L1 Adaptive Control

L1 Adaptive Control Theory

DC21_Hovakimyan-Cao-FM.indd 1 7/15/2010 8:24:14 AM

Advances in Design and ControlSIAM’s Advances in Design and Control series consists of texts and monographs dealing with all areas of design and control and their applications. Topics of interest include shape optimization, multidisciplinary design, trajectory optimization, feedback, and optimal control. The series focuses on the mathematical and computational aspects of engineering design and control that are usable in a wide variety of scientific and engineering disciplines.

Editor-in-ChiefRalph C. Smith, North Carolina State University

Editorial BoardAthanasios C. Antoulas, Rice UniversitySiva Banda, Air Force Research LaboratoryBelinda A. Batten, Oregon State UniversityJohn Betts, The Boeing Company (retired)Stephen L. Campbell, North Carolina State University Michel C. Delfour, University of Montreal Max D. Gunzburger, Florida State University J. William Helton, University of California, San Diego Arthur J. Krener, University of California, DavisKirsten Morris, University of WaterlooRichard Murray, California Institute of TechnologyEkkehard Sachs, University of Trier

Series VolumesHovakimyan, Naira, and Cao, Chengyu, L1 Adaptive Control Theory: Guaranteed Robustness with Fast AdaptationSpeyer, Jason L., and Jacobson, David H., Primer on Optimal Control TheoryBetts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second EditionShima, Tal and Rasmussen, Steven, eds., UAV Cooperative Decision and Control: Challenges and Practical ApproachesSpeyer, Jason L. and Chung, Walter H., Stochastic Processes, Estimation, and ControlKrstic, Miroslav and Smyshlyaev, Andrey, Boundary Control of PDEs: A Course on Backstepping DesignsIto, Kazufumi and Kunisch, Karl, Lagrange Multiplier Approach to Variational Problems and ApplicationsXue, Dingyü, Chen, YangQuan, and Atherton, Derek P., Linear Feedback Control: Analysis and Design with MATLABHanson, Floyd B., Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, and Computation Michiels, Wim and Niculescu, Silviu-Iulian, Stability and Stabilization of Time-Delay Systems: An Eigenvalue- Based ApproachIoannou, Petros and Fidan, Barıs, Adaptive Control TutorialBhaya, Amit and Kaszkurewicz, Eugenius, Control Perspectives on Numerical Algorithms and Matrix ProblemsRobinett III, Rush D., Wilson, David G., Eisler, G. Richard, and Hurtado, John E., Applied Dynamic Programming for Optimization of Dynamical SystemsHuang, J., Nonlinear Output Regulation: Theory and ApplicationsHaslinger, J. and Mäkinen, R. A. E., Introduction to Shape Optimization: Theory, Approximation, and ComputationAntoulas, Athanasios C., Approximation of Large-Scale Dynamical SystemsGunzburger, Max D., Perspectives in Flow Control and OptimizationDelfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Analysis, Differential Calculus, and OptimizationBetts, John T., Practical Methods for Optimal Control Using Nonlinear ProgrammingEl Ghaoui, Laurent and Niculescu, Silviu-Iulian, eds., Advances in Linear Matrix Inequality Methods in ControlHelton, J. William and James, Matthew R., Extending H1 Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives

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Society for Industrial and Applied MathematicsPhiladelphia

L1 Adaptive Control TheoryGuaranteed Robustness with Fast Adaptation

Naira HovakimyanUniversity of Illinois

Urbana, Illinois

Chengyu CaoUniversity of Connecticut

Storrs, Connecticut


is a registered trademark.

Copyright © 2010 by the Society for Industrial and Applied Mathematics.

10 9 8 7 6 5 4 3 2 1

All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA.

Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended.

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MATLAB and Simulink are registered trademarks of The MathWorks, Inc. For MATLAB product information, please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA, 508-647-7000, Fax: 508-647-7001, [email protected], www.mathworks.com.

Figure 1 in the preface appears courtesy of the United States Army.

Figure 2 in the preface appears courtesy of the Naval Postgraduate School.

Figures 3 & 4 in the preface appear courtesy of NASA.

Figure 6.1 used with permission from NASA.

Library of Congress Cataloging-in-Publication Data Hovakimyan, Naira. L1 adaptive control theory : guaranteed robustness with fast adaptation / Naira Hovakimyan, Chengyu Cao. p. cm. -- (Advances in design and control ; 21) Includes bibliographical references and index. ISBN 978-0-898717-04-4 1. Adaptive control systems. 2. Robust control. I. Cao, Chengyu. II. Title. TJ217.H68 2010 629.8’36--dc22 2010013646


To my parents Emma and Viktor, and to my sister Anna with love and gratitudeNH

To my wife Xingwei and our son Lucas Bochao, as well as to our parents Jinrong, Guangju, Runkuan, and Yageng with love and gratitude

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Contents

Foreword xi

Preface xiii

1 Introduction 11.1 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Two Different Architectures of Adaptive Control . . . . . . . . . . . . 4

1.2.1 Direct MRAC . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Direct MRAC with State Predictor . . . . . . . . . . . . 61.2.3 Tuning Challenges . . . . . . . . . . . . . . . . . . . . . 7

1.3 Saving the Time-Delay Margin . . . . . . . . . . . . . . . . . . . . . 81.4 Uniformly Bounded Control Signal . . . . . . . . . . . . . . . . . . . 12

2 State Feedback in the Presence of Matched Uncertainties 172.1 Systems with Unknown Constant Parameters . . . . . . . . . . . . . . 17

2.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 182.1.2 L1 Adaptive Control Architecture . . . . . . . . . . . . . 182.1.3 Analysis of the L1 Adaptive Controller: Scaling . . . . . 192.1.4 Design of the L1 Adaptive Controller: Robustness and

Performance . . . . . . . . . . . . . . . . . . . . . . . . 252.1.5 Simulation Example . . . . . . . . . . . . . . . . . . . . 292.1.6 Loop Shaping via State-Predictor Design . . . . . . . . . 31

2.2 Systems with Uncertain System Input Gain . . . . . . . . . . . . . . . 352.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 362.2.2 L1 Adaptive Control Architecture . . . . . . . . . . . . . 372.2.3 Performance Bounds of the L1 Adaptive Controller . . . 392.2.4 Performance in the Presence of Nonzero Trajectory

Initialization Error . . . . . . . . . . . . . . . . . . . . . 442.2.5 Time-Delay Margin Analysis . . . . . . . . . . . . . . . 472.2.6 Gain-Margin Analysis . . . . . . . . . . . . . . . . . . . 592.2.7 Simulation Example: Robotic Arm . . . . . . . . . . . . 59

2.3 Extension to Systems with Unmodeled Actuator Dynamics . . . . . . 672.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 672.3.2 L1 Adaptive Control Architecture . . . . . . . . . . . . . 682.3.3 Analysis of the L1 Adaptive Controller . . . . . . . . . . 70

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2.3.4 Simulation Example: Rohrs’ Example . . . . . . . . . . . 762.4 L1 Adaptive Controller for Nonlinear Systems . . . . . . . . . . . . . 80

2.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 802.4.2 L1 Adaptive Control Architecture . . . . . . . . . . . . . 812.4.3 Analysis of the L1 Adaptive Controller . . . . . . . . . . 832.4.4 Simulation Example: Wing Rock . . . . . . . . . . . . . 90

2.5 L1 Adaptive Controller in the Presence of Nonlinear UnmodeledDynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942.5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 942.5.2 L1 Adaptive Control Architecture . . . . . . . . . . . . . 952.5.3 Analysis of the L1 Adaptive Controller . . . . . . . . . . 972.5.4 Simulation Example . . . . . . . . . . . . . . . . . . . . 105

2.6 Filter Design for Performance and Robustness Trade-Off . . . . . . . 1112.6.1 Overview of Stochastic Optimization Algorithms . . . . . 1132.6.2 LMI-Based Filter Design . . . . . . . . . . . . . . . . . 114

3 State Feedback in the Presence of Unmatched Uncertainties 1213.1 L1 Adaptive Controller for Nonlinear Strict-Feedback Systems . . . . 121

3.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 1213.1.2 L1 Adaptive Control Architecture . . . . . . . . . . . . . 1223.1.3 Analysis of the L1 Adaptive Controller . . . . . . . . . . 1253.1.4 Simulation Example . . . . . . . . . . . . . . . . . . . . 136

3.2 L1 Adaptive Controller in the Presence of Unmatched Uncertainties . 1403.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 1413.2.2 L1 Adaptive Control Architecture . . . . . . . . . . . . . 1423.2.3 Analysis of the L1 Adaptive Controller . . . . . . . . . . 1453.2.4 Simulation Example . . . . . . . . . . . . . . . . . . . . 154

3.3 Piecewise-Constant Adaptive Laws for Systems with UnmatchedUncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1593.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 1593.3.2 L1 Adaptive Control Architecture . . . . . . . . . . . . . 1603.3.3 Analysis of the L1 Adaptive Controller . . . . . . . . . . 1643.3.4 Simulation Example . . . . . . . . . . . . . . . . . . . . 172

4 Output Feedback 1794.1 L1 Adaptive Output Feedback Controller for First-Order Reference

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1794.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 1804.1.2 L1 Adaptive Control Architecture . . . . . . . . . . . . . 1804.1.3 Analysis of the L1 Adaptive Output Feedback Controller 1834.1.4 Design for the L1-Norm Condition . . . . . . . . . . . . 1894.1.5 Simulation Example . . . . . . . . . . . . . . . . . . . . 190

4.2 L1 Adaptive Output Feedback Controller for Non-SPR ReferenceSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1924.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 1924.2.2 L1 Adaptive Control Architecture . . . . . . . . . . . . . 1934.2.3 Analysis of the L1 Adaptive Controller . . . . . . . . . . 198

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4.2.4 Simulation Example: Two-Cart Benchmark Problem . . . 207

5 L1 Adaptive Controller for Time-Varying Reference Systems 2115.1 L1 Adaptive Controller for Linear Time-Varying Systems . . . . . . . 211

5.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 2115.1.2 L1 Adaptive Control Architecture . . . . . . . . . . . . . 2125.1.3 Analysis of L1 Adaptive Controller . . . . . . . . . . . . 2145.1.4 Simulation Example . . . . . . . . . . . . . . . . . . . . 218

5.2 L1 Adaptive Controller for Nonlinear Systems with UnmodeledDynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2235.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 2235.2.2 L1 Adaptive Control Architecture . . . . . . . . . . . . . 2245.2.3 Analysis of the L1 Adaptive Controller . . . . . . . . . . 2265.2.4 Simulation Example . . . . . . . . . . . . . . . . . . . . 234

6 Applications, Conclusions, and Open Problems 2416.1 L1 Adaptive Control in Flight . . . . . . . . . . . . . . . . . . . . . . 241

6.1.1 Flight Validation of L1 Adaptive Control at NavalPostgraduate School . . . . . . . . . . . . . . . . . . . . 243

6.1.2 L1 Adaptive Control Design for the NASA AirSTARFlight Test Vehicle . . . . . . . . . . . . . . . . . . . . . 254

6.1.3 Other Applications . . . . . . . . . . . . . . . . . . . . . 2596.2 Key Features, Extensions, and Open Problems . . . . . . . . . . . . . 260

6.2.1 Main Features of the L1 Adaptive Control Theory . . . . 2606.2.2 Extensions Not Covered in the Book . . . . . . . . . . . 2606.2.3 Open Problems . . . . . . . . . . . . . . . . . . . . . . . 261

A Systems Theory 263A.1 Vector and Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . 263

A.1.1 Vector Norms . . . . . . . . . . . . . . . . . . . . . . . 263A.1.2 Induced Norms of Matrices . . . . . . . . . . . . . . . . 264

A.2 Symmetric and Positive Definite Matrices . . . . . . . . . . . . . . . 265A.3 L-spaces and L-norms . . . . . . . . . . . . . . . . . . . . . . . . . 265A.4 Impulse Response of Linear Time-Invariant Systems . . . . . . . . . . 267A.5 Impulse Response of Linear Time-Varying Systems . . . . . . . . . . 268A.6 Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

A.6.1 Autonomous Systems . . . . . . . . . . . . . . . . . . . 269A.6.2 Time-Varying Systems . . . . . . . . . . . . . . . . . . . 270

A.7 L-Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272A.7.1 BIBO Stability of LTI Systems . . . . . . . . . . . . . . 273A.7.2 BIBO Stability for LTV Systems . . . . . . . . . . . . . 276

A.8 Linear Parametrization of Nonlinear Systems . . . . . . . . . . . . . . 278A.9 Linear Time-Varying Representation of Systems with Linear

Unmodeled Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 281A.10 Linear Time-Varying Representation of Systems with Linear

Unmodeled Actuator Dynamics . . . . . . . . . . . . . . . . . . . . . 283A.11 Properties of Controllable Systems . . . . . . . . . . . . . . . . . . . 285

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A.11.1 Linear Time-Invariant Systems . . . . . . . . . . . . . . 285A.11.2 Linear Time-Varying Systems . . . . . . . . . . . . . . . 286

A.12 Special Case of State-to-Input Stability . . . . . . . . . . . . . . . . . 287A.12.1 Linear Time-Invariant Systems . . . . . . . . . . . . . . 287A.12.2 Linear Time-Varying Systems . . . . . . . . . . . . . . . 288

B Projection Operator for Adaptation Laws 291

C Basic Facts on Linear Matrix Inequalities 295C.1 Linear Matrix Inequalities and Convex Optimization . . . . . . . . . . 295C.2 LMIs for Computation of L1-Norm of LTI Systems . . . . . . . . . . 296C.3 LMIs for Stability Analysis of Systems with Time Delay . . . . . . . . 297C.4 LMIs in the Presence of Uncertain Parameters . . . . . . . . . . . . . 297

Bibliography 299

Index 315

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Foreword

This book has been inspired by the problems of adaptive flight control. Research in thisfield started with attempts to develop adaptive autopilots for supersonic aircraft in the mid1950s.An interesting perspective on the early development is given in the proceedings of theself-adaptive flight control symposium at Wright Development Center in March 1959 [62].The papers describing Honeywell’s self-oscillating adaptive controller [153] and the modelreference adaptive controller (MRAC) based on the MIT rule [174] are particularly note-worthy. At the symposium it was announced that flight tests would be performed on theF-101A (the MIT system) and on an X-15 test vehicle (Honeywell’s system) [61]. Resultsof the tests appeared after a few years [22, 113]. The interest in adaptive flight control gen-erated a lot of research on adaptive control in industry and academia. However, a crash ofthe X-15 [164], which was partially attributed to the adaptive system, gave adaptive flightcontrol a bad reputation.

