical systems we deal with�

The �rst section of the appendix covers the �Lagrangian formulation�� while thelast one gives some connections with the �Hamiltonian formulation��

caveat A complete derivation of the EL equations of motion� in the generalityneeded in this book �i�e�� covering mechanical� electrical and electromechanical sys�tems� would take us to far away from our main objective of control design� Thematerial is� therefore� not meant to be selfcontained� and is presented in a form thathighlights the main ideas� rather than mathematical rigor� Detailed derivations ofthe EL equations may be found in ��$�� and ��$��

� Generalized coordinates and velocities

The con�guration of a physical system is generally described by a set of quantitiescalled coordinates� For a single mass particle in space� the coordinates needed todescribe the con�guration could be a three dimensional vector of quantities describingthe position of the particle relative to some reference point in a coordinate system�e�g� x�� y�� and z�coordinates in a Cartesian coordinate system�

From a dynamic point of view� a physical system can be considered as consistingof several particles� with interconnections between particles� giving constraints onthe behavior of the system� and relations between the coordinates which would beindependent without the interconnections�

#�!

�� B� Derivation of the Euler�Lagrange equations

Since a physical system is often an isolated part of a much larger system� thesurroundings will also impose constraints on the behavior� In a more general setting�a physical system can be considered as consisting of a set of subsystems� with inter�connections giving internal constraints and possibly dependence between coordinates�and additional constraints given by the environment�

The choice of coordinates for a physical system with many interconnected subsys�tems is often somewhat arbitrary� but in general a subset of the total set of coordinatescan be associated with each of the subsystems�

For systems in static equilibrium� the con�guration coordinates describe the sys�tem completely� but if the system is dynamic� an extra set of dynamic variables� whichgives information about how the con�guration of the system is changing� is neededto describe the system� The �rst derivatives of the coordinates �the velocities canbe chosen as this extra set of dynamic variables� or another set of dynamic variables�called the momenta� can alternatively be chosen� The momenta and the velocitiesare said to be associated variables��

When considering a system as a set of interconnected subsystems� it may bepossible that the constraints of the system allows for a reduction of the numberof variables used for describing the system� because the constraints give relationsbetween the various variables which are not independent� This fact motivates us todistinguish two di�erent types of constraints� holonomic and non�holonomic�

Holonomic constraints are expressed as relations between coordinates� or relationson di�erential form �relations between velocities which can be integrated to yieldrelations between coordinates� Holonomic constraints are expressible in the form

fj�q�� qn� t � � j � �� m �B��

where qi� i � �� n are the coordinates of the system�

The m relations above can then be used to reduce the number of coordinates to�n�m�

For a system which has only holonomic constraints� it is possible to select a setof independent coordinates such that the constraint equations are no longer needed�This means that if there are n coordinates and m holonomic constraints� a set of�n � m independent coordinates can be derived� These coordinates together withtheir associated dynamic variables �velocities or momenta� constitute a set of �n�mvariables� which describe the dynamic motion of the system uniquely� The minimumnumber of N � �n�m that can be found is the degrees of freedom of the system� andwhen a system is described by a set of coordinates which eliminate the constraints�these coordinates are generally called the generalized coordinates of the system� withassociated generalized dynamic variables�

�The coordinates could also have a set of associated variables� called forces�

Generalized coordinates and velocities ��

Constraints which can not be the written on the form �B�� as relations betweenthe coordinates� are called non�holonomic� An example is a constraint of the form�di�erential equation

f� 'q�� 'qn� t �

which can not be integrated to the form �B�� This is the case for electric machineswith commutators�

Another type of non�holonomic constraints can be written on the form

f�q�� qn� t

which can be used to describe the motion of gas particles within a container� wherethe walls limit the motion of the particles�

For physical systems with non�holonomic constraints� it is not possible to �nd aset of N generalized coordinates� with N the number of degrees of the system� Thenumber of coordinates must be equal to the degrees of freedom plus the number ofnon�holonomic constraints� This means that there will coordinates which are notindependent� In this text� the derivation of the EL equation will be based on theuse of generalized coordinates� an it will not apply to non�holonomic systems� Thereare other methods for the derivation of the dynamic equations of motion for non�holonomic systems� but this topic will not be considered here� and the interestedreader should consult ��

Another case that is sometimes encountered in the modeling of physical systems�is the use of quasi coordinates� It is possible to have many di�erent sets of truecoordinates for instance q�� qN and q�� q

�N � but with this type of coordinates�

it is always possible to �nd linear relations between the di�erent sets� expressed as

qi �NXj �

aijq�j� i � �� N

If it is not possible to �nd such relations between the di�erent sets of coordinates�but instead relations of the form

'qi �NXj �

bij 'q�j� i � �� N

with� �bij �� q�i��qj or ��bij��qk �� bik��qj can be found� then the coordinatesq�� q

�N are called quasi coordinates� An example of this is the relation between

rotor currents and terminal currents in a commutator machine� or currents in an

�The given relations between the coe cients ensure that the di�erential equations are not inte�grable�


AC machine� which has been rotated from their natural frame� where they can beintegrated to physical charges� to another frame of reference� In both these cases� inte�gration of the currents will result in quasi coordinates without any physical meaning�

The main problem with quasi coordinates in the modeling of electromechanicalsystems� is that sometimes the coordinates are not explicitly used in the derivation ofthe model� only the velocities� Thus� a set of velocities which are not the true velocitiescan be used in the modeling procedure by mistake� but this is never discovered� sincethere is no need for evaluating the coordinates�

In the following� the use of quasi coordinates in the derivation of the EL equa�tions will not be considered� In the systems considered in this book� the generalizedvelocities will be chosen based on physical interpretation� and they can be integratedto give physically interpretable true coordinates�

To summarize the above� the dynamic motion of a physical holonomic systemwith N degrees of freedom can be completely described by a set of generalized in�dependent coordinates q��t� � � � � qN�t describing the con�guration of the system asa function of time� and a set of N dynamic variables� given either as generalizedvelocities 'q��t� � � � � 'qN �t� or generalized momenta p��t� � � � � pN�t� Thus� the state�

of a dynamic system can be presented in a N dimensional space� The coordinatesq��t� � � � � qN�t evolve in the conguration space of the system� and the space formedof the coordinates q��t� � � � � qN �t� p��t� � � � � pN�t is called the N �dimensional phasespace of the system�

Once a choice of independent generalized coordinates has been made a state func�tion� that characterizes the system� must be de�ned� Although other state functionscan be chosen� �e�g�� the Hamiltonian� we will select here the Lagrangian� that wedenote L�q� 'q� t� This choice is motivated by several reasons� �rst� the fact that theresulting equations of motion for the electrical portion of our systems will be identicalto those obtained from Kirchho��s laws� which is an appealing features for electricalengineers� Second� as we have seen in Chapter �� the EL formalism is more suitableto reveal the workless forces� a fundamental step in our PBC synthesis�

We should mention that the problem of selecting the proper set of independentvariables in a dynamical system always presents some di�culty� For instance� theLagrangian used in classical mechanics is de�ned as the di�erence between the kineticand the potential energy� that is

L�q� 'q� t � T �q� 'q� t� V�q� t

It turns out that to treat electromechanical systems this de�nition is not su�cientlygeneral� To handle these cases we should use instead the kinetic co�energy� Althoughthese functions coincide for linear systems� they will di�er in general�

�The state of a static system can be completely described by its N generalized coordinates� sincethere are no dynamic changes�

� Hamilton�s principle ��

� Hamilton�s principle

The path of a dynamic system will evolve in the con�guration space from one con��guration q�t� � �q��t�� qN�t��

� at time t � t�� to a new con�guration q�t� ��q��t�� qN�t��

� at time t � t�� The equations of motion governing this changeof con�guration can be found from di�erent principles�

They can be derived from basic physical relations like force laws� which form aset of di�erential principles� concerning incremental changes in the system�

An alternative way is the use of a variational method� based on integral principles�These principles relates to the gross motion of the system�

Hamilton�s principle is considered to be one of the most important integral princi�ples� It can be derived from d�Alembert�s principle and the principle of virtual work�but it is a more general principle than the former� and proves to be signi�cant formore than just mechanical systems�

Hamilton�s principle states that the actual dynamic path of a system described bya state function L�q� 'q� t� from time t� to t� is such that the line integral

I �

Z t�

t�

L�q� 'q� tdt �B�

is an extremum for this path�

0

'qi�t

'qi�t�

'qi�t�

qi�tqi�t� qi�t�

Pathwhich

makes�I�

Figure B�� The path of motion according to Hamilton�s principle�

� B� Derivation of the Euler�Lagrange equations

Thus� with Hamilton�s principle we have that the derivation of the equations ofmotion has been reduced to an optimization problem� The main mathematical tool tosolve this problem is the calculus of variations� which may be viewed as an extensionof ordinary calculus� which is used to �nd extrema of functions� to the case whenwe are interested in the extrema of functionals� �which are functions of functions� forinstance the integral �B� above�

� From Hamilton�s principle to the EL equations

Using standard techniques of calculus of variations� it can be shown that for thedynamic path to be and extremum� the �rst variation of the line integral I must beequal to zero� that is

�I � �

Z t�

t�

L�q� 'q� tdt �

subject to the end constraints

�q�t� � �q�t� �

where � means a time�independent variation� analogous to the di�erential used inordinary di�erential calculus��

Evaluating this variation� and setting it equal to zero� yields

�I �

Z t�

t�

NXi �

��L�qi

� d

dt

�L� 'qi

��qidt � � i � �� N �B�!

Now� the extremum �I � must hold for all variations �qi of any coordinate qi�Furthermore� since the N coordinates qi are independent� the only way one can satisfy�B�! is by setting the term in brackets equal to zero for all i� yielding

�L�qi

� d

dt

��L� 'qi

�� i � �� N �B�#

Equations �B�# are known as the ��rst di�erential equations of EL� or in short� �ELequations�� This equations yield a set of ordinary di�erential equations that describethe dynamics of any conservative system with independent coordinates�

�For readers unfamiliar with calculus of variations� it is useful to think of the problem of �ndingan extremum as one of �nding the points of zero slope of I � which are obtained by setting somedi�erential of I equal to zero�

�Euler obtained this equation heuristically in �� and Lagrange obtained it �incorrectly� in�� The correct derivation was done by P� du Bois�Reymond in ��

� EL equations for non�conservative systems �

EL equations for non�conservative systems

If there are non�conservative forces acting on the system� the derivation of the ELequations above can sometimes be carried through with a Lagrangian which has beenmodi�ed to take the nonconservative forces into account� We consider in the booktwo types of nonconservative forces� controls u and external disturbances Q� � whichare independent of the generalized coordinates and velocities� and dissipation� Theformer are easily incorporated in a nonconservative potential function

VNC�q� t � ��u&Q��q

while the latter are added in a nonconservative kinetic potential� which is the Rayleighdissipation function F� 'q�

A nonconservative Lagrangian can now be formulated as

LNC�q� 'q� t � T �q� 'q &

Z t

�

F� 'qdt� �V�q & VNC�q� t

and the equations of motion can be derived from Hamilton�s principle with this newLagrangian� giving

d

dt

��LNC

� 'q

�� LNC

�q�

which upon evaluation yields

d

dt

��L� 'q

�� L

�q� u&Q� � �F

� 'q

Throughout the book we will refer to this set of di�erential equations as EL equations�

List of generalized variables

For the systems considered in this book� the generalized coordinates and velocities�with associated forces and momenta� will be chosen as shown in Table B��

� Hamiltonian formulation

In the Hamiltonian formulation we seek to describe the motion in terms of �rst orderdi�erential equations� This formulation is viewed here� for simplicity� as a consequenceof the EL equations and a simple change of variables� However� it is wellknown thata direct Hamiltonian formulation is possible for many systems� and that there are


Generalized variables� Electro�magnetic part Mechanical part

Generalized coordinates� qi electric charges mechanical displacementsGeneralized velocities� 'qi electric currents mechanical velocitiesGeneralized forces� Qi negative electric voltages mechanical forcesGeneralized momenta� pi� �i �ux linkages mechanical momenta

Table B�� De�nition of variables�

even systems which do not easily admit the EL formulation but admit a Hamiltonianone�

We �rst observe that the EL equations �B�� provides the time derivative of �LNC� �q

�leading to a second order di�erential equation in 'q� With the goal of obtaining �rstorder di�erential equations� de�ne the conjugate variables p �called the generalizedconjugate momenta

p��

�LNC�q� 'q� t

� 'q�B��

Assume that this relation allows for a global� unique solution for 'q in terms of theremaining quantities 'q � g�q� p� t� �This is true under some rather weak conditionsof strict convexity and �quadratic boundedness� of LNC with respect to 'q� Let usnow de�ne a useful scalar quantity

G�q� 'q� t �� L�NC�q� 'q� t

� 'q'q � LNC�q� 'q� t

De�ne now the Hamiltonian as

H�q� p� t�� G�q� g�q� p� t� t

This is� of course� the well known Legendre transformation� Notice that� by de�nitionG and H are numerically equal� but are expressed with di�erent arguments�

We now proceed from the second order di�erential equations of �B�� to �rstorder state space form� The development is quite standard� we summarize it forcompleteness and to stress the role of the Rayleigh dissipation function� From thede�nitions of g�G and H and some lengthy� but straightforward derivations� we get

�'q'p

��

� I�I

��H�q�p�t�

�q�H�q�p�t�

�p

��

�

�F�q� �q�t�� q

�

where we choose to maintainF in terms of 'q because it typically relates more naturallyto velocities than to momenta�

� Hamiltonian formulation ��

Before closing this section let us evaluate the rate of change of the Hamiltonianalong solution trajectories� this gives

d

dt�H�q� p� t � � 'q�

�F�q� 'q� t

� 'q� d

dtLNC�q� 'q� t

which is our usual power balance equation�

Appendix C

Background material

The de�nitions below concern the minima of a vector function� Let f � IRn �� IR bea smooth function then we de�ne the following ��$��

De�nition C�� Critical Point A point x� � IRn is called critical point of f�x ifand only if �f

�x�x� �

De�nition C�� Minimum A point x� � IRn is a local minimum of f�x if there isa neighbourhood B of x

� with � � �� such that f�x � f�x� for all x � B �

De�nition C�� Absolute or global minimum A point x� � IRn is an absolute orglobal minimum of f�x if f�x � f�x� for all x � IRn�

De�nition C�� Unique minimum A point x� � IRn is an unique minimum of f�xif there are no other local minima of f�x in IRn�

De�nition C�� Strict minimum A point x� � IRn is a strict local minimum of f�xif there exists a neighbourhood B of x� with � � � such that f�x � f�x� forall x � B �

Theorem C�� Mean value� Assume that f � IRn �� IR is di�erentiable at each pointx of an open set S � IRn� Let x and y be two points of S such that the line segmentL�x� y � S� Then there exists a point z of L�x� y such that

f�y� f�x �

��f

�x�z

��y � x

�

Lemma C�� $�� Let f�x � IRn �� IR and B� � IRn� such that

�i f� �

#�!