The early development of adaptive control was dominated by experiments, and therewas very little support from theory [14]. Advances made in stability theory and systemidentification inspired development of the theory for adaptive systems. Understanding ofMRAC improved significantly when the Lyapunov theory was applied to stability analy-sis [25, 139]. The paper [56] by Goodwin, Ramage, and Caines, which gave conditions forglobal stability, was a landmark. There was also an analogous development of the self-tuningregulator (STR). Stability proofs were given in [48, 114]. A key assumption was that theparameter estimates must be bounded. Projection methods were introduced to ensure this.The assumptions required for global stability, however, were restrictive, as illustrated by theexample of Rohrs [147], which showed lack of robustness in the presence of unmodeled dy-namics. Various fixes, like filtering, projection, and normalization, were introduced to copewith the difficulties. The relation between the STR and the MRAS was also clarified [48].

Significant advances in applications of adaptive control in specific areas resultedin commercial products. The ship-steering autopilot Steermaster [80], developed inthe 1970s, is still on the market (http://www.es.northropgrumman.com/solutions/steermaster/index.html). The company First Control (http://www.firstcontrol.se/) developed systems with STR as the primary control algo-rithm. Thousands of systems have been installed in the process industry since 1985. Typicalapplications are rolling mills and continuous casting. Use of adaptive feedforward turnedout to be particularly useful.

In the late 1980s, there was a reasonable understanding of adaptive systems, manybooks appeared, and many applications were presented [13]. Adaptive controllers workedwell in specific applications, but they did not become the widely used universal controllersthat many of us dreamed about. Interest in adaptive control in academia declined in the 1990s

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because the easy problems had been solved and ideas for tackling the difficult problems werelacking. Backstepping [100] was the last major development.

Interest in adaptive control for aerospace applications reemerged at the turn of thecentury. The driving forces were requirements for reconfiguration and damage control anda strong desire to simplify extensive and costly verification and validation procedures. TheAir Force, the Navy, and NASA worked with industry and academia to develop adaptivetechniques for air vehicles and munition. Major flight tests were also performed [161,175].

The first results on L1 adaptive control were presented by Cao and Hovakimyan at the2006 American Control Conference [27], and the journal paper appears 2 years later [29].This book, written by the creators of L1 adaptive control, gives a comprehensive accountof the ideas and the theory and glimpses of flight control applications. L1 adaptive controlcan be viewed as a modified model reference adaptive control scheme where the basicarchitecture is based on the internal model principle.

The key theoretical results are bounds on the L∞-norms of the errors in model statesand control signals. The theory gives a nice way to explore adaptation rates, a long-standingopen problem in adaptive control. A main result is that the error norms are (uniformly) in-versely proportional to the square root of the adaptation gains. High values of the adaptationgains are thus advantageous. Another feature is that the control signal is filtered to avoidhigh frequencies in the control signals. The filter is also used to shape the nominal response.The conditions are given in terms of L1-norms of certain transfer functions that involvethe filter and the largest values of the unknown parameters. The performance bounds arethe key to establishing performance guarantees for adaptive control. Because of the internalmodel structure of the controller, it admits a good delay margin even for high adaptationgains [34]. In practice, however, the largest adaptation gains are limited by the computa-tional power and the high-frequency sensor noise. Design of the filter is crucial; it can,however, be handled with linear theory. Even if L1 adaptive control was inspired by flightcontrol, the concept of course can also be applied to other systems which require adaptation.L1 adaptive control has been tested in several applications, most notably flight control foraircraft, missiles, and spacecraft. The flight tests cover control surface and sensor failuresand other sources of unmodeled dynamics. Current flight tests are performed on a subscalecommercial jet at NASA.

Günter Stein, who gave a balanced account of adaptive flight control in the 1980s,summarized the state of the art as follows: The main point made is that for conventionalflight control problems, adaptive control is the losing alternative in a historical competitionwith explicit airdata scheduling. The flight tests of L1 adaptive control show great promise,but only time will tell if it will be a viable alternative to gain scheduling. The materialpresented in this book is a good start for those who want to explore an alternative to gainscheduling.

Lund, Sweden

February 2010

Karl Johan Åström

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Preface

This book gives a comprehensive overview of the recently developed L1 adaptivecontrol theory with detailed proofs of the fundamental results. The key feature of L1 adap-tive control architectures is the guaranteed robustness in the presence of fast adaptation.This is possible to achieve by appropriate formulation of the control objective with the un-derstanding that the uncertainties in any feedback loop can be compensated for only withinthe bandwidth of the control channel. By explicitly building the robustness specification intothe problem formulation, it is possible to decouple adaptation from robustness via continu-ous feedback and to increase the speed of adaptation, subject only to hardware limitations.With L1 adaptive control architectures, fast adaptation appears to be beneficial both for per-formance and robustness, while the trade-off between the two is resolved via the selectionof the underlying filtering structure. The latter can be addressed via conventional methodsfrom classical and robust control. Moreover, the performance bounds of L1 adaptive controlarchitectures can be analyzed to determine the extent of the modeling of the system that isrequired for the given set of hardware.

The book is organized into six chapters and has an appendix that summarizes the mainmathematical results, used to develop the proofs.

Chapter 1 starts with a brief historical overview of adaptive control theory. It pro-ceeds with an introduction to the main ideas of the L1 adaptive controller. Two equivalentarchitectures of model reference adaptive controllers (MRAC) are considered next, and thechallenges of tuning these schemes are discussed. The chapter proceeds with analysis of astable scalar system with constant disturbance and introduces the main idea of the L1 adap-tive controller. Two key features are analyzed in detail: the closed-loop system’s guaranteedphase margin and the uniform bound for its control signal.

Chapter 2 presents the L1 adaptive controller for systems in the presence of matcheduncertainties. It starts from linear systems with constant unknown parameters and developsthe proofs of stability and the performance bounds. The results in this section prove that theL1 adaptive controller leads to guaranteed, uniform, and decoupled performance boundsfor both the system’s input and output signals. First, it is proved that with fast adaptation thestate and the control signal of the closed-loop nonlinear L1 adaptive system follow the samesignals of a reference linear time-invariant (LTI) system for all t ≥ 0. As compared to theoriginal reference system of MRAC, which assumes perfect cancelation of uncertainties,the reference LTI system in L1 adaptive control theory assumes only partial cancelationof uncertainties, namely, those that are within the bandwidth of the control channel. De-spite this, and because it is an LTI system, its response scales uniformly with changes inthe initial conditions, reference inputs, and uncertain parameters. Therefore, the responseof the closed-loop nonlinear L1 adaptive system also scales with all the changes in initial

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conditions, reference inputs, and uncertainties. Next, it is proved that this LTI referencesystem can be designed to achieve the desired control specifications. This step is the keyto the trade-off between performance and robustness and is reduced to tuning the structureand the bandwidth of a stable strictly proper bandwidth-limited linear filter. Thus, the com-plete performance bounds of the nonlinear L1 adaptive controller are presented via twoterms: the first is inversely proportional to the rate of adaptation, while the second dependsupon the bandwidth of a linear filter. This decoupling between adaptation and robustnessis the key feature of the L1 adaptive controller. The chapter proceeds by extending theclass of systems to accommodate an uncertain system input gain, time-varying parame-ters, and disturbances. A rigorous proof for a lower bound of the time-delay margin of theclosed-loop L1 adaptive system is provided in the case of open-loop linear systems withunknown constant parameters. This lower bound is computed from an LTI system, using itsphase margin and its gain crossover frequency. The loop transfer function of this LTI systemhas a decoupled structure, which allows for tuning its phase margin or time-delay marginvia the selection of the bandwidth-limited filter. The chapter proceeds by considering un-modeled actuator dynamics and nonlinear systems in the presence of unmodeled dynamicsand uses the well-known Rohrs’ example to provide further insights into the L1 adaptivecontroller. Other benchmark applications are discussed. An overview of tuning methodsfor the design of this filter for a performance–robustness trade-off is presented toward theend, and as an example, an LMI-based solution is described with certain (conservative)guarantees.

Chapter 3 extends the L1 adaptive controller to accommodate unmatched uncertain-ties. It starts with nonlinear strict-feedback systems, for which the L1 adaptive backsteppingscheme is presented. The chapter proceeds with an extension to multi-input multi-output(MIMO) nonlinear systems in the presence of general unmatched uncertainties and unmod-eled dynamics or, alternately, unknown time- and state-dependent nonlinear cross-coupling,which cannot be controlled by recursive (backstepping-type) design methods. Two differ-ent adaptive laws are introduced, one of which, being piecewise constant, is directly relatedto the sampling parameter of the CPU. There are certain advantages to this new type ofadaptive law. It updates the parametric estimate based on the hardware (CPU) providedspecification. At the sampling times, the adaptive law reduces one of the components ofthe identification error to zero, with the residual being proportional to the sampling intervalof integration. This implies that by increasing the rate of sampling, one can reduce the in-fluence of the residual term on the performance bounds. The uniform performance boundsare derived for the control signal and the system state as compared to the correspondingsignals of a bounded closed-loop reference system, which assumes partial cancelation ofuncertainties within the bandwidth of the control channel. This MIMO architecture hasbeen applied to NASA’s Generic Transport Model (GTM), which is part of the AirSTARsystem, and to Boeing’s X-48B blended wing body aircraft. Appropriate references areprovided.

Chapter 4 presents the output feedback solution. It starts by considering first-orderreference systems for performance specifications. Next, it considers more general referencesystems that do not verify the SPR property for their input-output transfer function. In thesecond case, the piecewise-constant adaptive law is invoked for compensation of the effectof uncertainties on the system’s regulated output. Similar to state-feedback architectures,a closed-loop reference system is considered, which assumes partial cancelation of uncer-tainties within the bandwidth of the control channel. However, unlike the state-feedback

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case, the sufficient condition for stability in this case couples the system uncertainty with thedesired reference-system behavior and the filter design. The two-cart benchmark example isdiscussed as an illustration of this extension. Also, the flight tests at the Naval PostgraduateSchool are based on the solutions from this chapter.

Chapter 5 presents an extension to accommodate linear time-varying (LTV) referencesystems. This extension is critical for practical applications. For example, in flight con-trol, quite often the performance specifications across the flight envelope are different atdifferent operational conditions. This leads to a time-varying reference system, the analy-sis of which cannot be captured by the tools developed in previous chapters. Appropriatemathematical tools for addressing this class of systems are presented in the appendices.The chapter also presents a complete solution for nonlinear systems in the presence ofunmodeled dynamics. The uniform performance bounds of the system state and the con-trol signal are computed with respect to the corresponding signals of an LTV referencesystem, which meets different transient specifications at different points of the operationalenvelope.

Chapter 6 summarizes some of the further extensions not captured within this book,gives an overview of the applications and the flight tests that have used this theory, andstates the open problems and the challenges for future thinking. Appropriate references areprovided.

The book concludes with appendices, where basic mathematical facts are collected tosupport the main proofs.

The book can be used for teaching a graduate-level special-topics course in robustadaptive control.

Notations

The book interchangeably uses time-domain and frequency-domain language for represen-tation of signals and systems. For example, ξ (t) and ξ (s) denote the function of time andits Laplace transform, respectively. However, this should not confuse the reader, as all theequations in the book are written using only one argument, either t or s. There are no mixednotations in any of the equations of the book. By smoothly moving from one form of rep-resentation to another, we streamlined the analysis and proofs. Whenever needed, thoroughexplanations are provided. Unless otherwise noted, || · || will be used for the 2-norm of avector. Finally, L(ξ (t)) is used to denote the Laplace transform of the time signal ξ (t).

AcknowledgmentsThe most valuable performance metric for every feedback solution is the experimentalvalidation of it. We are especially grateful to Isaac Kaminer and Vladimir Dobrokhodovfrom the Naval Postgraduate School for sharing their challenging problems with us and fortheir insights into the performance of the solutions. Their numerous successful flight testsin the context of very challenging multiconstraint applications in Camp Roberts, CA, since2006 have built strong credibility in this theory. The invaluable insights shared with us byKevin Wise and Eugene Lavretsky from Boeing and by Brett Ridgely and Richard Hindmanfrom Raytheon lie behind the assumptions, solutions, and techniques in this book. KevinWise and Eugene Lavretsky motivated many of the problems and posed a number of good

L1book2010/7/22page xvi

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xvi Preface

questions to us. Kevin provided tremendous encouragement, and if not for his efforts oftransitioning adaptive control into industry products and his feedback from that experience,this theory would not have been developed. We are grateful to Randy Beard from BrighamYoung University, Jonathan How from MIT, and Ralph Smith from North Carolina StateUniversity for their experimental results in different applications.

We would like to acknowledge a number of research scientists, postdoctoral fel-lows, and graduate students who have worked with us at different times and whose Ph.D.dissertations and technical papers contribute to the chapters in this book. Among these,we are especially thankful to Vijay Patel of the Indian Ministry of Defense for his helpand support with flight control applications during his sabbatical stay at Virginia Tech inthe initial stages of the theory’s development. Graduate students Jiang Wang, Dapeng Li,Enric Xargay, Zhiyuan Li, Evgeny Kharisov, Hui Sun, and Kwang-Ki Kim contributedto the development of the results in this book while working on their dissertations. Weare profoundly grateful to Evgeny Kharisov for his support with the editing of the bookand his careful check of all the proofs and examples and for generating a large num-ber of new simulations to support the theoretical claims. We would like to acknowledgeTyler Leman for his efforts in transitioning the theory to the X-48B platform and AlanGostin for developing the MATLAB-based toolbox in support of this theory. Enric Xar-gay’s contributions to the flight tests of NASA’s GTM (AirSTAR) model deserve specialmention. His feedback from that experience significantly enhanced the understanding ofthe theory. Enric’s help with the last proofreading of the book was extremely useful andconstructive.

We had invaluable comments from Karl J. Åström regarding the structure and thelayout of the book. His recommendations regarding the additional simulations and exampleshelped to provide deeper insights for the reader. He also helped us with the editing of thehistorical overview. We are grateful to Roberto Tempo for numerous useful discussions onthe design and the optimization problems of the methods in this book. Geir Dullerud fromthe University of Illinois at Urbana-Champaign provided a number of useful comments.Several discussions with Allen Tannenbaum and Tryphon Georgiou have influenced ourthinking over the years. We are thankful to Xiaofeng Wang for extending the theory toevent-triggered networked systems and to Tamer Basar for his interest in quantization ofthe L1 adaptive controller.