�� C� Background material

�ii �f�x� �

�iii ��f�x�

� In� � � � � � for all x � B�

then f�x has a unique strict minimum at the origin� locally in B�� If B� � IRn thenthe minimum is global and unique� �

The lemma C�$ above appears too strong in some cases� Notice that a necessarycondition for �iii is that f�x � O�kxk� in the ball B�� specially when globality isto be assured it is desirable to �nd milder conditions� The lemma below establishesweaker su�cient conditions for a function f�x to have a global minimum at theorigin�

Lemma C� �� Let f�x � IRn � IR be a C� function� Assume�

� f�x � � for all x � IRn� x �� and f� �

��f�x�x

� � for all x �� IRn

Then the function f�x is globally positive de�nite with an unique and global mini�mum at x � � �

Condition � implies that f�x is positive de�nite with � a strict global minimum�Nevertheless� it is important to remark that this condition alone does not imply theuniqueness of the minimum� Condition implies that is the only critical point�hence that is also an unique minimum of f�x�

Appendix D

Proofs

� Proofs for the PI D controller

� Properties of the storage H��q� �q� ��

A Positive de�niteness of H��q� 'q� �

Let us partition H� as H� � W� &W� &W� &W� where

W� ��

�'q�D 'q &

�

��q�K �

p�q & ��q�D 'q� �D��

W� ��

��q�K �

p�q & Ug & �q�K �p�qd � �� &

�

�qd � �� K

�p�qd � �� & c�� D�

W� ��

�'q�D 'q &

�

#��KdB

�� D 'q� �D�!

W� ��

#�q�K �

p�q &�

#��KdB

�� &�

#'q�D 'q� �D�#

Under the conditions of proposition !� �� W� is positive de�nite �� W� is positivede�nite if

�

sk�pmdM

� �

while

rkdmbMdM

� � �D��

insures W� to be positive de�nite� Thus� V is positive de�nite for � su�ciently small�

#��

�� D� Proofs

B Time derivative of H��q� 'q� �

Using the properties P�� P�� and after some straightforward calculations one ob�tains that the time derivative of H� along the trajectories of �!��#� �!�� is boundedby

'H� ��k�qk� � ��k 'qk� � ��k�k� � �

�bmdm#

� dM � kck�k � kck�qk�k 'qk� & z��

��

$� k�qkk�k

��Q�

� k�qkk�k

��

� k�kk 'qk

��Q�

� k�kk 'qk

�)�

�kdmam#bM

� �kdm

�k�k�

�D��

where we de�ned

Q��

�k�pm � kg �k�pM � kdm � kg

�k�pM � kdm � kgkdmam��bM

�� Q�

��

� kdmam��bM

�aMdM�aMdM bmdm

��

and the constants

��

�bmdm

� ��

�k�pm

� ��

kdmam#bM

� �D�$

We derive now su�cient conditions for 'H� to be negative semide�nite in ��q� 'q� �with � � � If

�k�pm � kgkdmam

bM�k�pM & kdM & kg

� � � �D��

we have Q� � � In a similar way� �for all bmbM

�� we have that Q� � if

kdmamdm �amdM ��

� �� D��

The third right hand term of �D�� is negative if�

�

kc

��

#bmdm � dM

�� kxk �D��

where the left hand side is positive due to �!�$�� Finally� the last term in �D�� isnegative if

kdmam#bMkdM

� � �D��

while �D�� D�� and �D�� are satis�ed for � su�ciently small� Therefore� 'H� islocally negative semide�nite with � � �

�Observe that we give this condition in terms of the original state x� instead of x�� This inorder to derive the domain of attraction �and prove the semiglobal stability claim that requires �arbitrarily small� in the coordinates x�

�� Lyapunov stability of the PI�D ��

� Lyapunov stability of the PI�D

A Domain of attraction

To de�ne the domain of attraction we will �rst �nd some positive constants ��

such that

��kxk� V �x ��kxk�� D��

Notice that

V � W� � �

#

�k�pmk�qk� &

kdmbM

k�k��&

�

#

�dmk 'qk� & �

kiMkzk�

��

To obtain the lower bound in terms of x we need the following inequality�� kiM

�

�k�vk� &

�k�im��

� kiM�

�k�qk� kzk�

which leads to

V � ��

nhk�pm &

k�im�kiM

� ik�qk� & dmk 'qk�

o& �

�

nkdmbMk�k� &

h�

kiM� �

ik�vk�

oso we de�ne �� as

��

�

#min

�k�pm &

�k�im�kiM

� �

��kdmbM

� dm�

��

kiM� �

� ��

In a similar manner� an upperbound on V is

V ��

#

�k�pm & kg

&

�

h� dM & k�pM

i�k�qk� &

��&

�

dM

�k 'qk�

&�

��dM &

kdmbM

�k�k� & �

kimkzk��

Now using Kp�� K �

p &��Ki we have

kzk� �� &

kiM�

� �k��k� & kiM

�k�qk�

� �k�qk� & k��k��

so we de�ne

�� max

n��& �

�dM

� ��

h�dM &

kdMbm

i� ��

�k�pm & kg

& �

�

��dM & k�pM

& �

kim

o�

From �D�� and �D�� we conclude that the domain of attraction contains the set

kxk c��

�

kc

��

bmdm � dM

�r��

�� D��!

� D� Proofs

� Proof of positive de niteness of f��qp� de ned in

��

We will establish the proof that

f��qp��

nXi �

�k�i

Z �qpi

�

sat�xidxi

�& Vp��qp & qp�� Vp�qp�� q�p

�Vp�qp

�qp�

is positive de�nite and radially unbounded by verifying the conditions of Lemma C��so as to de�ne a kmin

�ithat ensures this to be the case�

Condition ��To prove that f��qpi � for all �qp �� IRn we shall prove �rst that there exists� � such that

nXi �

k�i

Z �qpi

�

sat�xdx � minifk�ig sat��

�k�qpk� � k�qpk � � �D��#

nXi �

k�i

Z �qpi

�

sat�xdx � minifk�ig sat��

k�qpk � k�qpk � �� D��

Let � be a constant that satis�es inequalities �!� � and �!�! � For the sake ofclarity we consider two cases separately�

Case �� k�qpk � �

Notice that in this case we have that j�qpij � �� i � n� then using P�� and�!� � we get

nXi �

k�i

Z �qpi

�

sat�xdx �nXi �

k�isat��

��q�pi � min

ifk�ig sat��

�k�qpk��

�D��

Case � k�qpk � �

Within this case we shall consider three di�erent cases�

case a� j�qpij � � �i � nFirst notice that

nXi �

k�isat��

��q�pi � min

ifk�ig sat��

�k�qpk� � min

ifk�ig sat��

k�qpk�

Using �!� � and P�� D�� follows�

� Proof of positive de�niteness of f��qp de�ned in ��

case b� j�qpij � � �i � nFrom P�� and �!�! notice that

nXi �

k�i

Z �qpi

�

sat�xdx �nXi �

k�isat��

j�qpij � min

ifk�ig sat��

nXi �

j�qpij��D��$

then �D�� easily follows observing that k�qpk Pn

i � j�qpij�case c� j�qpij � �� j�qpj j � � �i� j � n� i �� j

Without loss of generality we can take i n� and � j � n� � then a simpleanalysis along the lines of cases a and b� shows that �D�� holds as well in thiscase�

Now we prove that� for all � � there exists constants �� IR such that

Vp�qp� Vp�qp�� q�p�Vp�qp��qp

�qp� ��

�k�qpk� � k�qpk � �

�k�qpk � k�qpk � �� D��

On one hand� notice that using Lemma C�$ it follows from � �� that

Vp�qp� Vp�qp�� q�p�Vp�qp�qp

�qp� � �kg k�qpk�� D��

on the other hand� invoking the Mean Value Theorem we have that � � IRn suchthat

Vp�qp�� Vp�qp ��Vp�qp�qp

��

��qp� � qp kvkqp� � qpk�

then using � �� we can write

Vp�qp� Vp�qp�� Vp�qp�qp

�qp��qp � � kvk�qpk� �D�

Since �D�� and �D� hold for all �qp � IRn� then �D�� holds with � � �kg�

and� � �kv� We then conclude from �D��#� �D�� and �D�� that

f��qp �

�%%�%%

�mini fk�ig sat��

�� kg

�

�k�qpk� � k�qpk � �


�� kv

�k�qpk � k�qpk � ��

From here it�s easy to see that condition � is satis�ed provided

minifk�ig � kmin

�i� max

��kgsat��

�#kvsat��

��D� �

�� D� Proofs

holds with � as in P�� kg and kv de�ned by � �� and � �� respectively�

Condition ��Taking the partial derivatives of f��q we get

�f

��qp��qp � K�

��

sat��qp�sat��qp�

��sat��qpn

��&

�Vp��qp

��qp� �Vp�qp

�qp��

Now� taking the norm and using the triangle inequality we get

�f��qp ��qp �

K�

��


��sat��qpn

��

�Vp��qp��qp� �Vp

�qp�qp�

��D�

On one hand� from � �� and using the Mean Value Theorem we have thatfor all � �

��Vp��qp

��qp� �Vp�qp

�qp�

�� kg k�qpk if k�qpk � �� kv if k�qpk � �

and on the other hand since K� is diagonal and using P�� we obtainK�

��


��sat��qpn

��

��

mini fk�ig sat��

k�qpk if k�qpk � �

mini fk�ig sat�� if k�qpk � ��

Thus� we are able to write

�f��qp ��qp �

�%�%


�� kg

�k�qpk if k�qpk � �

�mini fk�ig sat�� kv if k�qpk � �

which happens to hold provided �D� � is satis�ed�

In the case of sat�x � tanh�x we have that P�� is true with �� kv

kg� Substi�

tution of this � in �D� � implies �!�#��

� The BP transformation


For the proof of Proposition �� the following lemma is needed�


Lemma D��

dDe�qm

dqm�� U & U� � �D� !

�

Proof� Note that the di�erential equation ��!� has the unique solution De�qm �e UqmDe� e

�Uqm ��#��

De�qm � D�e �qm � e UqmDe� e

�Uqm � e �U�qmDe� eU�qm

� e U�qm e UqmDe� � De� eU�qm e Uqm

� e U�qm e Uqm � I

� U� & U �

The third implication follows from the fact that U and De�qm do not commute�

unless dDe�qm�dqm

� � see ��!�� and this implies that f�U � e U�qm e Uqm and

De�qm cannot commute� unless f�U � I�

�

Proof of Proposition ��From ��! it follows that

'qe � e UqmP�� 'ze

-qe � e UqmP�� -ze & U e UqmP��

� 'qm 'ze

Inserting these two equations into �� and multiplying from the left by e �Uqm

results in

e �UqmDe�qm eUqmP��

� -ze & e �UqmDe�qmU e UqmP�� 'qm 'ze

& e �UqmW��qm 'qm e UqmP�� 'ze & e �UqmW��qm 'qm

& e �UqmRe eUqmP��

� 'ze � e �UqmMeu �D� #

FromDe�qm � e UqmDe� e�Uqm� and since ��! implies that e UqmRe � Re e

Uqm

��#$�� notice that

e �UqmDe�qm eUqm � De� �D� �

e �UqmDe�qmU e Uqm � De� U

e �UqmW��qm eUqm ��

� e �Uqm �U e UqmDe� e

�Uqm

� e UqmDe� U e �Uqm e Uqm

� UDe� �De� U �D� �

e �UqmRe eUqm � Re �D� $

�� D� Proofs

In addition� it follows from ��!! that W��qm � e UqmW�� This implies thate �UqmW��qm � W�� which is constant with respect to qm� Finally� inserting�D� �D� $ into �D� #� gives

De� P�� -ze & UDe� P

�� 'qm 'ze &W�� 'qm &ReP

�� 'ze � e �UqmMeu

For the transformed mechanical system .m� it follows that

'q�e W��qm 'qe � 'z�e P�� e U�qmW��qm e

UqmP�� 'ze

�D�� 'z�e P

�� e �UqmW��qm e

UqmP�� 'ze

�D�� 'z�e P

�� UDe� �De� U �P��

� 'ze

� 'z�e P�� UDe� P

�� 'ze

� 'z�e P�� De� U

�P�� 'ze

W�� qm 'qe � W�

� �qm eUqmP��

� 'ze

� W�� P��

� 'ze

This completes the proof� �

�� A Lemma on the BP Transformation

Lemma D�� Unless U � � the velocities 'z � � 'z�e � 'qm�� introduced by the BP trans�

formation cannot be derived from a transformation z � Z�q of the generalized coor�dinates q � �q�e � qm�

��

Proof� The transformation from the generalized electrical velocities 'qe and thegeneralized mechanical velocity 'qm to 'z � � 'z�e � 'qm�

� is�'ze'qm

��

�P� e

�Uqm �

� �'qe'qm

�If z � Z�q� then

�Z

�q�

�P� e

�Uqm �

�

since 'z � �Z�q

'q� From this� it can be seen that ze must be of the form

ze � Ze�q � P� e�Uqmqe & c� c � IRne

Taking the total time derivate gives

'ze ��Ze

�q'q � �P� e

�UqmU 'qmqe & P� e�Uqm 'qe

� Proof of Eqs� �� and ��

from which it follows that since the BP transformation �see ��! is de�ned as'ze � P� e

�Uqm 'qe� it must be true that

�P� e�UqmU 'qmqe � � �qe � IRne

For this to hold� U must be the zero matrix� since P� e�Uqm is nonsingular� and

consequently for U �� there is no transformation ze � Ze�q such that 'ze ��Ze�q

'q� �

Proof of Eqs� �� and ��

�� A theorem on positivity of a block matrix

For use in the following proofs� a theorem on positivity of a block matrix is needed�The results is given in terms of the block elements on the diagonal of the matrix andtheir corresponding Schur complements� A proof of this theorem can be found in��#��

Theorem D�� An arbitrarily partitioned Hermitian� matrix of the form

Q �

�Q�� Q��

QH�� Q��

�is positive de�nite �Q � if and only if either�

Q�� Q�� QH

��Q�� Q��

or �Q�� Q�� Q��Q

�� Q

H��

�

For a necessary and su�cient condition on the matrix to be positive semide�nitewhen one of the block matrices Q�� or Q�� is positive de�nite �and hence invertible�the requirement to the Schur complements can be relaxed from greater than zero�to greater than or equal to zero� For necessary and su�cient conditions of positivesemide�niteness in the case where none of the block matrices are invertible� see ��#��

�� Proof of Eq� ��

Proof� It must be shown that

M �

� R&K� 'qd ��S�qm� 'qd

��S��qm� 'qd Re

�� I� �

�Superscript H is used to denote the conjugate transpose of a complex matrix�

�� D� Proofs

with R � diagfRe� Rmg� K� 'qd � diagfK�� 'qmdI�� K�� 'qdg and

S�qm� 'qd �

�� npLsrJ eJ npqm 'qmd

�� npLsr 'q

�sdJ eJ npqm

�� IR��

For the use of the theorem in Section #�� it must be checked if there exists a � � such that M� �I� � with the given de�nition of K� 'qd�

Since Re � minfRs� RrgI�� under the assumption that � � � minfRs� Rrg�which ensures invertibility of Re � �I�� the theorem in the previous section can beused� and it must only be checked if there exists a � within these limits such that

R &K � �

#S fRe � �I�g�� S� � �I� �

Writing out this expression� it follows that��fRs �K�� qmd�� gI� ��

�� fRr � g I� �� Rm �K�� qd��

��

��

�

��

�� npLsrJ eJnpqm qmd

�� npLsr q

�sdJ e

Jnpqm

��

�Rs� I� ��

Rr� I�

�

�

��

�npLsrJ e�Jnpqm qmd �� npLsrJ e

�Jnpqm qsd

�

�

��fRs �K�� qmd�� g I� ��

�� fRr � g I� �� Rm �K�� qd��

��

��

�

��

��npLsrRr� J e

Jnpqm qmd

��

npLsrRr� q�sdJ e

J npqm

��

�

��

�npLsrJ e�Jnpqm qmd �� npLsrJ e

�Jnpqm qsd

�

�

��fRs �K�� qmd�� g I� ��

�� fRr � g I� �� Rm �K�� qd��

��

��

�

��

n�pL�sr

Rr� q�mdI� �� n�pL

�sr

Rr� qmd qsd

��

�n�pL

�sr

Rr� qmd q�sd ��

n�pL�sr

Rr� q�sd qsd

��


and it must be required that the matrix below be positive semide�nite��

nRs &K�� 'qmd� n�pL

�sr

��Rr� � 'q�md � �

oI� ��

n�pL�sr

��Rr� � 'qmd 'qsd

�� fRr � �g I� ��n�pL

�sr

��Rr� � 'q�sd 'qmd �� Rm &K�� 'qd� n�pL

�sr

��Rr� � 'q�sd 'qsd � �

��

Under the assumption that the # # upper left submatrix is positive de�nite andhence invertible� only the positive semide�niteness of its Schur complement must bechecked according to the theorem�

The upper # # matrix is invertible if and only if

Rs &K�� 'qmd�n�pL

�sr

#�Rr � �'q�md � �

This condition can be satis�ed by choosing the gain as

K�� 'qmd��

n�pL�sr

#��'q�md & k�� Rr� k� � �D� �

For each choice of �� there will be a corresponding � � � minfRs� Rrg such thatthe requirement above is satis�ed� but as �� Rr� � � �

Calculation of the Schur complement for the upper # # matrix� results in

Rm &K�� 'qd�n�pL

�sr

#�Rr � �'q�sd 'qsd

� n�pL�srh

Rs &K�� 'qmd� n�pL�sr

��Rr� � 'q�md � �

i��Rr � ��

'q�sd 'qsd 'q�md � � �D� �

Now� since

Rs &K�� 'qmd�n�pL

�sr

#�Rr � �'q�md � � �

#��Rs & k� � ��Rr � � & n�pL�sr 'q

�md�Rr � � � ��

#��Rr � �

�D� � can be rewritten as

Rm&K�� 'qd

�n�pL�sr

#

�#��Rs & k� � � & n�pL

�sr 'q

�md

#��Rs & k� � ��Rr � � & n�pL�sr 'q

�md�Rr � � � ��

�'q�sd 'qsd � �

�� D� Proofs

Choosing K�� 'qd as

K�� 'qd �n�pL

�sr

#��'q�sd 'qsd

gives the requirement

Rm &n�pL

�sr

#

��

�� #��Rs & k� � � & n�pL

�sr 'q

�md

#��Rs & k� � ��Rr � � & n�pL�sr 'q

�md�Rr � � � ��


A rearrangement of the terms �nally gives

Rm &n�pL

�sr

#

�#��Rs & k� � ��Rr � � � �� & n�pL

�sr 'q

�md�Rr � � � ��

#��Rs & k� � ��Rr � � & n�pL�sr�� 'q

�md�Rr � � � ��


From this equation it can be seen that there exists a � � minfRs� Rr� Rmg suchthat the above inequality is satis�ed� at least for any � ��

��Rr� As �� approaches

its upper limit� � goes to zero�

This bound on �� becomes the restricting bound� However� � goes to zero withthe mechanical damping Rm� even if �� can be chosen independent of Rm� Thisdependence on Rm can be avoided by adding a constant k� � to K�� 'qd� giving

K�� 'qd��

n�pL�sr

#��'q�sd 'qsd & k��

�

Rr� k� �

�


Proof� For the proof of Eq� �� #�� it must be shown that

�R�qm� 'qm &K� 'qm�es � �I� �

with

�R�qm� 'qm &K� 'qm�es �

�RsI� &K�� 'qmI�

��npLsrJ eJnpqm 'qm

��npLsrJ e�Jnpqm 'qm RrI�

�for some � � � with the given choice of the nonlinear gain K�� 'qm�

Using the results from Section #�� the fact that Rr � and calculating the Schurcomplement of the lower matrix� gives the requirement

Rs &K�� 'qm�n�pL

�sr

#�Rr � �'q�m � �

Using the results from the derivation of �D� �� it follows that the requirement isful�lled for some � � minfRs� Rrg if

K�� 'qm��

n�pL�sr

#�'q�m & k�� Rr� k� �

�

� Derivation of Eqs� �� and ��

Derivation of Eqs� �� and ��

In this section it will be shown how �� and �� are derived for a generaltorque reference ��


The starting point is �� #

Ploss � u� 'qs � 'qm

The control u is �rst eliminated from the above expressions by using �� whichcan be rewritten as

u � Ls-qs & Lsr eJ npqm -qr & npLsr 'qmJ e Jnpqm 'qr &Rs 'qs

� Lr-qr & Lsr e�Jnpqm -qs � npLsr 'qmJ e �Jnpqm 'qs &Rr 'qr

The derivative of the rotor currents� -qr� can be eliminated from the stator equationby using the last of the equations above� This results in

u �

�Ls � L�

sr

Lr

�-qs &Rs 'qs & np

L�sr

Lr

'qmJ 'qs

&

��LsrRr

LrI� & npLsr 'qmJ

�e Jnpqm 'qr

Substitution of the expression above together with � npLsr 'q�s J e Jnpqm 'qr in Ploss�

and use of the fact that z�J z � � �z � IR� �skew�symmetry gives

Ploss � 'q�s u� 'qm

�

�Ls � L�

sr

Lr

�'q�s -qs &Rs 'q

�s 'qs & np

L�sr

Lr'qm 'q�s J 'qs � LsrRr

Lr'q�s e J npqm 'qr

&npLsr 'qm 'q�s J e Jnpqm 'qr � npLsr 'qm 'q�s J e J npqm 'qr

�

�Ls � L�

sr

Lr


�s 'qs � LsrRr

Lr'q�s e Jnpqm 'qr

�

�Ls � L�

sr

Lr


�s 'qs � LsrRr

L�r

'q�s�e Jnpqm�r � Lsr 'qs

��

�Ls � L�

sr

Lr

�'q�s -qs &

�Rs &Rr

L�sr

L�r

�'q�s 'qs � LsrRr

L�r

'q�s e Jnpqm�r �D�!

which is identical to ��

�� D� Proofs


Under the assumption of perfect control� i�e� that the stator current tracking errorhas converged to zero� the desired functions for 'qs and -qs de�ned in �� can besubstituted into �� For convenience they are rewritten here as

'qs � 'qsd ��

Lsr

�� &

Lr'�

Rr�

�I� &

Lr

np��J

�e Jnpqm�rd

�� CI� &DJ � e J npqm�rd

-qs � -qsd ��

Lsr

$�RrLr

-�� RrLr'��

R�r

��I� &

�� Lr

'��np��

� &Lr

np��'�

�J

�

&np 'qmLsr

�� Lr

np��I� &

�� &

Lr'�

Rr�

�J

�)e Jnpqm�rd

&�

Lsr

�� &

Lr'�

Rr�

�I� &

Lr

np��J

�e Jnpqm

�'��I� &

Rr

np��J

��rd

��

Lsr

��Lr

-�Rr�

� Lr 'qm��

� &'�� LrRr

n�p��

�I�

&

�Lr

np��'� & np 'qm &

npLr 'qm '�Rr�

&Rr

np��

�J

�e Jnpqm�rd

�� AI� &BJ � e Jnpqm�rd

where the constants A� B� C and D have been introduced to simplify later calcula�tions� In the above calculations '�rd from �� !� was used�

The above expressions substituted into �D�! results in

Ploss �

�Ls � L�

sr

Lr

��rd �CI� �DJ �AI� &BJ �rd

&

�Rs &Rr

L�sr

L�r

��rd �CI� �DJ �CI� &DJ �rd

�RrLsr

L�r

��rd �CI� �DJ �rd

�

�Ls � L�

sr

Lr

��AC &BD��rd�rd

&

�Rs &Rr

L�sr

L�r

��C� &D�

��rd�rd �

RrLsr

L�r

C��rd�rd

where the skew�symmetry of J has been used in the last transition� and the constants


AC� BD� C� and D� are given as

AC ��

L�sr

�Lr

-�Rr�

� Lr 'qm��

� &'�� LrRr

n�p��

�� &

Lr'�

Rr�

�

��

L�sr

�Lr

-�Rr�

� Lr 'qm��

� &'�� LrRr

n�p��

&L�r-� '�

R�r

�� L�

r 'qm '�Rr��

� &Lr

'��

Rr�� L�

r'�

n�p��

�

BD ��

L�sr

�Lr

np��'� & np 'qm &

npLr 'qm '�Rr�

&Rr

np��

�Lr

np��

��

L�sr

�L�r

n�p�� '� &

Lr 'qm��

� &L�r 'qm

'�Rr��

� &LrRr

n�p��

�

C� ��

L�sr

�� &

Lr'�

Rr�&

L�r'��

R�r

��

�

D� ��

L�sr

L�r

n�p��

Use of the above expressions results in

AC &BD ��

L�sr

�Lr

-�Rr�

&'��

&L�r-� '�

R�r

��&

Lr'��

Rr�� L�

r'�

n�p�� &

L�r

n�p�� '�

�

C� &D� ��

L�sr

�� &

Lr'�

Rr�&

L�r'��

R�r

��&

L�r

np��

�

By the use of the above results and the fact that ��rd�rd � �� Ploss is �nally

�� D� Proofs

found to be

Ploss �LsLr

L�srRr

-�� &Ls

L�sr

'�� &L�rLs

L�srR

�r

-� '� &LrLs

L�srRr

'��

L�rLs

'�n�pL

�sr

��

&L�rLs

n�pL�sr

�� '� � �

Rr

-�� Lr

R�r

-� '� &Lr

'�n�p

��

Lr

n�p�� '�

&Rs

L�sr

�� &

LrRs

L�srRr

'�� &L�rRs

L�srR

�r

'�� &

L�rRs

n�pL�sr

�� &

Rr

n�p��

�L�rLs � L�

srLr

L�srR

�r

-� '� &LsLr � L�

sr

Rr

-��

&LsRr & LrRs

L�srRr

'�� &RrLrLs & L�

rRs

L�srRr

'��

&LrL

�sr � L�

rLs

n�pL�sr

��'��

&Rs

L�sr

��

&

�L�rLs � LrL

�sr

n�pL�sr

'�� &L�rRs &RrL

�sr

n�pL�sr

��

��

��

which is identical to �� when � � d�

� Boundedness of all signals for indirect FOC


Starting from �� we have

'd � ��KP &

KI

p

�-qm

� ��KP &

KI

p

��v�J �rd & v�J �� L

� ��KP &

KI

p

��d � L �

� � �d�

� e�J �d��

The rest of the proof consist in expressing e �J �d�� as a bounded signal� whichensures a bounded nonlinear feedback since it is multiplied by d�


A Boundedness of e �J �d��

Since �� r � �rd and �Rr

�� (Rr �Rr

'�� '�r � '�rd

� �Rr��r � �rd� �Rrv & �Rr�rd

� �Rr��& �Rr��rd � v

The term �rd � v can be replaced by

�rd � v � � �

(Rr

'�rd � � �

��dJ �rd

so that

'�� Rr�� Rr

�

��dJ�rd

J �rd is bounded since

J �rd � J � e J �d��

�� e J �d

� �

�

so that the derivative of �� becomes

'�� Rr��

�Rr

��d� e J �d

� �

�

From this equation� ��t follows from the convolution integral

��t� e�Rrt��

�

Z t

�

e�Rr�t�s� �Rrd�s eJ �d�s�

� �

�ds

� ��Rr�(Rr

Z t

�

e�Rr�t�s� '�d�sJ e J �d�s�ds��

�

� ��Rr�(Rr

Z t

�

e�Rr�t�s�de J �d�s�

dsds

��

�

� ��Rr�(Rr

e�RrtZ t

�

eRrsde J �d�s�

dsds

��

�Integration by parts results in an expression for the integralZ t

�

eRrsde J �d�s�

dsds � eRrs e J �d�s� jt� �

Z t

�

e J �d�s�RreRrsds

� eRrt e J �d�t� � e J �d�� Rr

Z t

�

e J �d�s�eRrsds

�� D� Proofs

So that� with initial conditions �r� � � �rd� � ��

e �J �d�t��t� e �J �d�t�e�Rrt��

� ��Rr�(Rr

e�Rrt e �J �d�t��eRrt e J �d�t� � e J �d�� Rr

Z t

�

e J �d�s�eRrsds��

�

�

� ��Rr�(Rr

�I� � e�Rrt e �J ��d�t��d�� Rr

Z t

�

e �J ��d�t��d�s��e�Rr�t�s�ds��

�

�

� � �I� � e �RrtI��J ��d�t��d�� Rr�

(Rr

��

�&

�Rr�(Rr

Rr

Z t

�

e�Rr�t�s��

cos z� sin z

�ds

where z�t� s is de�ned as

z�t� s�� d�t� �d�s

�

B Elimination of �� from the expression for ��d

For elimination of e �J �d�� in the equation

'd � ��KP &

KI

p

��d � L �

� � �d�

� e �J �d��

�the expression

e �J �d�� e�Rrt�f��tf��t

��

�Rr�(Rr

��

�&

�Rr�(Rr

Rr

Z t

�

e�Rr�t�s��

cos z� sin z

�ds

can be used� where the functions f�� f� � L depend on the initial values �� d� �so that