Irene Gregory from the NASA Langley Research Center (LaRC), Dave Doman andMichael Bolender from the Wright-Patterson Air Force Research Laboratory (AFRL),and John Burken from the NASA Dryden Flight Research Center have shared differ-ent mid- to high-fidelity models with us, explaining the particular challenges and objec-tives of those applications. We would like to thank Irene with a special mention for herefforts in transitioning the theory to the flight tests on the GTM at NASA LaRC. We alsowould like to acknowledge Kevin Cunningham, Austin Murch, and the rest of the staffof the AirSTAR flight test facility for sharing their insights into flight dynamics and fortheir support with implementation of the control law on the GTM. Our interactions withJohnny Evers from the Eglin AFRL, Siva Banda from the Wright-Patterson AFRL, andthe research groups they lead were always full of inspiration. The financial support forthis research was initially provided by the Air Force Office of Scientific Research, and itwas leveraged into various programs across the country, including Certification of AdvancedFlight Critical Systems at Wright-Patterson AFRL and NASA’s Integrated Resilient AircraftControl.

L1book2010/7/22page xvii

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Preface xvii

Last, but not least, we would like to thank our families for their unconditional ded-ication, love, and support, and to whom—with our humble gratitude—we dedicate thisbook.

Champaign, IllinoisStorrs, Connecticut

March 2010

Naira HovakimyanChengyu Cao

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xviii Preface

Figure 1: SIG Rascal 110 research aircraft(Camp Roberts, CA).

Figure 2: Flight tests with Naval Postgraduate School(Fort Hunter Liggett, CA).

L1book2010/7/22page xix

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Preface xix

Figure 3: AirSTAR T1 and T2 research aircraft(NASA Wallops Flight Facility, VA).

Figure 4: AirSTAR Mobile Operations Station(NASA Wallops Flight Facility, VA).

L1book2010/7/22page 1

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Chapter 1

Introduction

1.1 Historical OverviewResearch in adaptive control was motivated by the design of autopilots for highly agileaircraft that need to operate at a wide range of speeds and altitudes, experiencing largeparametric variations. In the early 1950s adaptive control was conceived and proposed asa technology for automatically adjusting the controller parameters in the face of changingaircraft dynamics [61, 126]. In [14], that period is called the brave era because “there wasa very short path from idea to flight test with very little analysis in between.” The tragicflight test of the X-15 was the first trial of an adaptive flight control system [164]. It clearlyindicated a lack of depth in understanding the robustness properties of adaptive feedbackloops.

The initial results in adaptive control were inspired by system identification [115],which led to an architecture consisting of an online parameter estimator combined withautomatic control design [16,81]. Two architectures of adaptive control emerged: the directmethod, where only controller parameters were estimated, and the indirect method, whereprocess parameters were estimated and the controller parameters were obtained using somedesign procedure. To achieve identifiability, it was necessary to introduce a condition ofpersistency of excitation [15] in order to guarantee that the parameter estimates converge.The relationships between the architectures were clarified in [48].

The progress in systems theory led to fundamental theory for development of stableadaptive control architectures (see [18,20,49,57,102,103,127,132,150,151,159] and refer-ences therein). This was accompanied by several examples, including Rohrs’example, chal-lenging the robustness of adaptive controllers in the presence of unmodeled dynamics, [147].Although [147] included a rigorous proof of the existence of two infinite-gain operators in theclosed-loop adaptive system, the explanation given for the phenomena observed in the sim-ulations, which was based on qualitative considerations, was not complete. A thorough ex-planation was provided in later papers by Åström [12] and Anderson [5]. Nevertheless, withhis example, Rohrs brought up an important point: the available adaptive control algorithmsto that date were unable to adjust the bandwidth of the closed-loop system and guarantee itsrobustness. The results and conclusions of this paper led to an ideological controversy, andother authors started to investigate the robustness and convergence of adaptive controllers.

1


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2 Chapter 1. Introduction

The works of Ioannou and Kokotovic [72–74], Peterson and Narendra [142], Kresselmeierand Narendra [96], and Narendra and Annaswamy [130] deserve special mention. In thesepapers, the authors analyzed the causes of instability and proposed damping-type modifica-tions of adaptive laws to prevent them. The basic idea of all the modifications was to limitthe gain of the adaptation loop and to eliminate its integral action. Examples of these mod-ifications are the σ -modification [74] and the e-modification [130]. All these modificationsattempted to provide a solution to the problem of parameter drift; however, they did notdirectly address the architectural problem identified by Rohrs. We notice that lack of robust-ness of adaptive controllers has been analyzed in robust control literature [55].An incompleteoverview of robustness and stability issues of adaptive controllers can be found in [5].

On the other hand, an example presented in [185] demonstrated that the system out-put can have overly poor transient tracking behavior before ideal asymptotic convergencetakes place. In [182], the author proved that it may not be possible to optimize L2 and L∞performance simultaneously by using a constant adaptation rate. Following these results,modifications of adaptive controllers were proposed in [43, 163] that render the trackingerror arbitrarily small in terms of both mean-square and L∞-bounds. Further, it was shownin [42] that the modifications proposed in [43,163] could be derived as a linear feedback ofthe tracking error, and the improved performance was obtained only due to a nonadaptivehigh-gain feedback. In [159], a composite adaptive controller was proposed, which suggestsa new adaptation law using both tracking error and prediction error and leads to less oscil-latory behavior in the presence of high adaptation gains as compared to model referenceadaptive control (MRAC). In [125], a high-gain switching MRAC technique was introducedto achieve arbitrary good transient tracking performance under a relaxed set of assumptionsas compared to MRAC, and the results were shown to be of existence type only. In [131],a multiple model switching scheme was proposed to improve the transient performanceof adaptive controllers. In [10], it was shown that an arbitrarily close transient bound canbe achieved by enforcing a parameter-dependent persistent excitation condition. In [101],computable L2- and L∞-bounds for the output tracking error signals were obtained for aspecial class of adaptive controllers using backstepping. The underlying linear nonadaptivecontroller possesses a parametric robustness property. However, for a large parametric un-certainty it requires high-gain feedback. In [136], dynamic certainty equivalent controllerswith unnormalized estimators were used for adaptation, which permit derivation of a uni-form upper bound for the L2-norm of the tracking error in terms of the initial parameterestimation error. In the presence of sufficiently small initial conditions, the author provedthat the L∞-norm of the tracking error is upper bounded by the L∞-norm of the referenceinput. In [9, 50, 137], a differential game theoretic type H∞ approach was investigated forachieving arbitrarily close disturbance attenuation for tracking performance, albeit at theprice of increased control effort. In [187], a new certainty-equivalence-based adaptive con-troller was presented using a backstepping-type controller with a normalized adaptive lawto achieve asymptotic stability and guarantee performance bounds comparable with the tun-ing functions scheme, without the use of higher-order nonlinearities. References [128,129]developed the supervisory control approach that defines a fast switching scheme betweencandidate controllers leading to guaranteed performance bounds. However, robustness ofthese schemes to unmodeled dynamics appears to be limited by the frequency of switch-ing [6, 66].

As compared to the linear systems theory, several important aspects of the transientperformance analysis seem to be missing in these efforts. First, the bounds are computed


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1.1. Historical Overview 3

for tracking errors only, not for control signals. Although the latter can be deduced from theformer, it is straightforward to verify that the ability to adjust the former may not extend tothe latter in case of nonlinear control laws. Second, since the purpose of adaptive control isto ensure stable performance in the presence of modeling uncertainties, one needs to ensurethat (admissible) changes in reference commands and system dynamics due to possible faultsor unexpected uncertainties do not lead to unacceptable transient deviations or oscillatorycontrol signals, implying that a retuning of adaptation parameters is required. Finally, oneneeds to ensure that the modifications or solutions, suggested for performance improvementof adaptive controllers, are not achieved via high-gain feedback.

In brief summary, the development of the theory of adaptive control over the years hastaken rather the trend of defining a larger and larger class of systems, for which a Lyapunovproof can be done for asymptotic stability. At which location should the uncertainty appear,what should be the degree of mismatch, how should the adaptive law be modified, etc., to geta negative definite (semidefinite) derivative of the associated candidate Lyapunov functionfor a new class of systems? These questions or one of them is present in almost every paperaddressing the next stage of development in the theory of adaptive control. Significant effortshave been reported on relaxation of the matching conditions by extending the backstepping-design approach to a broader class of systems, including strict-parametric feedback andfeedforward systems [9, 45, 97, 100, 137, 138], analysis of robustness of these schemes tounmodeled dynamics [4,8,70,71,77,134], extensions to output feedback with an objective toachieve global or semiglobal output feedback stabilization [76,84,98,99,119,120], extensionto systems with time-varying parameters [68, 122, 135, 186], relaxation of the relative de-gree [strictly positive real (SPR)] requirement via input-filtered transformations [118,121],extension to nonminimum phase systems [69, 75], etc.

These fundamental results provide sufficient conditions on the bounds of uncertaintiesand initial conditions, which would guarantee that with the given adaptive feedback archi-tecture, the signals in the feedback loop remain bounded. Though very important, whendealing with practical applications, boundedness, ultimate boundedness, or even asymp-totic convergence are weak properties for nonlinear (adaptive) feedback systems. On onehand, unmodeled dynamics, latencies, and noise require precise quantification of the ro-bustness and the stability margins of the underlying feedback loop. On the other hand,performance requirements in real applications necessitate a predictable response for theclosed-loop system, dependent upon the changes in system dynamics. In adaptive control,the nature of the adaptation process plays a central role in both robustness and performance.Ideally, one would like adaptation to correctly respond to all the changes in initial condi-tions, reference inputs, and uncertainties by quickly identifying a set of control parametersthat would provide a satisfactory system response. This, of course, demands fast estimationschemes with high adaptation rates and, as a consequence, leads to the fundamental questionof determining the upper bound on the adaptation rate that would not result in poor robust-ness characteristics. We notice that the results available in the literature consistently limitedthe rate of variation of uncertainties, by providing examples of destabilization due to fastadaptation [75, p. 549], while the transient performance analysis was continually reducedto persistency of excitation-type assumptions, which, besides being a highly undesirablephenomenon, cannot be verified a priori. The lack of analytical quantification of the rela-tionship between the rate of adaptation, the transient response, and the robustness marginsled to gain-scheduled designs of adaptive controllers, examples of which are the successfulflight tests of the late 1990s by the Air Force and Boeing [175, 176]. The flight tests relied


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on intensive Monte Carlo analysis for determination of the best rate of adaptation for var-ious flight conditions. It was apparent that fast adaptation was leading to high frequenciesin control signals and increased sensitivity to time delays. The fundamental question wasthus reduced to determining an architecture, which would allow for fast adaptation withoutlosing robustness. It was clearly understood that such an architecture can reduce the amountof gain scheduling, and possibly eliminate it, as fast adaptation—in the presence of guar-anteed robustness—should be able to compensate for the negative effects of rapid variationof uncertainties on the system response.

The L1 adaptive control theory addressed precisely this question by setting an archi-tecture in place for which adaptation is decoupled from robustness. The speed of adaptationin these architectures is limited only by the available hardware, while robustness is re-solved via conventional methods from classical and robust control. The architectures ofL1 adaptive control theory have guaranteed transient performance and guaranteed robust-ness in the presence of fast adaptation, without introducing or enforcing persistence ofexcitation, without any gain scheduling in the controller parameters, and without resortingto high-gain feedback. With L1 adaptive controller in the feedback loop, the response ofthe closed-loop system can be predicted a priori, thus significantly reducing the amount ofMonte Carlo analysis required for verification and validation of such systems. These fea-tures of L1 adaptive control theory were verified—consistently with the theory—in a largenumber of flight tests and in mid- to high-fidelity simulation environments [19, 35, 36, 46,51, 58, 59, 67, 82, 83, 88, 94, 104–106, 110, 124, 140, 141, 170, 172].

To facilitate the development of L1 adaptive control theory, in the next section we in-troduce two equivalent architectures of MRAC, which lead to the same error dynamics fromthe same initial conditions. We later use one of these structures as a basis for developmentof the main results in this book.

1.2 Two Different Architectures of Adaptive ControlIn this section we present two different, but equivalent, architectures of adaptive control.Although their implementation is different, they both lead to the same error dynamics fromthe same initial conditions. The difference in their implementation principle is the key tothe development of L1 adaptive control architectures in this book.

1.2.1 Direct MRAC

Let the system dynamics propagate according to the following differential equation:

x(t) = Amx(t)+b(u(t)+k�x x(t)

), x(0) = x0 ,

y(t) = c�x(t) ,(1.1)

where x(t) ∈ Rn is the state of the system (measured),Am ∈ R

n×n is a known Hurwitz matrixthat defines the desired dynamics for the closed-loop system, b, c ∈ R

n are known constantvectors, kx ∈ R

n is a vector of unknown constant parameters, u(t) ∈ R is the control input,and y(t) ∈ R is the regulated output. Given a uniformly bounded piecewise-continuous


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1.2. Two Different Architectures of Adaptive Control 5

reference input r(t) ∈ R, the objective is to define an adaptive feedback signal u(t) such thaty(t) tracks r(t) with desired specifications, while all the signals remain bounded.

The MRAC architecture proceeds by considering the nominal controller

unom(t) = −k�x x(t)+kgr(t) , (1.2)

where

kg � 1

c�A−1m b

. (1.3)

This nominal controller assumes perfect cancelation of the uncertainties in (1.1) and leadsto the desired (ideal) reference system

xm(t) = Amxm(t)+bkgr(t) , xm(0) = x0 ,

ym(t) = c�xm(t) ,(1.4)

where xm(t) ∈ Rn is the state of the reference model. The choice of kg according to (1.3)

ensures that ym(t) tracks step reference inputs with zero steady-state error.The direct model reference adaptive controller is given by

u(t) = −k�x (t)x(t)+kgr(t) , (1.5)

where kx(t) ∈ Rn is the estimate of kx . Substituting (1.5) into (1.1) yields the closed-loop

system dynamics

x(t) = (Am−bk�x (t))x(t)+bkgr(t) , x(0) = x0 ,

y(t) = c�x(t) ,

where kx(t) � kx(t)−kx denotes the parametric estimation error.Letting e(t) � xm(t) − x(t) be the tracking error signal, the tracking error dynamics

can be written ase(t) = Ame(t)+bk�x (t)x(t) , e(0) = 0. (1.6)

The update law for the parametric estimate is given by

˙kx(t) = −�x(t)e�(t)Pb, kx(0) = kx0 , (1.7)

where � ∈ R+ is the adaptation gain and P = P� > 0 solves the algebraic Lyapunov

equationA�mP +PAm = −Q

for arbitrary Q =Q� > 0. The block diagram of the closed-loop system is given in Fig-ure 1.1.

Consider the following Lyapunov function candidate:

V (e(t), kx(t)) = e�(t)Pe(t)+ 1

�k�x (t)kx(t) . (1.8)


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Its time derivative along the system trajectories (1.6)–(1.7) is given by

V (t) = −e�(t)Qe(t)+2e�(t)Pbk�x (t)x(t)+ 2

�k�x (t) ˙kx(t)

= −e�(t)Qe(t)+2k�x (t)

(1

�

˙kx(t)+x(t)e�(t)Pb

)= −e�(t)Qe(t) ≤ 0.