h� �d�t�

��

ie �J �d�� e�Rrtf��t d

�& �e

�Rrtf��t &�Rr

(Rr

d�t

��Rr�(Rr

Rr

Z t

�

e�Rr�t�s��d�t

�cos z & � sin z

�ds

which �nally leads to

'd � ��KP &

KI

p

��d � L & e�Rrtf��t

d�� e�Rrtf��t

��Rr

(Rr

d�t &�Rr�(Rr

Rr

Z t

�



�ds

�


By use of the di�erential operator p�� d�dt� this can be written as�

p&

�pKP &KI

p

��

�Rr

(Rr

��d ��

KP &KI

p

��L � e�Rrtf��t

d�

& �e�Rrtf��t

��Rr�(Rr

Rr

Z t

�


�cos�z & � sin z

�ds

�

so that d is the output of a LTI operator G�p with a nonlinear feedback b�t

d�t � � pKP &KI

p� & �pKP &KI�� Rr

�Rr

� �

�Rr�Rr

(Rr

Z t

�



�ds& e�Rrtf��t

d�

�

&pKP &KI

p� & �pKP &KI�� Rr

�Rr

��L & �e�Rrtf��t

C Feedback gain calculation

The bounded signal b�t which is multiplied with d is

b�t � ��RrRr

(Rr

Z t

�

e�Rr�t�s� cos zds� e�Rrtf��t�

�

and it follows that b�t � b �t & b��t with

kb �tk �j�Rr

(Rr

j� b��t � L�

�

�� Proofs

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Index

A

AC motors

variable reluctance � � � � � � � � � � � � ! �

AC motors � � � � � � � � � � � � � � � � � � � � � � � ��

Blondel�Park transformable � � � ��

control challenges � � � � � � � � � � � � � �$

controllers

digital implementation � � � � � � ! �

on converter level � � � � � � $$� ! �

overview � � � � � � � � � � � � � � � �� $�

PBC � � � � � � � � � � � � � � ��! �� !#

decoupling conditions �� ! ! �

dynamics

damping injection � � � � � � � � � � � �

passive decomposition � � � � � � � ��

strict passi�ability � � � � � � � � � � �

general model � � � � � � � � � � � � � �$

Lagrangian � � � � � � � � � � � � � � � � � �

generalized coordinates � � � � � � � � ��

nonlinear control � � � � � � � � � � � � � � ��

nonlinear magnetics � � � � � � � � � � �! �

not PB�transformable � � � � � � � � � ! �

observers � � � � � � � � � � � � � � � � � � � � � ! �

PM stepper motor � � � � � � � �$� ! �

scalar control � � � � � � � � � � � � � � � � � ��

switched reluctance � � � � � � � � � � � ��

synchronous � � � � � � � � � � � � � � �$� ! �

varying parameters � � � � � � � �� ! �

vector control � � � � � � � � � � � � � � � � � ��

Adaptation

of load torque � � � � � � � � � � � � � � � � �!!�

rotor parameters � � � � � � � � � ! !� !!�

stator parameters � � � � � � � � ! !� !!#

Approximate di�erentiation � � � � � � � !

EL controller with � � � � � � � � � � see Econtroller��

�lter � � � � � � � � � � � � � � � � � � � � � � � �� !

BBackstepping � � � � � � � � � � � � � � � � � � � � � $!� � � � � � � � � � � � � see Flux norm referenceBlondel�Park

strong transformability � � � � � � � � ��transformation � � � � � � � � � � � � � � � � � de�nition of � � � � � � � � � � � � � � � � � ��

Brushless DC motor � � � � � � � � � � � � � � �$

CComplex notation � � � � � � � � � � � � � � � � ��Coordinate transformations � � � � � � � !��

Copernicus � � � � � � � � � � � � � � � � � � � !��rotations � � � � � � � � � � � � � � � � � � � � � �!��

Coordinatesquasi � � � � � � � � � � � � � � � � � � � � � � � � � #��

Copernicus � � � � � � � � � � � � � � � � � � � � � � � !��Current harmonics � � � � � � � � � � � � � � � �! �

DDC motors � � � � � � � � � � � � � � � � � � � � � � � ��Diesel�%turbine�electric propulsion � ��Directaxis � � � � � � � � � � � � � � � � � � � � � � � ! �dm� dM � � � � � � � � � � � � � � � � � � � � � � � � � � � � #DTC� direct torque control � � � � � � � �! �

EEL controllers

saturated � � � � � � � � � � � � � � � � � � � ��stability proofs for � � � � � � ��$� #��

with approximate di�erentiation��with virtual robot � � � � � � � � � � � � � ��#

EL equations

�!�

�� INDEX

and Hamilton�s principle � � � � � � #��de�nition of � � � � � � � � � � � � � � � �$� #��

EL parametersAC motors � � � � � � � � � � � � � � � � � � � � �!de�nition of � � � � � � � � � � � � � � � � � � � ��of rigid�joint manipulator � � � � � � ��switched � � � � � � � � � � � � � � � � � � � � � � �!$

EL plant � � � � � � � � � � � � � � � � � � � � � � � � � � #�EL system

de�nition of � � � � � � � � � � � � � � � � � � � ��EL systems

interconnection of � � � � � � � � � � � $� $switched� modeling of � � � � �!�� # underactuated � � � � � � � � � � � � � � � � � ��underdamped � � � � � � � � � � � � � � � � � � ��

Experimental setup �!�!!�� !��!��# $� #!�#!$

FFeedback linearization � � � � � � � � � � � � $

experimental results � � � � � � � � � � �!�#induction motor control � � � � � � � !��observerbased � � � � � � � � � � � � � � � �!�$robustness of � � � � � � � � � � � � � � � � � !��

Flatnessof induction motor model � � � � � $

Fluxcontrol of � � � � � � � �!� !�� ! $� !!�norm reference � � � � � � �!� ! �� !!�referenceselection of � � � � � � � � � � � � � � � � � !��

sensors � � � � � � � � � � � � � � � � � � � � � � � !��weakening� � � � � � � � � � � � � � � �!!�� !!�

FOC � � � � � � � � � � � � � � � ! �� !� � !�!!� direct � � � � � � � � � � � � � � � � � � � � !�#� !��improvement of � � � � � � � � � � � � � !��

indirect � � � � � � � � � � � � � ! $� !�#� !��boundedness of signal � � � � � � � # �discretetime � � � � � � � � � � � � � � � #! robustness � � � � � � � � � � � � � �# !#��stability of � � � � � � � � � � � � � � � � � � !�!tuning � � � � � � � � � � � � � � � � � �#��# �

induction motors � � � � � � � � � � � � � �!�

ForcesCoriolis� centrifugal � � � � � � � � � � � � !external � � � � � � � � � � � � � � � � � � � � � � � ��non�conservative � � � � � � � � � � � � � � #��

Frictionpassivity of LuGre model � � � � � ��

Fuzzy logicmotor control � � � � � � � � � � � � � � � � � $�

GGeneralized coordinates � � � � � � � � � � �#�!

electromechanical systems � � � � � #��Gradient estimator

passivity of � � � � � � � � � � � � � � � � � � � � $$

HHamilton

principle of � � � � � � � � � � � � � � � � � � � #�$Highgain control

of currents � � � � � � � � � � � � � � � � � � � � !�!remarks to � � � � � � � � � � � � � � � � � � � �!�

IIgnorable coordinates � � � � � � � � � � � � � ��Induction motor

control problem � � � � � � � � � � � � � � � ! �controllersdqimplementation � � � � � !!$� !#�experimental results � � � �!�!!� �!�#� # $� #!$

observerbased � � � � � � � � � !#�!�!PBC � � � � � � � � � � � � � � � � � � � � � � � �! $position control � � � � � � � � � � � � � !� sensorless � � � � � � � � � � � � � � � � � � � !��speed control � � � � � � � � � � �!� � ! $

currentfed � � � � � � � � � � ! �� !��#!��ux tracking � � � � � � � � � � � � � � � � � � ! �measured variables � � � � � � � � � � � � !��model � � � � � � � � � � � � � � � � � � � � !� ! ��model � � � � � � � � � � � � � � � � � � � !� ab�model � � � � � � � � � � � � � � � !�$� ! dq�model � � � � � � � � � � � � � � � � � � � !�$bilinear � � � � � � � � � � � � � � � � � � � � � !�#change of coordinates � � � � � � � !��

INDEX ��

currentfed � � � � � � � � � � � � � � � � � !�!discretetime � � � � � � � � � � � � � � � #!�geometric properties � � � � � � � � !�#Newton�s second law form � � � !�!passivity properties � � � � � � � � � !�!

observers � � � � � � � � � � � � � � � � �!#�� !#�adaptive � � � � � � � � � � � � � � � � � � � � $

parameter estimation � � � � �!!#� !!�rotor time constant � � � � � � � � � � � !��speed tracking � � � � � � � � � � � � � � � � ! �squirrelcage � � � � � � � � � � � � � !�!� !��state estimation � � � � � � � � � � � � � � � $

Integral actionin stator currents � � � � � � � � � � � � � !!!