Hence, the equilibrium of (1.6)–(1.7) is Lyapunov stable, i.e., the signals e(t), kx(t) arebounded. Since x(t) = xm(t) − e(t), and xm(t) is the state of a stable reference model,then x(t) is bounded. To show that the tracking error converges asymptotically to zero, wecompute the second derivative of V (e(t), kx(t)) as

V (t) = −2e�(t)Qe(t) .

It follows from (1.6) that e(t) is uniformly bounded, and hence V (t) is bounded, implyingthat V (t) is uniformly continuous. Application of Barbalat’s lemma (see A.6.1) yields

limt→∞ V (t) = 0,

which consequently proves that e(t) → 0 as t → ∞. Thus, x(t) asymptotically converges toxm(t). This in turn implies that y(t) = c�x(t) asymptotically converges to ym(t) = c�xm(t),which follows r(t) with desired specifications.

Notice that asymptotic convergence of parametric estimation errors to zero is notguaranteed. The parametric estimation errors are guaranteed only to stay bounded.

1.2.2 Direct MRAC with State Predictor

Next, we consider a reparameterization of the above architecture using a state predictor (oridentifier), given by

˙x(t) = Amx(t)+b(u(t)+ k�x (t)x(t)) , x(0) = x0 ,

y(t) = c�x(t) ,(1.9)

where x(t) ∈ Rn is the state of the predictor. The system in (1.9) replicates the system

structure from (1.1) with the unknown parameter kx replaced by its estimate kx(t). Bysubtracting (1.1) from (1.9), we obtain the prediction error dynamics (or identification errordynamics), independent of the control choice,

˙x(t) = Amx(t)+bk�x (t)x(t) , x(0) = 0,

where x(t) � x(t)−x(t) and kx(t) � kx(t)−kx . Notice that these error dynamics are identicalto the error dynamics in (1.6).

Next, let the adaptive law for kx(t) be given as

˙kx(t) = −�x(t)x�(t)Pb , kx(0) = kx0 , (1.10)


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1.2. Two Different Architectures of Adaptive Control 7

where � ∈ R+ is the adaptation rate and A�

mP +PAm = −Q,Q=Q� > 0. This adaptivelaw is similar to (1.7) in its structure, except that the tracking error e(t) is replaced by theprediction error x(t). The choice of the Lyapunov function candidate

V (x(t), kx(t)) = x�(t)P x(t)+ 1

�k�x (t)kx(t)

leads toV (t) = −x�(t)Qx ≤ 0,

implying that the errors x(t) and kx(t) are uniformly bounded. Notice, however, that with-out introducing the feedback signal u(t) one cannot apply Barbalat’s lemma to concludeasymptotic convergence of x(t) to zero. Both x(t) and x(t) can diverge at the same rate,keeping x(t) uniformly bounded.

If we use (1.5) in (1.9), with account of (1.10), the closed-loop state predictor replicatesthe bounded reference system of (1.4):

˙x(t) = Amx(t)+bkgr(t) , x(0) = x0 ,

y(t) = c�x(t) .

Hence, Barbalat’s lemma can be invoked to conclude that x(t) → 0 as t → ∞. The blockdiagram of the closed-loop system with the predictor is given in Figure 1.2.

Figures 1.1 and 1.2 illustrate the fundamental difference between the direct MRACand the predictor-based adaptation. In Figure 1.2, the control signal is provided as input toboth systems, the system and the predictor, while in Figure 1.1 the control signal serves onlyas input to the system. This feature is the key to the development of L1 adaptive controlarchitectures with quantifiable performance bounds.

1.2.3 Tuning Challenges

From the above Lyapunov analysis, we notice that the tracking error can be upper boundedin the following way:

‖e(t)‖(= ‖x(t)‖) ≤√

V (t)

λmin(P )≤√

V (0)

λmin(P )= ‖kx(0)‖√

λmin(P )�, ∀ t ≥ 0.

This bound shows that the tracking error can be arbitrarily reduced for all t ≥ 0(including the transient phase) by increasing the adaptation gain � [100]. However, fromthe control law in (1.5) and the adaptive laws in (1.7) and (1.10), it follows that largeadaptive gains result in high-gain feedback control, which manifests itself in high-frequencyoscillations in the control signal and reduced tolerance to time delays. Moreover, applicationsrequiring identification schemes with time scales comparable with those of the closed-loopdynamics appear to be extremely challenging due to undesirable interactions of the twoprocesses [5]. Due to the lack of systematic design guidelines to select an adequate adaptationgain, tuning of such applications is being commonly resolved by either computationallyexpensive Monte Carlo simulations or trial-and-error methods following some empiricalguidelines or engineering intuition.As a consequence, proper tuning of MRAC architectures(or their equivalent state-predictor-based reparameterizations) represents a major challengeand has largely remained an open question in the literature.


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r u

kx

x

xm

e

x(t) = Amx(t)+b(u(t)+k�x x(t))y(t) = c�x(t)

xm(t) = Amxm(t)+bkgr(t)ym(t) = c�xm(t)

˙kx (t) = −�x(t)e�(t)Pb

u(t) = −k�x (t)x(t)+kgr(t)

System

Reference Model

Adaptation Law

Control Law

Figure 1.1: Closed-loop direct MRAC architecture.

r

u

kx

x

x

x

x(t) = Amx(t)+b(u(t)+k�x x(t))y(t) = c�x(t)

˙x(t) = Amx(t)+b(u(t)+ k�x (t)x(t))y(t) = c�x(t)

˙kx (t) = −�x(t)x�(t)Pb

u(t) = −k�x (t)x(t)+kgr(t)

System

State Predictor

Adaptation Law

Control Law

Figure 1.2: Closed-loop MRAC architecture with state predictor.

1.3 Saving the Time-Delay MarginNext we will introduce the key ideas of the L1 adaptive controller, which enables fastadaptation with guaranteed robustness. We will start with a simple stable scalar system withconstant disturbance, which can be analyzed by resorting to tools from classical control. Wenotice that in this case, MRAC reduces to a linear (model-following) integral controller.Since the closed-loop system remains linear, we use the Nyquist criterion to analyze stabilityand robustness of this system. Taking advantage of its linear structure, we present (i) someof the benefits of L1 adaptive control architectures and (ii) different concepts and toolsthat will be used throughout the book. In particular, we will show that fast adaptationof L1 adaptive control architectures is beneficial for robustness. We will also derive theuniform performance bounds of the L1 adaptive controller, for both the state and the controlsignal, and show the role of the bandwidth-limited filter of the L1 architecture in obtainingthese uniform bounds.

Toward that end, consider the scalar system

x(t) = −x(t)+ θ +u(t) , x(0) = x0 , (1.11)


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1.3. Saving the Time-Delay Margin 9

�s

1s+1

xxm u

θ

Figure 1.3: Closed-loop system with MRAC-type integral controller.

where θ is the unknown constant to be rejected by the control input u(t). Let the objectivebe stabilization of the origin. For this system, the MRAC architecture described in (1.4)and (1.5) reduces to an integral controller of the structure

u(t) = −θ (t) , (1.12)

where θ (t) is the estimate of θ , given by

˙θ (t) = −�(xm(t)−x(t)) , θ (0) = θ0 , � > 0, (1.13)

and xm(t) is the reference signal, generated by the system

xm(t) = −xm(t) , xm(0) = x0 .

We notice that this reference system is obtained from the original system (1.11) by substitu-tion of the ideal nominal controller unom(t) = −θ into it, thus assuming perfect cancelationof the uncertain parameter θ in the system (1.11). The block diagram of the closed-loopsystem is shown in Figure 1.3.

The loop transfer function of this system (with negative feedback) is

L1(s) = �

s(s+1). (1.14)

Because the closed-loop system remains linear time-invariant (LTI), one can use standardtools from classical control to analyze the stability margins of this system. The two mostcommonly used stability margins are the gain and the phase margin. From Figure 1.4(a),it is obvious that the Nyquist plot of L1(s) never crosses the negative part of the real line;therefore, the closed-loop system has infinite gain margin (gm = ∞). The gain crossoverfrequency ωgc can be computed from

|L1(jωgc)| = �

ωgc

√ω2gc+1

= 1,

which leads to the phase margin

φm = π + L1(jωgc) = arctan

(1

ωgc

).

Careful analysis indicates that increasing � leads to higher gain crossover frequency andconsequently reduces the phase margin. The reduction of phase margin with large � can


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also be observed in Figure 1.4(a). So, if increasing � improves the tracking performancefor all t ≥ 0, including the transient phase, then it obviously hurts the robustness (or rel-ative stability) of the closed-loop system. Thus, the adaptation rate � is the key to thetrade-off between performance and robustness. Since tracking and robustness cannot beachieved simultaneously, there is nothing surprising about this, but we would like to ex-plore if the architecture can be modified so that the trade-off between tracking and robust-ness is resolved differently and the adaptation gain � can be safely increased for transientperformance improvement without hurting the robustness of the closed-loop system (seeSection 1.2.3).

To obtain the L1 adaptive controller for this system, the controller in (1.12)–(1.13)will be modified in two ways. First, we introduce the state predictor,

˙x(t) = −x(t)+ θ (t)+u(t) , x(0) = x0 ,

which leads to the following prediction error dynamics, independent of the control choice:

˙x(t) = −x(t)+ θ (t) , x(0) = 0, (1.15)

where x(t) � x(t) − x(t) and θ (t) � θ (t) − θ . The parametric estimate, given by (1.13), isthus replaced by

˙θ (t) = −�x(t) , θ (0) = θ0 , � > 0.

Next, instead of choosing the adaptive controller as u(t) = −θ (t), we use a low-pass filteredversion of θ (t),

u(s) = −C(s)θ (s) , (1.16)

where u(s) and θ (s) are the Laplace transforms of u(t) and θ (t), respectively, and C(s) isa bounded-input bounded-output (BIBO) stable strictly proper transfer function subject toC(0) = 1 with zero initialization for its state-space realization. The block diagram of thissystem is given in Figure 1.5. In the foregoing analysis, we further consider a first-orderlow-pass filter

C(s) = ωc

s+ωc ; (1.17)

however, similar results can be obtained using more complex filters. The loop transferfunction of this system (with negative feedback) is

L2(s) = �C(s)

s(s+1)+�(1−C(s)). (1.18)

Notice that in the absence of the filter, i.e., with C(s) = 1, the controller in (1.16) reducesto the MRAC-type integral controller introduced earlier, and (1.18) reduces to (1.14), thatis, L2(s) = L1(s).

Although (1.18) has a more complex structure than (1.14), the Nyquist plot in Fig-ure 1.4(b) shows that the phase and the gain margins of the L1 controller are not significantlyaffected by large values of �. The effect of the adaptive gain on the robustness margins ofthe two closed-loop systems is clearly presented in Figure 1.6. The figure shows that, whilethe phase margin of the MRAC-type integral controller vanishes as one increases the adap-tation gain �, the L1 adaptive controller has a guaranteed bounded-away-from-zero phaseand gain margins in the presence of fast adaptation.


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1.3. Saving the Time-Delay Margin 11

−1 −0.8 −0.6 −0.4 −0.2 0

−1

−0.8

−0.6

−0.4

−0.2

0

Nyquist Diagram

Real Axis

Imag

inar

y A

xis

Γ=10Γ=100Γ=1000

(a) Integral controller

−1.2 −1 −0.8 −0.6 −0.4 −0.2 0

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

Nyquist Diagram

Real Axis

Imag

inar

y A

xis

Γ=10Γ=100Γ=1000

(b) L1 controller

Figure 1.4: Nyquist plots for the loop transfer functions.

−C(s) 1s+1

1s+1

−�s

u

x

xθ

θ

Figure 1.5: Closed-loop system with L1 adaptive controller.

Further, notice that as �→ ∞, the expression in (1.18) leads to the following limitingloop transfer function:

L2l(s) = C(s)

1−C(s)= ωc

s. (1.19)

This loop transfer function has an infinite gain margin (gm = ∞) and a phase margin ofφm = π/2. However, from Figure 1.6(a), we notice that the gain margin is always finite andactually converges to gm = 6.02 dB with the increase of�. We note that the (high-frequency)dynamics of the adaptation loop do not appear in the limiting loop transfer function in (1.19).Then, since the phase crossover frequency tends to infinity as the adaptation gain� increases,this limiting loop transfer function cannot be used to analyze the gain margin of the closed-loop system with the L1 adaptive controller. However, the gain crossover frequency staysin the low-frequency range, where the limiting loop transfer function in (1.19) is a goodapproximation of the actual loop transfer function in (1.18). Consequently, the limiting looptransfer function can be used to analyze the phase margin of the closed-loop adaptive system.

One can equivalently measure the robustness of the system by computing its time-delay margin, which is defined as the amount of time delay that brings the system to the


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0 200 400 600 800 10000

5

10

15

Adaptive gain

Gai

n m

argi

n (d

B)

L1

(a) gain margin

0 200 400 600 800 10000

20

40

60

80

100

Adaptive gain

Pha

se m

argi

n (d

eg)

L1PI

(b) phase margin

Figure 1.6: Effect of high adaptation gain on the stability margins.

verge of instability. The additional phase lag in the system due to a time delay τ is given byφτ (ω) = (e−τjω)= −τω. Recalling the definition of the phase margin, one can computethe time-delay margin T as the amount of delay introduced in the system that reduces thephase margin to zero:

φm = T ωgc ⇒ T = φm

ωgc.

From (1.19), it follows that ωgc = ωc, which further implies that the L1 adaptive controllerhas the following time-delay margin as �→ ∞:

T = φm

ωgc= π

2ωc.

Hence, we observe that the L1 adaptive controller, defined by (1.16), retains guaran-teed robustness in the presence of large values of�, while the MRAC-type integral controllerobviously loses its phase margin in the presence of fast adaptation.

1.4 Uniformly Bounded Control SignalNext, we analyze a key property of the L1 adaptive controller, which is inherently related tothe robustness features discussed above. We start by considering the following closed-loopstructure:

xref (s) = 1

s+1

(θ

s+uref (s)

)+ x0

s+1,

uref (s) = −C(s)θ

s.

(1.20)

This system is constructed from (1.11) and (1.16) by using θ/s = L(θ ) instead of θ (s)in (1.16) and, hence, represents a closed-loop architecture using the ideal nonadaptive ver-sion of the L1 controller. We will refer to this system as a (closed-loop) reference system, asit is with respect to this system that we are able to compute uniform performance bounds.