Kkc � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � #kg � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � #km� kM � � � � � � � � � � � � � � � � � � � � � � � � � � � � $�kv � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � #KYP lemma � � � � � � � � � � � � � � � � � � � � � � #�

LL�

spaces � � � � � � � � � � � � � � � � � � � � � � � � #$$stability � � � � � � � � � � � � � � � � � � � � � � #$$

Lagrangiande�nition of � � � � � � � � � � � � � � � � � � � ��$function � � � � � � � � � � � � � � � � � � � � � � � ��nonconservative � � � � � � � � � � �!�� #

Levitated ballmodel of � � � � � � � � � � � � � � � � � � � � � � � !

Levitating ball � � � � � � � � � � � � � � � � � � � � ��Load torque � � � � � � � � � � � � � �� !!�� !##

MM � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��Manifold designs � � � � � � � � � � � � � � � � � � $�Marine vessel

model of � � � � � � � � � � � � � � � � � � � � � � � !�stabilization of � � � � � � � � � � � � � � � � � $#

Mechanical commutation � � � � � � � � � � ��

NNestedloop � � � � � � � � � � � � � � � � � � � � � � ! �

Neural networkmotor control � � � � � � � � � � � � � � � � � $�

Nonholonomicdouble integrator � � � � � � � � � � � � � !��

Nonlinear magnetics � � � � � � � � � � � � � � ��

OOBFL � � � � � � � � � � � � � � � � � � � � � � � � � � � � !�$Observers

nonlinear � � � � � � � � � � � � � � � � � � � � � !��

PParameters

�xed estimate � � � � � � � � � � � � � � � � �!�!varying � � � � � � � � � � � � � � � � � � � � � � � !

Park�s transformation � � � � � � � � � � � � � ��Partial damping

GAS with � � � � � � � � � � � � � � � � � � � � � � �Passive decomposition � � � � � � � � � �� Passive systems

interconnection of � � � � � � � � � � � � �#� Passivity

invariance of � � � � � � � � � � � � � �$� #$�PBC

AC motorsgeometric perspective � � � � � � � ! #

induction motors � � � � � � � � � � � � � �! $dqmodel � � � � � � � � � � � � � � !�� !!$compatibility with indirect FOC!�

currentfed � � � � � � � � � � � � � � � � � !� discretetime � � � � � � � � � � � � � � � #!!observerbased � � � � � � � � � !# !�!observerless � � � � � � � � � � � � � � � � ! $position control � � � � � � � � � � � � � !!�

motor controloverview � � � � � � � � � � � � � � � $$ $�

nestedloop � � � � � � � � � � � � � � !� � ! �reluctance motor � � � � � � � � � � � � � �! �stepper motors � � � � � � � � � � � � � � � �! �synchronous motors � � � � � � � � � � � ! �

PD controllerpassivity of � � � � � � � � � � � � � � � � � � � � $�

�� INDEX

physical interpretation of � � � � � � �#�Permanent magnet motors � � � � � � � � ��PI�D controller

passivity of � � � � � � � � � � � � � � � � � � � � ��stability of � � � � � � � � � � � � � � � � � � � � � ��

PID controlpassivity of � � � � � � � � � � � � � � � � � � � � �

Position control � � � � � � � � � � � � � � ! �� !!�Power consumption

minimization of � � � � � � � � � � � � � � � !!�Power e�ciency � � � � � � � � � � � � � � �$� !!�

Qq�

de�nition of � � see Notation sectionQuadratureaxis � � � � � � � � � � � � � � � � � � ! �

RRayleigh dissipation function

de�nition of � � � � � � � � � � � � � � � � � � � ��Resistances

timevarying � � � � � � � � � � � � � � � � � � ��RLC network

as passive operator � � � � � � � � � � � �#$�Robot

simulation parameters of � � � � � � � ��Robots

�exible�jointsmodel of � � � � � � � � � � � � � � � � � � � � � !#stabilization of � � � � � � � � � � � � � � � �!

rigid�jointsbounded controls � � � � � � � � � � � � �$!

Robustadaptive tracking controller � � � � ��

Rotational sensors � � � � � � � � � � � � � � � � !��Rotor resistance

adaptation of � � � � � � � � � � � � � � � � � ! !variations in � � � � � � � � � � � � � � � � � � !

Rotor time constant�Tr � � � � � � � � � � � �!��

SSaturation function

de�nition of � � � � � � � � � � � � � � � � � � � ��!Sensorless control � � � � � � � � � � � � � � � � � !��

Singular perturbation � � � � � � � � � � � � � $�Sliding modes

converter control � � � � � � � � � � � � � � $�induction motor control � � � � � � � $�

Slip speed � � � � � � � � � � � � � � � � � � � ! �� !��Speed

control of � � � � � � � � � � � � � � � � � � � � � ! $reference� 'qm��t � � � � � � � � � � � � � �! �

Speed control � � � � � � � � � � � � � � � � � � � � �! �Squirrelcage � � � � � � � � � � � � � � � � � � � � � !��Squirrelcage rotor

currents in � � � � � � � � � � � � � � � � � � � �!��substitution of � � � � � � � � � � � � � � � � �$

Stepper motor � � � � � � � � � � � � � � � � � � � � �$Storage function � � � � � � � � � � � � � � � � � � ! Switched reluctance motor � � � � � � � � ��Synchronous motor � � � � � � � � � � �$� ! �

TTORA

model of � � � � � � � � � � � � � � � � � � � � � � � ! parameters of � � � � � � � � � � � � � � � � � � ��stabilization of � � � � � � � � � � � � � � � � � � with bounded inputs � � � � � � � � � ��

Torque ripple � � � � � � � � � � � � � � � � � � � � � ! �

UUnderactuated

AC motors � � � � � � � � � � � � � � � � � � � � � EL systems � � � � � � � see EL systems

UnderdampedEL systems � � � � � � � see EL systems

VVoltage transducers � � � � � � � � � � � � � � � !��

WWindings

concentrated � � � � � � � � � � � � � � � � � � ! �sinusoidally distributed � � � � � � � �$

Worklessfactorization of � � � � � � � � � � � � � � � !#!

Workless forces � � � � � � � � � � � � � � � � � � � !�!factorization of � � � � � � � � � � � � � � � !#!

INDEX ��

ZZero dynamics

periodic � � � � � � � � � � � � � � � � � � � � � � ! #Zero�sequence � � � � � � � � � � � � � � � � � � � � ! �

passivity based control of euler-lagrange systems

Documents

Transcript of passivity based control of euler-lagrange systems