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1.4. Uniformly Bounded Control Signal 13

Notice that the reference controller uref (s) = −C(s) θs, as compared to the nominal con-

troller unom(s) = − θs

of MRAC, assumes only partial cancelation of uncertainties, i.e., itcompensates only for the uncertainties within the bandwidth of C(s). This reference systemdefines the best achievable performance with the L1 adaptive architecture. The response ofthis closed-loop reference system can be written as

xref (s) = 1

s(s+1)(1−C(s))θ + x0

s+1.

Similarly, the response of the system in (1.11) with the L1 controller in (1.16) takes theform (in the frequency domain)

x(s) = 1

s+1

(θ

s−C(s)θ (s)

)+ x0

s+1= 1

s+1

((1−C(s))

θ

s−C(s)θ (s)

)+ x0

s+1,

where θ (s) and θ (s) are the Laplace transforms of θ (t) and θ (t), respectively. Notice that

xref (s)−x(s) = 1

s+1C(s)θ (s) . (1.21)

Also, it follows from (1.15) that

x(s) = 1

s+1θ (s) , (1.22)

which allows for rewriting (1.21) as

xref (s)−x(s) = C(s)x(s) . (1.23)

Moreover, notice that

θ (s) = θ (s)− θ/s = −�x(s)/s+ θ0/s− θ/s .

Substituting the above expression into (1.22) and solving for x(s) leads to

x(s) = − 1

s2 + s+� (θ − θ0) .

We can now take the inverse Laplace transform of x(s) for � > 1/4 to obtain

x(t) = − θ − θ0√�−1/4

e− 12 t sin(

√�−1/4 t) . (1.24)

This expression yields the following uniform upper bound on the prediction error:

|x(t)| ≤ |θ − θ0|√�−1/4

, ∀ t ≥ 0.

Letting γ0 � |θ − θ0|/√�−1/4, we can write

|x(t)| ≤ γ0 , ∀ t ≥ 0.

Notice that lim�→∞ γ0 = 0. Also notice from (1.24) that limt→∞ x(t) = 0.


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From (1.23) we can also derive the following uniform upper bound:

|xref (t)−x(t)| =∣∣∣∣∫ t

0hc(τ )x(t− τ )dτ

∣∣∣∣≤∫ t

0|hc(τ )x(t− τ )|dτ

≤ γ0

∫ t

0|hc(τ )|dτ ≤ γ0

∫ ∞

0|hc(τ )|dτ , ∀ t ≥ 0,

where hc(t) is the impulse response ofC(s). In particular, for theC(s) in (1.17), the impulseresponse can be explicitly computed, leading to the following uniform upper bound:

|xref (t)−x(t)| ≤ γ0 , ∀ t ≥ 0.

This implies that the error between the closed-loop system with the L1 adaptive controllerand the closed-loop reference system, which uses the reference controller, can be uniformlybounded by a constant inverse proportional to the square root of the rate of adaptation.

Similarly, using (1.16), (1.20), and (1.22), we can derive

uref (s)−u(s) = C(s)θ (s) = C(s)(s+1)x(s) . (1.25)

DenotingHu(s) �C(s)(s+1) and letting hu(t) be the impulse response forHu(s), we obtainthe following upper bound:

|uref (t)−u(t)| ≤ γ0

∫ ∞

0|hu(τ )|dτ , ∀ t ≥ 0. (1.26)

Because C(s) is strictly proper and BIBO stable, Hu(s) � C(s)(s+1) is proper and BIBOstable, and hence it has uniformly bounded impulse response, that is

∫∞0 |hu(τ )|dτ <∞.

Further, since lim�→∞ γ0 = 0, we can conclude from (1.26) that the time history of theL1 adaptive controller can be rendered arbitrarily close to the one of the reference controllerfor all t ≥ 0 by increasing the rate of adaptation �.

Notice that without the low-pass filter, i.e., with C(s) = 1, equation (1.25) reduces to

uref (s)−u(s) = (s+1)x(s) .

From this expression, it is obvious that the transfer function from x(t) to uref (t) −u(t) isimproper, and hence, in the absence of the filter C(s), one cannot uniformly upper bound|uref (t)−u(t)| as we did in (1.26).

This simple analysis illustrates the role of C(s) toward obtaining a uniform perfor-mance bound for the control signal of the L1 adaptive control architecture, as comparedto its nonadaptive version (which is uniformly bounded by definition). We further noticethat this uniform bound is inverse proportional to the square root of the rate of adaptation,similar to the tracking error. Thus, both performance bounds can be systematically improvedby increasing the rate of adaptation. The remaining issue is the design of the low-pass lin-ear filter C(s) to ensure that the reference system in (1.20) achieves desired performancespecifications in the presence of unknown θ .

In the remainder of this book, we will show that, similar to this simple exampleand with appropriate extension of the above described concepts to nonlinear closed-loopadaptive systems, the L1 adaptive control theory shifts the tuning issue from determiningthe rate of the nonlinear gradient minimization scheme to the design of a linear strictly


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1.4. Uniformly Bounded Control Signal 15

proper and stable filter, implying that the trade-off between performance and robustnessof the closed-loop adaptive system can be systematically addressed using well-establishedtools from classical and robust control.

Finally, we note that the uniform bounds for the system’s state and control signals areexpressed in terms of the impulse response of proper BIBO-stable transfer functions, whichcorrespond to the L1-norms of the underlying systems. Consequently, the correspondingcontrol architectures are referred to as L1 adaptive controllers.


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Chapter 2

State Feedback in the Presenceof Matched Uncertainties

In this chapter we present the full state-feedback solution for several different classes ofsystems in the presence of matched uncertainties. We start with linear systems with con-stant unknown parameters and present the L1 adaptive controller for this class of systems.We derive the performance bounds of this controller and show that these can be clearlydecoupled into two distinct components, the adaptation and the robustness bounds. Theadaptation bounds can be improved by increasing the rate of adaptation, while the ro-bustness bounds can be appropriately addressed via known methods from linear-systemstheory. We proceed by considering linear time-varying systems in the presence of unknownsystem input gain and present a new architecture for this class of systems. We analyzethe time-delay margin for the case of constant unknown parameters and provide a guaran-teed lower bound for it via the phase margin of an auxiliary LTI system. Then, we extendthe L1 adaptive controller to nonlinear systems in the presence of unmodeled dynamicsand analyze several well-known benchmark examples from the literature. We further dis-cuss various methods for the design of the underlying filter toward achieving the desiredperformance-robustness trade-off for the closed-loop adaptive system with the L1 adaptivecontroller and provide a conservative, but guaranteed, solution via linear matrix inequali-ties (LMIs).

2.1 Systems with Unknown Constant ParametersThis section considers LTI systems in the presence of unknown constant parameters. TheL1 adaptive controller ensures uniformly bounded transient and steady-state tracking forboth of the system’s signals, input and output, as compared to the same signals of abounded reference LTI system, which assumes partial cancelation of uncertainties withinthe bandwidth of the control channel. The time histories of the signals of this referenceLTI system can be made arbitrarily close to the signals of a different LTI system, calleddesign system, the output of which can be used for control specifications [29]. This decou-pling of the performance bounds between adaptation and robustness is further illustrated insimulations.

17


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18 Chapter 2. State Feedback in the Presence of Matched Uncertainties

2.1.1 Problem Formulation

Consider the class of systems

x(t) = Ax(t)+b(u(t)+ θ�x(t)) , x(0) = x0 ,

y(t) = c�x(t) ,(2.1)

wherex(t) ∈ Rn is the system state vector (measured);u(t) ∈ R is the control signal;b, c ∈ R

n

are known constant vectors; A is the known n× n matrix, with (A, b) controllable; θ isthe unknown parameter, which belongs to a given compact convex set θ ∈ � ⊂ R

n; andy(t) ∈ R is the regulated output. In this section we present an adaptive control solution,which ensures that the system output y(t) follows a given piecewise-continuous boundedreference signal r(t) with quantifiable transient and steady-state performance bounds.

2.1.2 L1 Adaptive Control Architecture

Consider the control structure

u(t) = um(t)+uad(t) , um(t) = −k�mx(t) , (2.2)

where km ∈ Rn renders Am �A−bk�m Hurwitz, while uad(t) is the adaptive component, to

be defined shortly. The static feedback gain km leads to the following partially closed-loopsystem:

x(t) = Amx(t)+b(θ�x(t)+uad(t)) , x(0) = x0 ,

y(t) = c�x(t) .(2.3)

For the linearly parameterized system in (2.3), we consider the state predictor

˙x(t) = Amx(t)+b(θ�(t)x(t)+uad(t)) , x(0) = x0 ,

y(t) = c�x(t) ,(2.4)

where x(t) ∈ Rn is the state of the predictor and θ (t) ∈ R

n is the estimate of the parameter θ ,governed by the following projection-type adaptive law:

˙θ (t) = �Proj(θ (t),−x�(t)Pbx(t)) , θ (0) = θ0 ∈� , (2.5)

where x(t) � x(t) − x(t) is the prediction error, � ∈ R+ is the adaptation gain, and P =

P�> 0 solves the algebraic Lyapunov equationA�mP +PAm= −Q for arbitrary symmetric

Q = Q� > 0. The projection is confined to the set � (see Definition B.3). The Laplacetransform of the adaptive control signal is defined as

uad(s) = −C(s)(η(s)−kgr(s)

), (2.6)

where r(s) and η(s) are the Laplace transforms of r(t) and η(t) � θ�(t)x(t), respectively,kg � −1/(c�A−1

m b), and C(s) is a BIBO-stable and strictly proper transfer function withDC gain C(0) = 1, and its state-space realization assumes zero initialization.


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2.1. Systems with Unknown Constant Parameters 19

L1 Adaptive Controller

x(t) = Amx(t)+b(uad(t)+ θ�x(t))y(t) = c�x(t)

˙x(t) = Amx(t)+b(uad(t)+ θ�(t)x(t))y(t) = c�x(t)

˙θ (t) = �Proj(θ (t),−x�(t)Pbx(t))

C(s)kgr u x

x

xθ�x

Figure 2.1: Closed-loop L1 adaptive system.

The L1 adaptive controller is defined via the relationships in (2.2), (2.4)–(2.6), withkm and C(s) verifying the following L1-norm condition:

λ� ‖G(s)‖L1L < 1, (2.7)

whereG(s) �H (s)(1−C(s)) , H (s) � (sI−Am)−1b , L� max

θ∈� ‖θ‖1 . (2.8)

The L1 adaptive control architecture with its main elements is represented in Figure 2.1.

2.1.3 Analysis of the L1 Adaptive Controller: Scaling

Closed-Loop Reference System

Consider the following nonadaptive version of the adaptive control system in (2.1), (2.2),(2.6), which defines the closed-loop reference system for the class of systems in (2.1):

xref (t) = Axref (t)+b(θ�xref (t)+uref (t)) , xref (0) = x0 ,

uref (s) = −C(s)(θ�xref (s)−kgr(s))−k�mxref (s) ,

yref (s) = c�xref (s) .

(2.9)

As compared to the nominal controller of MRAC in (1.2), the controller in (2.9) attemptsto compensate only for uncertainties in the system that are within the bandwidth of C(s).The block diagram of this system is given in Figure 2.2. Notice that C(s) = 1 leads to thereference model of MRAC, which was introduced in (1.4).

Lemma 2.1.1 If ‖G(s)‖L1L < 1, then the system in (2.9) is bounded-input bounded-state(BIBS) stable with respect to r(t) and x0.

Proof. From the definition of the closed-loop reference system in (2.9), it follows that

xref (s) =H (s)kgC(s)r(s)+G(s)θ�xref (s)+xin(s) ,


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System with unknown parameter

(sI−A)−1b c�kg

k�m

C(s)

θ�

θ�

xref yrefurefr

Figure 2.2: Closed-loop reference system.

where xin(s) � (sI −Am)−1x0. Recalling the fact that H (s), C(s), and G(s) are properBIBO-stable transfer functions, it follows from (2.9) that for all τ ∈ [0, ∞) the followingbound holds:

‖xref τ‖L∞ ≤ ‖H (s)kgC(s)‖L1‖rτ‖L∞ +‖G(s)θ�‖L1‖xref τ‖L∞ +‖xinτ‖L∞ .

Since Am is Hurwitz, xin(t) is uniformly bounded. Definition A.7.4 and the relationshipsin (2.7) and (2.8) imply that

‖G(s)θ�‖L1 = maxi=1,...,n

‖Gi(s)‖L1

n∑j=1

|θj | ≤ ‖G(s)‖L1L < 1. (2.10)

Consequently,

‖xref τ‖L∞ ≤ ‖H (s)kgC(s)‖L1

1−‖G(s)θ�‖L1

‖rτ‖L∞ + ‖xinτ‖L∞1−‖G(s)θ�‖L1

. (2.11)

Since r(t) and xin(t) are uniformly bounded, and (2.11) holds uniformly for all τ ∈ [0, ∞),xref (t) is uniformly bounded. Boundedness of yref (t) follows from its definition. This com-pletes the proof. �

Transient and Steady-State Performance

The following error dynamics can be derived from (2.3) and (2.4):

˙x(t) = Amx(t)+bθ�(t)x(t) , x(0) = 0, (2.12)

where θ (t) � θ (t)− θ . Letting η(t) � θ�(t)x(t), with η(s) being its Laplace transform, theerror dynamics in (2.12) can be written in the frequency domain as

x(s) =H (s)η(s) . (2.13)

Lemma 2.1.2 The prediction error in (2.12) is uniformly bounded:

‖x‖L∞ ≤√

θmax

λmin(P )�, θmax � 4max

θ∈� ‖θ‖2 , (2.14)

where λmin(P ) is the minimum eigenvalue of P .


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Proof. Consider the following Lyapunov function candidate:

V (x(t), θ (t)) = x�(t)P x(t)+ 1

�θ�(t)θ (t) .

Using Property B.2 of the projection operator, we can upper bound the derivative of theLyapunov function along the trajectories of the system as

V (t) = ˙x�(t)P x(t)+ x�(t)P ˙x(t)+ 1

�

( ˙θ�(t)θ (t)+ θ�(t) ˙

θ (t))

= x�(t)(A�mP +PAm)x(t)+2x�(t)Pbθ�(t)x(t)+ 2

�θ�(t) ˙

θ (t)

= −x�(t)Qx(t)+2x�(t)Pbθ�(t)x(t)+2θ�(t)Proj(θ (t),−x(t)x�(t)Pb)

= −x�(t)Qx(t)+2θ�(t)(x(t)x�(t)Pb+Proj(θ (t),−x(t)x�(t)Pb)

)≤ −x�(t)Qx(t) ,

which implies that x(t) and θ (t) are uniformly bounded. Next, since x(0) = 0, it followsthat

λmin(P )‖x(t)‖2 ≤ V (t) ≤ V (0) = θ�(0)θ (0)

�.

The projection operator ensures that θ (t) ∈�, and therefore

θ�(0)θ (0)

�≤ 4maxθ∈� ‖θ‖2

�,

which leads to the following upper bound:

‖x(t)‖2 ≤ θmax

λmin(P )�.

Since ‖ · ‖∞ ≤ ‖ ·‖, and this bound is uniform, the bound above yields

‖xτ‖L∞ ≤√

θmax

λmin(P )�,

which holds for every τ ≥ 0. The result in (2.14) immediately follows from this inequality,as it holds uniformly in τ . �

We notice that the bound in (2.14) is derived independently of uad(t). This impliesthat both x(t) and x(t) can diverge at the same rate, maintaining a uniformly bounded errorbetween the two. Next, we prove that, with the adaptive feedback given by (2.6), the stateof the predictor remains bounded and consequently leads to asymptotic convergence of thetracking error x(t) to zero.

Lemma 2.1.3 If uad(t) is defined according to (2.6), and the condition in (2.7) holds, thenwe have the following asymptotic result:

limt→∞ x(t) = 0. (2.15)


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Proof. To prove asymptotic convergence of x(t) to zero, one needs to ensure that x(t)in (2.4), with uad(t) given by (2.6), is uniformly bounded.

First, we notice that

x(s) =G(s)η(s)+kgH (s)C(s)r(s)+xin(s) ,


‖xτ‖L∞ ≤ ‖G(s)‖L1‖ητ‖L∞ +‖kgH (s)C(s)‖L1‖rτ‖L∞ +‖xinτ‖L∞ . (2.16)

Next, applying the triangular relationship for norms to the bound in (2.14), we have

|‖xτ‖L∞ −‖xτ‖L∞| ≤√

θmax

λmin(P )�.

The projection in (2.5) ensures that θ (t) ∈�, and hence ‖ητ‖L∞ ≤ L‖xτ‖L∞ . Substitutingfor ‖xτ‖L∞ yields

‖ητ‖L∞ ≤ L(

‖xτ‖L∞ +√

θmax

λmin(P )�

). (2.17)

Then, the bounds on ‖xτ‖L∞ and ‖ητ‖L∞ in (2.16) and (2.17), with account of the stabilitycondition in (2.7), lead to

‖xτ‖L∞ ≤λ√

θmaxλmin(P )� +‖kgH (s)C(s)‖L1‖rτ‖L∞ +‖xinτ‖L∞

1−λ .

Since the bound on the right-hand side is uniform, x(t) is uniformly bounded. Applicationof Barbalat’s lemma leads to the asymptotic result in (2.15). �

Remark 2.1.1 The above presented proof can be straightforwardly extended to accom-modate faster prediction error dynamics by considering a more general structure for thepredictor as compared to (2.4):

˙x(t) = Amx(t)+b(θ�(t)x(t)+uad(t))−Kspx(t) , x(0) = x0 ,

y(t) = c�x(t) ,

where Ksp ∈ Rn×n can be used to assign faster poles for (Am−Ksp). The idea of having

different poles for the prediction error dynamics as compared to the original H (s) = (sI−Am)−1b was first introduced in [27].

To streamline the subsequent derivation of the performance bounds for the class ofsystems considered in this section, we first note that (Am,b) is the state-space realizationof H (s), and since the pair (A, b) is controllable and Am = A− bk�m , then (Am,b) is alsocontrollable. Hence, Lemma A.12.1 implies that there exists co such that

H1(s) � C(s)1

c�o H (s)c�o (2.18)

is proper and BIBO stable.


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Theorem 2.1.1 For the system in (2.1) and the controller defined via (2.2) and (2.4)–(2.6),subject to the L1-norm condition in (2.7), we have

‖xref −x‖L∞ ≤ γ1√�

, ‖uref −u‖L∞ ≤ γ2√�

, (2.19)

limt→∞‖xref (t)−x(t)‖ = 0, lim

t→∞‖uref (t)−u(t)‖ = 0, (2.20)

where

γ1 � ‖C(s)‖L1

1−‖G(s)‖L1L

√θmax

λmin(P ),

γ2 � ‖H1(s)‖L1

√θmax

λmin(P )+‖C(s)θ� +k�m‖L1γ1 .

Proof. The response of the closed-loop system in (2.3) with the L1 adaptive controllerin (2.6) can be written (in the frequency domain) as

x(s) =H (s)kgC(s)r(s)+G(s)θ�x(s)−H (s)C(s)η(s)+xin(s) .

Also, from the definition of the closed-loop reference system in (2.9), it follows that

xref (s) =H (s)kgC(s)r(s)+G(s)θ�xref (s)+xin(s) .

The two expressions above and the prediction error dynamics in (2.13) lead to

xref (s)−x(s) =G(s)θ�(xref (s)−x(s))+C(s)x(s) , (2.21)

which, along with Lemma A.7.1, implies that

‖(xref −x)τ‖L∞ ≤ ‖G(s)θ�‖L1‖(xref −x)τ‖L∞ +‖C(s)‖L1‖xτ‖L∞ .

Then, the bounds in (2.10) and (2.14) lead to the uniform upper bound

‖(xref −x)τ‖L∞ ≤ ‖C(s)‖L1

1−‖G(s)‖L1L‖xτ‖L∞ ≤ ‖C(s)‖L1

1−‖G(s)‖L1L

√θmax

λmin(P )�, (2.22)

which proves the first bound in (2.19). Moreover, it follows from (2.21) that

xref (s)−x(s) = (I−G(s)θ�)−1C(s)x(s) .

From the Final Value Theorem and Lemma 2.1.3, we know that

lims→0

sx(s) = limt→∞ x(t) = 0.

Then, since the reference system is BIBS stable (see Lemma 2.1.1), (I−G(s)θ�)−1 is stable,and hence, the Final Value Theorem and the limit for x(t) above lead to

limt→∞ (xref (t)−x(t)) = lim

s→0s (xref (s)−x(s))

= lims→0

s(I−G(s)θ�)−1C(s)x(s)

= lims→0

(I−G(s)θ�)−1C(s) (sx(s))

= 0,

which leads to the first limit in (2.20).


�

�

�

�

�

�

�

�


To derive the second bound in (2.19), we first notice that the expressions in (2.2),(2.6), and (2.9) lead to the following relationship:

uref (s)−u(s) = C(s)η(s)− (C(s)θ� +k�m)(xref (s)−x(s)) . (2.23)

Then, it follows from the error dynamics in (2.13) and the definition of H1(s) in (2.18) that

C(s)η(s) =H1(s)x(s) ,

which implies that

uref (s)−u(s) =H1(s)x(s)− (C(s)θ� +k�m)(xref (s)−x(s)) ,

and, consequently, the following bound holds:

‖(uref −u)τ‖L∞ ≤ ‖H1(s)‖L1‖xτ‖L∞ +‖C(s)θ� +k�m‖L1‖(xref −x)τ‖L∞ .

Hence, the bounds in (2.14) and (2.22) lead to the second upper bound in (2.19). To showthe second limit in (2.20), we notice that the boundedness and the uniform continuity ofx(t) and η(t) imply that ˙x(t) is bounded and uniformly continuous, and application ofBarbalat’s lemma leads thus to limt→∞ ˙x(t) = 0. Consequently, limt→∞ η(t) = 0. Then,the second limit in (2.20) follows immediately from the expression in (2.23) and the firstlimit in (2.20). �

Remark 2.1.2 Since C(0) = 1, application of the Final Value Theorem to the closed-loopreference system in (2.9) in the case of constant r(t) ≡ r leads to

limt→∞yref (t) = c�H (0)C(0)kgr = r .

Remark 2.1.3 Notice that if we setC(s) = 1, which corresponds to the MRAC architecture,the norm of H1(s) is reduced to

‖H1(s)‖L1 =∥∥∥∥ 1

c�o H (s)c�o∥∥∥∥

L1

,

which is not bounded, since c�o H (s) is strictly proper. Therefore, one cannot conclude auniform performance bound for the control signal of MRAC similar to the one in (2.19).

Remark 2.1.4 Theorem 2.1.1 implies that, by increasing the adaptive gain �, the time his-tories of x(t) and u(t) can be made arbitrarily close to xref (t) and uref (t) for all t ≥ 0. Becausexref (t) and uref (t) are signals of an LTI system, then all the changes in initial conditions,reference inputs, and parametric uncertainties will lead to uniform scaled changes in thetime histories of these signals and consequently also in the time histories of the correspond-ing signals x(t) and u(t) of the L1 adaptive nonlinear closed-loop system. Thus, the controlobjective is reduced to selection of km and C(s) to ensure that the LTI reference system hasthe desired response.


�

�

�

�

�

�

�

�


c�kg (sI−A)−1b

k�m

C(s)

C(s)

θ�

θ�

xdes ydesudesr

Modified system

Figure 2.3: Design system for control specifications.

2.1.4 Design of the L1 Adaptive Controller: Robustness andPerformance

Notice that the closed-loop reference system in (2.9) depends upon the vector θ of unknownparameters, and hence it cannot be used for introducing the transient specifications. Next,we consider the following LTI system, which will be referred to as a design system, with itsoutput free of uncertainties:

xdes(s) = C(s)kgH (s)r(s)+xin(s) ,

udes(s) = kgC(s)r(s)−C(s)θ�xdes(s)−k�mxdes(s) , (2.24)

ydes(s) = c�xdes(s) . (2.25)

The block diagram of this system is shown in Figure 2.3. As compared to the referencesystem in Figure 2.2, we note that the filter C(s) is also a part of the system definition.

Lemma 2.1.4 Subject to (2.7), the following upper bounds hold:

‖ydes −yref ‖L∞ ≤ λ

1−λ‖c�‖1(‖kgH (s)C(s)‖L1‖r‖L∞ +‖xin‖L∞

), (2.26)

‖xdes −xref ‖L∞ ≤ λ

1−λ(‖kgH (s)C(s)‖L1‖r‖L∞ +‖xin‖L∞

),

‖udes −uref ‖L∞ ≤ λ

1−λ‖C(s)θ� +k�m‖L1

·(‖kgH (s)C(s)‖L1‖r‖L∞ +‖xin‖L∞)

. (2.27)

Proof. It follows from (2.9) that

xref (s) = (I−G(s)θ�)−1(H (s)kgC(s)r(s)+xin(s)) .

Since this reference system is BIBS stable, (I−G(s)θ�)−1 is stable, and hence it can beexpanded into convergent series:

(I−G(s)θ�)−1 = I+∞∑i=1

(G(s)θ�)i .


�

�

�

�

�

�

�

�


Therefore, the reference system can be rewritten as

xref (s) =(

I+∞∑i=1

(G(s)θ�)i)

(kgH (s)C(s)r(s)+xin(s))

= xdes(s)+∞∑i=1

(G(s)θ�)i(kgH (s)C(s)r(s)+xin(s)) .

Since ‖G(s)θ�‖L1 ≤ ‖G(s)‖L1L� λ < 1, then

‖xdes −xref ‖L∞ ≤∞∑i=1

λi(‖kgH (s)C(s)‖L1‖r‖L∞ +‖xin‖L∞ )

= λ

1−λ (‖kgH (s)C(s)‖L1‖r‖L∞ +‖xin‖L∞ ) .

(2.28)

Recalling the definitions of yref (t) = c�xref (t) and ydes(t) = c�xdes(t), the bound in (2.28)leads to the bound in (2.26).

From (2.9) and (2.24), one can derive

udes(s)−uref (s) = −(C(s)θ� +k�m)(xdes(s)−xref (s))

and further use the bound in (2.28) to obtain (2.27). �

Taking into consideration that xin(t) is exponentially decaying, the control objec-tive can be achieved via proper selection of the static feedback gain km and the low-pass filter C(s). In particular, the design of km and C(s) needs to ensure that C(s)c�H (s)(which does not depend on the unknown parameters) has the desired transient and steady-state performance characteristics, while simultaneously guaranteeing a small value of λ(or ‖G(s)‖L1 ). In general, km is chosen so that the state matrixAm specifies desired closed-loop dynamics, while the bandwidth-limited filterC(s) is designed to track reference signalsand compensate for the undesirable effects of the uncertainties within a prespecified rangeof frequencies.

In the case when C(s) is a low-pass filter, the systemG(s), which was defined in (2.8)asG(s) =H (s)(1−C(s)), can be seen as the cascade of a low-pass systemH (s) and a high-pass system (1−C(s)). Then, if the bandwidth of C(s), which approximately correspondsto the cut-off frequency of (1−C(s)), is designed to be larger than the bandwidth of H (s),the resulting G(s) is a “no-pass filter” with small L1-norm. The illustration is given inFigure 2.4. Hence, it follows that ‖G(s)‖L1 can be rendered arbitrarily small by

(i) increasing the bandwidth of the low-pass filter C(s) for a given set of closed-loopperformance specifications (in terms of the state matrix Am). This solution leadsto small design bounds and, therefore, yields a closed-loop adaptive systems withdesired behavior. However, low-pass filters with high bandwidths may result in high-gain feedback and thus lead to closed-loop systems with overly small robustnessmargins and susceptible to measurement noise.

(ii) reducing the bandwidth ofH (s) by slowing down the eigenvalues of the matrixAm fora given filter design. With this solution, a certain amount of performance is sacrificedto maintain a desired level of robustness.


�

�

�

�

�

�

�

�


Figure 2.4: Cascaded systems.

It is important to emphasize that the use of a static feedback gain km in the controlsignal is not necessary. In fact, in L1 adaptive control architectures, the desired closed-loopdynamics are specified through the state matrixAm of the predictor. However, if the desiredclosed-loop dynamics are far from the actual open-loop system dynamics, the upper boundon the uncertain parameter vector θ might be large, which may lead to high-gain feedbacksolutions as one tries to satisfy the L1-norm condition. Under these circumstances, the useof a static feedback gain can be useful for achieving designs with the desired performanceand satisfactory robustness margins.

The lemma below tries to illustrate some of the points discussed above by showing thatin the case of a first-order low-pass filter, the L1-norm of the system G(s) can be renderedarbitrarily small by increasing the bandwidth of this filter.

Lemma 2.1.5 LetC(s) =ωc/(s+ωc). For the arbitrary strictly proper BIBO-stable systemH (s), the following is true:

limωc→∞‖(1−C(s))H (s)‖L1 = 0.

Proof. Notice that

(1−C(s))H (s) = sH (s)

s+ωc .

Therefore

‖(1−C(s))H (s)‖L1 ≤∥∥∥∥ 1

s+ωc∥∥∥∥

L1

‖sH (s)‖L1 = 1

ωc‖sH (s)‖L1 .

From the fact that H (s) is strictly proper and stable, it follows that sH (s) is proper andstable, which implies that ‖sH (s)‖L1 exists and is bounded. This leads to

limωc→∞‖(1−C(s))H (s)‖L1 ≤ lim

ωc→∞1

ωc‖sH (s)‖L1 = 0.

The proof is complete. �

Remark 2.1.5 Theorem 2.1.1 and Lemma 2.1.4 imply that the L1 adaptive controller cangenerate a system response to track (2.24) and (2.25) both in transient and steady-state ifwe select the adaptive gain large and minimize λ. Notice that udes(t) in (2.24) depends uponthe unknown parameter θ , while ydes(t) in (2.25) does not. This implies that for differentvalues of θ , the L1 adaptive controller will generate different control signals (dependent


�

�

�

�

�

�

�

�


on θ ) to ensure uniform system response (independent of θ ). This is exactly what one wouldexpect from an adaptive controller in terms of adapting to unknown parameters: the controlsignal changes dependent upon the uncertainties exactly so that the system output retainsuniform (scaled) performance during the transient and steady-state phases. This basicallystates that the L1 adaptive architecture controls an unknown system as an LTI feedbackcontroller would have done if the parameters were known.

Remark 2.1.6 It follows from Theorem 2.1.1 that in the presence of a large adaptivegain, the L1 adaptive controller and the system state approximate uref (t) and xref (t),respectively. Therefore, y(t) approximates the output response of the LTI systemc�(I−G(s)θ�)−1kgH (s)C(s) to the input r(t), and hence its transient performance specifi-cations, such as overshoot and settling time, can be derived for every value of θ . If we furtherminimize λ, it follows from Lemma 2.1.4 that y(t) approximates the output response of theLTI system C(s)c�H (s) to the input signal r(t). In this case, the L1 adaptive controllerleads to uniform transient performance of y(t) independent of the value of the unknownparameter θ . For the resulting L1 adaptive control signal one can characterize the transientspecifications such as its amplitude and rate of change for every θ ∈�, using udes(t).

Example 2.1.2 Next, we compare the performance of the L1 adaptive controller to that ofa linear high-gain feedback. Consider the scalar system dynamics

x(t) = θx(t)+u(t) , x(0) = 0,

where θ ∈ [θmin, θmax], and let the desired system have a pole at −2 and unity DC gain.Since for this example we have b = 1, it follows from the definition of H (s) that

H (s) = 1

s+2.

Then, in order to ensure that the desired system has unity DC gain and the closed-loopsystem tracks step-reference signals with zero steady-state error, a feedforward gain kg isneeded. So we set

kg = 2,

which leads to the desired system

kgH (s) = 2

s+2.

First, let the high-gain feedback controller be given by

u(t) = −kx(t)+kr(t) ,

leading to the following closed-loop dynamics:

x(t) = (θ −k)x(t)+kr(t).One needs to choose k > θmax to guarantee stability. We notice that both the steady-state error and the transient performance depend on the unknown parameter θ . By further


�

�

�

�

�

�

�

�


introducing a proportional-integral (PI) controller, one can achieve zero steady-state error.If one chooses

k� max{|θmax|, |θmin|} ,

the response of the closed-loop system (in the frequency domain) is given by

x(s) = k

s− (θ −k)r(s) ≈ k

s+k r(s) ,

which is obviously different from the performance specified by kgH (s). Next, we applythe L1 adaptive controller. Letting um = −2x, the partially closed-loop dynamics can bewritten as

x(t) = −2x+ (uad(t)+ θx(t)) .

Selecting kg = 2 and C(s) = ωcs+ωc with large ωc, and setting the adaptive gain � large, it

follows from (2.19) that

x(s) ≈ xref (s) ≈ xdes(s) = C(s)kgH (s)r(s) = ωc

s+ωc2

s+2r(s) ≈ 2

s+2r(s) ,

u(s) ≈ uref (s) = (−2−C(s)θ )xref (s)+C(s)2r(s) ≈ (−2− θ )xref (s)+2r(s) .

The first of these relationships implies that the control objective is met, while the secondstates that the L1 adaptive controller approximates uref (t), which partially cancels θ .

2.1.5 Simulation Example

Consider the system in (2.1) and let

A=[

0 1−1 −1.4

], b =

[01

], c =

[10

], θ =

[4

−4.5

].

Further, let�= {ϑ = [ϑ1, ϑ2]� ∈ R2 :ϑi ∈ [−10, 10], for all i= 1,2}, which leads toL= 20.

Letting km = 0, we implement the L1 adaptive controller following (2.2), (2.4)–(2.6). Let

� = 10000, C1(s) = ωc

s+ωc .

The L1-norm ‖G1(s)‖L1 = ‖H (s)(1 −C1(s))‖L1 can be calculated numerically. In par-ticular, Figure 2.5(a) shows λ1 = ‖G1(s)‖L1L with respect to the bandwidth of the low-pass filter ωc. Notice that for ωc > 50, we have λ1 < 1. Choosing ωc = 160 s−1 leads toλ1 = ‖G1(s)‖L1L≈ 0.3< 1, which yields improved performance bounds in (2.26)–(2.27).

The simulation results for the L1 adaptive controller are shown in Figures 2.6(a)and 2.6(b) for step reference inputs r = 25, 100, 400. We notice that the L1 adaptivecontroller leads to scaled control inputs and scaled system outputs for scaled referenceinputs. Figures 2.7(a) and 2.7(b) show the performance for the time-varying reference signalr(t) = 100cos(0.2t) without any retuning of the controller.

Next, consider the following design:

� = 400, C2(s) = 3ω2c s+ω3

c

(s+ωc)3.


�

�

�

�

�

�

�

�


0 20 40 60 80 1000

2

4

6

8

ωc rad/s

(a) λ1(ωc)

0 20 40 60 80 1000

2

4

6

8

10

12

ωc rad/s

(b) λ2(ωc)

Figure 2.5: λ1 and λ2 with respect to ωc.

0 2 4 6 8 100

100

200

300

400

500

time [s]

r=25r=100r=400

(a) Time history of y(t)

0 2 4 6 8 10−500

0

500

1000

1500

2000

2500

time [s]

r=25r=100r=400

(b) Time history of u(t)

Figure 2.6: Performance of L1 adaptive controller with C1(s) = 160s+160 for step-reference

inputs.

Figure 2.5(b) shows λ2 = ‖G2(s)‖L1L= ‖H (s)(1−C2(s))‖L1L as a function ofωc. Noticethat for ωc > 40, we have λ2 < 1. Setting ωc = 50 s−1 leads to λ2 = 0.71. The simulationresults are shown in Figures 2.8(a) and 2.8(b) for constant reference inputs r = 25, 100, 400,which are again scaled accordingly.

Remark 2.1.7 The example above illustrates that high-order filters C(s) may give the op-portunity to use relatively small adaptive gains. While a rigorous relationship between thechoice of the adaptive gain and the order of the filter cannot be derived, an insight into thiscan be gained from the following analysis. It follows from (2.1), (2.2), and (2.6) that

x(s) = kgH (s)C(s)r(s)+H (s)θ�x(s)−H (s)C(s)η(s)+xin(s) ,

while the state predictor can be rewritten as

x(s) = kgH (s)C(s)r(s)+H (s)(1−C(s))η(s)+xin(s) .

We note that the low-frequency component of the parameter estimate, C(s)η(s), is the inputto the actual system, while the complementary high-frequency component, (1−C(s))η(s),


�

�

�

�

�

�

�

�


0 10 20 30 40 50−100

−50

0

50

100

time [s]

(a) y(t) (solid) and r(t) (dashed)

0 10 20 30 40 50−500

0

500

time [s]


Figure 2.7: Performance of L1 adaptive controller withC1(s) = 160s+160 for r = 100cos(0.2t).

0 2 4 6 8 100

100

200

300

400

500

time [s]

r=25r=100r=400

(a) Time history of y(t)

0 2 4 6 8 10−500

0

500

1000

1500

2000

2500

time [s]

r=25r=100r=400


Figure 2.8: Performance of L1 adaptive controller withC2(s) = 7500s+503

(s+50)3 for step-reference

inputs.

goes into the state predictor. Since a low-pass filterC(s) can only attenuate frequency contentabove its bandwidth, L1 adaptive designs using filters with high bandwidths will requirelarge adaptive gains in order to generate frequencies beyond the bandwidths of the filters.A properly designed high-order C(s) can be more effective to serve the purpose of filteringwith reduced tailing effects and, hence, can generate similar λ with a smaller bandwidth.This further implies that a similar level of performance can be achieved with a smalleradaptive gain.

2.1.6 Loop Shaping via State-Predictor Design

To get further insights into the L1 adaptive controller, we consider the following first-ordersystem, corrupted by input and output disturbances:

x(t) = −x(t)+u(t)+σ (t) , x(0) = x0 ,

y(t) = x(t)+d(t) ,


�

�

�

�

�

�

�

�


C(s)

1s+1

1s+1

ksp

�s

r u

σ d

x y

x

σ

L1 controller

Figure 2.9: Closed-loop L1 system with modified predictor.

where x(t) ∈ R is the system state, y(t) ∈ R is the measured output, u(t) ∈ R is the controlinput, and σ (t), d(t) ∈ R are unknown bounded signals, representing input and outputdisturbances, respectively.

For the design of the L1 adaptive controller, following (2.18), we consider the fol-lowing state predictor:

˙x(t) = −x(t)+u(t)+ σ (t)−kspx(t) , x(0) = x0 ,

y(t) = x(t) ,(2.29)

where x(t) ∈ R is the predictor state, ksp ≥ 0 is a constant that can be tuned to shapethe frequency response of the closed-loop system, x(t) � x(t) − y(t), and σ (t) ∈ R is thedisturbance estimation, governed by the following adaptation law:

˙σ (t) = −�x(t) , � > 0.

Finally, let C(s) be a strictly proper stable transfer function with DC gain C(0) = 1, andassume zero initialization for its state-space realization. The L1 controller for this systemcan be defined as follows:

u(s) = −C(s)σ (s)+ r(s) ,

where u(s), σ (s), and r(s) are the Laplace transforms of u(t), σ (t), r(t), respectively. Theblock diagram of this system is given in Figure 2.9.

The modification of the state predictor in (2.29) affects the prediction error dynamics,

˙x(t) = −(1+ksp)x(t)+ σ (t) , x(0) = 0,

where σ (t) � σ (t) − σ (t). In the frequency domain, these error dynamics can be rewrit-ten as

x(s) =Hm(s)σ (s) , Hm(s) � 1

s+1+ksp.


�

�

�

�

�

�

�

�


Table 2.1: Closed-loop transfer functions for the system in Figure 2.9.

r(s) σ (s) d(s)

x(s) 1s+1

s2+(ksp+1)s+�(1−C(s))(s+1)P (s) −�C(s)

P (s)

y(s) 1s+1

s2+(ksp+1)s+�(1−C(s))(s+1)P (s)

s2+(ksp+1)s+�(1−C(s))P (s)

u(s) 1 −�C(s)P (s) −� (s+1)C(s)

P (s)

P (s) � s2 + (ksp +1)s+�

Notice that the closed-loop system is LTI, and therefore we can use classical controltools to investigate its properties [17]. We compute the transfer functions from the signalsr(t), σ (t), and d(t) to each of the outputs x(t), y(t), and the control u(t). These transferfunctions are summarized in Table 2.1. One can see that there are only six different transferfunctions. Moreover, the transfer functionsHxr (s) =Hyr (s) = 2/(s+1) andHur (s) = 1 donot depend upon the controller parameters. Therefore, we will consider only the followingfour transfer functions:

Hxσ (s) =Hyσ (s) = s2+(ksp+1)s+�(1−C(s))(s+1)P (s) , Hyd (s) = s2+(ksp+1)s+�(1−C(s))

P (s) ,

Huσ (s) =Hxd (s) = −�C(s)P (s) , Hud (s) = −�(s+1)C(s)

P (s) ,(2.30)

where P (s) � s2 + (ksp + 1)s+� is the characteristic polynomial of the adaptation loop.For the purpose of analysis, let

C(s) = 1

s+1.

Figure 2.10 shows the Bode plots for the transfer functions in (2.30) for the L1 adaptivecontroller (with ksp = 0) for different adaptation gains. One can see that the Bode plots of thetransfer functions Hyd (s), Huσ (s), and Hud (s) have a peak. The frequency location of thepeak depends upon�, and it moves to the right toward higher frequencies as one increases�.For this simple LTI closed-loop system, we can analytically explain this phenomenon byconsidering the common part of the denominator of the transfer functions, which is givenby P (s) = s2 + s+� = s2 +2ζωns+ω2

n, where ωn �√� and ζ � 1/(2

√�). One can see

that the damping ζ decreases with the increase of the adaptation gain �. This peak points toa sensitivity to noise and disturbances, which is typical as well for other adaptive controllersin the presence of high adaptation rates, including MRAC. However, from the Bode plotsofHuσ andHyσ , one can see that with the growth of �, while moving to the right, the peaknever crosses the horizontal axis, which is consistent with the theoretical results proved inSection 2.1.3. Moreover, with the L1 adaptive controller, this issue can be addressed byappropriate tuning of the state predictor, as will be illustrated shortly.

In fact, in the presence of the term −kspx(t) in the predictor dynamics, the newcharacteristic polynomial for the adaptation loop is given by P (s) � s2 + (ksp + 1)s+�.


�

�

�

�

�

�

�

�


10�2

10�1

100

101

102

103

�5

�10

�15

�20

�25

�30

�35

�40

�45

�50

0

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

109

103

10

(a) transfer function from σ to x

10�2

100

102

104

106

�10

�20

�30

�40

�50

0

10

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

109

103

10

(b) transfer function from d to y

10�2

100

102

104

106

�10

�20

�30

�40

�50

�60

�70

�80

�90

�100

0

10

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

109

103

10

(c) transfer function from σ to u

10�2

100

102

104

106

�10

�20

�30

�40

�50

0

10

20

30

40

50

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

109

103

10

(d) transfer function from d to u

Figure 2.10: Bode plots for the closed-loop transfer functions for different adaptation gains� and for ksp = 0.

By appropriate tuning of ksp, we can adjust the damping of the adaptation loop ζ = (ksp +1)/(2

√�) to an arbitrary desired value. Figure 2.11 shows the effect of the coefficient ksp

on the Bode plots for a fixed value of the adaptive gain � = 10000. One can see that bysetting the value of ksp appropriately, we can eliminate the peak described earlier in all thetransfer functions. Therefore, in the following discussion we consider ksp = 1.4

√�− 1,

which yields ζ = 0.7.Figure 2.12 shows the Bode plots for the closed-loop system with the modified (faster)

state predictor for different values of the adaptation gain �. One can see that the adaptivecontrol system provides good disturbance rejection within the bandwidth of the low-passfilter, and the control channel is not affected by high-frequency disturbance content. Adetailed discussion can be found in [90]. Notice that similar results in noise attenuation canalso be achieved via a higher order C(s).


�

�

�

�

�

�

�

�

2.2. Systems with Uncertain System Input Gain 35

10−2

10−1

100

101

102

103

−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

ksp = 100ksp = 10ksp = 0


10−2

10−1

100

101

102

103

−50

−40

−30

−20

−10

0

10

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

ksp = 100ksp = 10ksp = 0


10−2

100

102

104

106

−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

0

10

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

ksp = 100ksp = 10ksp = 0


10−2

100

102

104

106

−50

−40

−30

−20

−10

0

10

20

30

40

50

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

ksp = 100ksp = 10ksp = 0


Figure 2.11: Bode plots for the closed-loop transfer functions for different ksp for� = 10000.

2.2 Systems with Uncertain System Input GainThe results in Section 2.1 imply that fast adaptation ensures uniform performance boundsfor the system’s signals, both input and output, as compared to the corresponding signals ofa bounded-reference LTI system. This gives the opportunity to extend the class of systemsbeyond the one with constant unknown parameters and consider systems with time-varyingparameters and disturbances. In this section, we also incorporate uncertainty in the systeminput gain. We introduce a new control architecture, which gives an opportunity to com-pensate for the effect of the unknown system input gain on the output performance. Wealso derive the time-delay margin of this closed-loop architecture and analyze the effect ofnonzero initialization errors on the system’s performance [34].


�

�

�

�

�

�

�

�


10�2

10 1

100

101

102

103

�50

�45

�40

�35

�30

�25

�20

�15

�10

�5

0

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

109

103

10


10�2

10 �1

100

101

102

103

�50

�40

�30

�20

�10

0

10

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

109

103

10


10�2

100

102

104

106

�100

�90

�80

�70

�60

�50

�40

�30

�20

�10

0

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

109

103

10


10�2

100

102

104

106

�50

�45

�40

�35

�30

�25

�20

�15

�10

�5

0

5

Mag

nitu

de (

dB)

Bode Diagram

Frequency (rad/sec)

109

103

10


Figure 2.12: Bode plots for the closed-loop transfer functions for different adaptation gains� and for ksp = 1.4

√�−1.


Consider the following class of systems:

x(t) = Amx(t)+b(ωu(t)+ θ�(t)x(t)+σ (t)

), x(0) = x0 ,

y(t) = c�x(t) ,(2.31)

where x(t) ∈ Rn is the system state vector (measured); u(t) ∈ R is the control input; y(t) ∈ R

is the regulated output; b, c ∈ Rn are known constant vectors;Am is a known Hurwitz n×n

matrix specifying the desired closed-loop dynamics; ω ∈ R is an unknown constant withknown sign; θ (t) ∈ R

n is a vector of time-varying unknown parameters; and σ (t) ∈ R modelsinput disturbances.


�

�

�

�

�

�

�

�


Assumption 2.2.1 (Uniform boundedness of unknown parameters) Let

θ (t) ∈� , |σ (t)| ≤�0 , ∀ t ≥ 0,

where � is a known convex compact set and �0 ∈ R+ is a known (conservative) bound of

σ (t).

Assumption 2.2.2 (Uniform boundedness of the rate of variation of parameters) Letθ (t) and σ (t) be continuously differentiable with uniformly bounded derivatives:

‖θ (t)‖ ≤ dθ <∞ , |σ (t)| ≤ dσ <∞ , ∀ t ≥ 0.

Assumption 2.2.3 (Partial knowledge of uncertain system input gain) Let

ω ∈�0 � [ωl0 , ωu0 ] ,

where 0< ωl0 < ωu0 are given known upper and lower bounds on ω.

The control objective is to design a full-state feedback adaptive controller to ensurethat y(t) tracks a given bounded piecewise-continuous reference signal r(t) with quantifiableperformance bounds.


State Predictor

We consider the following state predictor:

˙x(t) = Amx(t)+b(ω(t)u(t)+ θ�(t)x(t)+ σ (t)

), x(0) = x0 ,

y(t) = c�x(t) ,(2.32)

which has the same structure as the system in (2.31); the only difference is that the unknownparameters ω, θ (t), and σ (t) are replaced by their adaptive estimates ω(t), θ (t), and σ (t).

Adaptation Laws

The adaptive process is governed by the following projection-based adaptation laws:

˙θ (t) = �Proj(θ (t),−x�(t)Pbx(t)), θ (0) = θ0 ,

˙σ (t) = �Proj(σ (t),−x�(t)Pb) , σ (0) = σ0 ,

˙ω(t) = �Proj(ω(t),−x�(t)Pbu(t)) , ω(0) = ω0 ,

(2.33)

where x(t) � x(t) − x(t), � ∈ R+ is the adaptation rate, and P = P� > 0 is the solution

of the algebraic Lyapunov equation A�mP +PAm = −Q for arbitrary Q=Q� > 0. In the

implementation of the projection operator we use the compact set � as given in Assump-tion 2.2.1, while we replace �0 and �0 by � and �� [ωl , ωu] such that

�0 <� , 0< ωl < ωl0 < ωu0 < ωu . (2.34)

The purpose of this definition for the projection bounds will be clarified in the analysis ofthe time-delay and the gain margins.


�

�

�

�

�

�

�

�



x = Amx+b (ωu+ θ�x+σ )y = c�x

˙x = Amx+b(ωu+ θ�x+ σ

)y = c�x

˙θ = �Proj(θ ,−x�Pbx)˙σ = �Proj(σ ,−x�Pb)˙ω = �Proj(ω,−x�Pbu)

η = ωu+ θ�x+ σ

kD(s)kgr u x

x

xω, θ , σ

Figure 2.13: Closed-loop adaptive system.

Control Law

The control signal is generated as the output of the following (feedback) system:

u(s) = −kD(s)(η(s)−kgr(s)) , (2.35)

where r(s) and η(s) are the Laplace transforms of r(t) and η(t) � ω(t)u(t)+ θ�(t)x(t)+ σ (t),respectively; kg � −1/(c�A−1

m b); and k > 0 and D(s) are a feedback gain and a strictlyproper transfer function leading to a strictly proper stable

C(s) � ωkD(s)

1+ωkD(s)∀ ω ∈�0 (2.36)

with DC gain C(0) = 1. One simple choice isD(s) = 1/s, which yields a first-order strictlyproper C(s) of the form

C(s) = ωk

s+ωk .

As before, let

L� maxθ∈� ‖θ‖1 , H (s) � (sI−Am)−1b , G(s) �H (s)(1−C(s)) . (2.37)

The L1 adaptive controller is defined via (2.32), (2.33), (2.35), subject to the followingL1-norm condition:

‖G(s)‖L1L < 1. (2.38)

The L1 adaptive control architecture with its main elements is represented in Figure 2.13.In the case of constant θ (t), the L1-norm condition can be simplified. For the specific

choice of D(s) = 1/s, it is reduced to

Ag �[Am+bθ� bω

−kθ� −kω]

, (2.39)

being Hurwitz for all θ ∈� and ω ∈�0.


�

�

�

�

�

�

�

�


2.2.3 Performance Bounds of the L1 Adaptive Controller


Consider the following closed-loop reference system, which again corresponds to the non-adaptive version of the L1 adaptive controller:

xref (t) = Amxref (t)+b(ωuref (t)+ θ�(t)xref (t)+σ (t)

), xref (0) = x0, (2.40)

uref (s) = C(s)

ω(kgr(s)−ηref (s)), (2.41)

yref (t) = c�xref (t), (2.42)

where r(s) and ηref (s) are the Laplace transforms of r(t) and ηref (t) � θ�(t)xref (t)+σ (t),respectively. The next lemma establishes stability of the closed-loop system in (2.40)–(2.42).

Lemma 2.2.1 If k and D(s) verify the L1-norm condition in (2.38), the closed-loop refer-ence system in (2.40)–(2.42) is BIBS stable with respect to r(t) and x0.

Proof. It follows from (2.40)–(2.42) that

xref (s) =G(s)ηref (s)+H (s)C(s)kgr(s)+xin(s) ,

where, similar to previous analysis, we have set xin(s) � (sI−Am)−1x0. Since Am is Hur-witz, xin(t) is uniformly bounded. Next, it follows from Lemma A.7.1 that

‖xref τ‖L∞ ≤ ‖G(s)‖L1‖ηref τ‖L∞ +‖H (s)C(s)kg‖L1‖r‖L∞ +‖xin‖L∞ .

Using the definition in (2.37), we have the following upper bound for ηref (t):

‖ηref τ‖L∞ ≤ L‖xref τ‖L∞ +‖στ‖L∞ .

Substituting and solving for ‖xref τ‖L∞ , one obtains

‖xref τ‖L∞ ≤ ‖H (s)C(s)kg‖L1‖r‖L∞ +‖G(s)‖L1�+‖xin‖L∞1−‖G(s)‖L1L

.

Then, because k and D(s) verify the condition in (2.38), ‖xref τ‖L∞ is uniformly boundedfor all τ ≥ 0. Hence, the closed-loop reference system in (2.40)–(2.42) is BIBS stable. �

Lemma 2.2.2 If θ (t) ≡ θ is constant, andD(s) = 1/s, then the closed-loop reference systemin (2.40)–(2.42) is BIBS stable with respect to r(t) andx0 if and only if the matrixAg in (2.39)is Hurwitz for all θ ∈� and ω ∈�0.

Proof. In the case of constant θ (t), the state-space realization of (2.40)–(2.42) is given by[xref (t)uref (t)

]= Ag

[xref (t)uref (t)

]+[

bσ (t)kkgr(t)−kσ (t)

],

xref (0) = x0 ,uref (0) = 0,

which is stable if and only if Ag is Hurwitz for all θ ∈� and ω ∈�0. �


�

�

�

�

�

�

�

�



The system dynamics in (2.31) and the state predictor in (2.32) lead to the followingprediction-error dynamics:

˙x(t) = Amx(t)+b(ω(t)u(t)+ θ�(t)x(t)+ σ (t)) , x(0) = 0, (2.43)

where θ (t) � θ (t) − θ (t), σ (t) � σ (t) − σ (t), and ω(t) � ω(t) −ω. Let η(t) � ω(t)u(t) +θ�(t)x(t) + σ (t), and let η(s) be the Laplace transform of η(t). Then, the error dynamicsin (2.43) can be rewritten in frequency domain as

x(s) =H (s)η(s) . (2.44)

Lemma 2.2.3 The prediction error x(t) is uniformly bounded,

‖x‖L∞ ≤√

θm

λmin(P )�, (2.45)

where

θm � 4maxθ∈� ‖θ‖2 +4�2 + (ωu−ωl)2 +4

λmax(P )

λmin(Q)

(dθ max

θ∈� ‖θ‖+dσ�)

. (2.46)

Proof. Consider the Lyapunov function candidate:

V (x(t), θ (t), ω(t), σ (t)) = x�(t)P x(t)+ 1

�

(θ�(t)θ (t)+ ω2(t)+ σ 2(t)

). (2.47)

First, we prove that

V (t) ≤ θm

�.

Since x(0) = x(0), we can easily verify that

V (0) ≤ 1

�

(4maxθ∈� ‖θ‖2 +4�2 + (ωu−ωl)2

)<θm

�.

Using the projection-based adaptive laws in (2.33), one can derive the following upperbound:

V (t) = −x�(t)Qx(t)+2x�(t)Pbω(t)u(t)+2x�(t)Pbθ�(t)x(t)

+2x�(t)Pbσ (t)+ 2

�

(θ�(t) ˙

θ (t)+ ω(t) ˙ω(t)+ σ (t) ˙σ (t))

− 2

�

(θ�(t)θ (t)+ σ (t)σ (t)

)= −x�(t)Qx(t)+2ω(t)

(x�(t)Pbu(t)+Proj(ω(t),−x�(t)Pbu(t))

)+2θ�(t)

(x(t)x�(t)Pb+Proj(θ (t),−x(t)x�(t)Pb)

)+2σ (t)

(x�(t)Pb+Proj(σ (t),−x�(t)Pb)

)− 2

�

(θ�(t)θ (t)+ σ (t)σ (t)

)≤ −x�(t)Qx(t)+ 2

�

(|θ�(t)θ (t)|+ |σ (t)σ (t)|

).

(2.48)


�

�

�

�

�

�

�

�


The projection operator ensures that θ (t) ∈ �, ω(t) ∈ �, |σ (t)| ≤ � for all t ≥ 0, andtherefore, the upper bounds in (2.2.2) lead to the following upper bound:

θ�(t)θ (t)+ σ (t)σ (t) ≤ 2(maxθ∈� ‖θ‖dθ +dσ�) .

Moreover, the projection operator also ensures that

maxt≥0

(1

�

(θ�(t)θ (t)+ ω2(t)+ σ 2(t)

))≤ 1

�

(4maxθ∈� ‖θ‖2 +4�2 + (ωu−ωl)2

),

which holds for all t ≥ 0.If at any time t1 > 0, one has V (t1) > θm/�, then it follows from (2.46) and (2.47)

that

x�(t1)P x(t1)> 4λmax(P )

�λmin(Q)

(dθ max

θ∈� ‖θ‖+dσ�)

,

and thus

x�(t1)Qx(t1) ≥ λmin(Q)x�(t1)P x(t1)

λmax(P )> 4

dθ maxθ∈� ‖θ‖+dσ��

. (2.49)

Hence, if V (t1)> θm/�, then from (2.48) and (2.49) we have

V (t1)< 0, (2.50)

and it follows from (2.50) that for all t ≥ 0,

V (t) ≤ θm

�.

Since λmin(P )‖x(t)‖2 ≤ x�(t)P x(t) ≤ V (t), then

‖x(t)‖2 ≤ θm

λmin(P )�, ∀ t ≥ 0,

which leads to (2.45). �We further notice that the bound in (2.45) is proportional to the square root of the rate

of variation of uncertainties and is inverse proportional to the square root of the adaptationgain.

Theorem 2.2.1 Given the system in (2.31) and the L1 adaptive controller defined via (2.32),(2.33), and (2.35), subject to the L1-norm condition in (2.38), we have

‖xref −x‖L∞ ≤ γ1√�

, ‖uref −u‖L∞ ≤ γ2√�

, (2.51)

where

γ1 � ‖C(s)‖L1

1−‖G(s)‖L1L

√θm

λmin(P ),

γ2 �∥∥∥∥C(s)

ω

∥∥∥∥L1

Lγ1 +∥∥∥∥H1(s)

ω

∥∥∥∥L1

√θm

λmin(P ),

and H1(s) = C(s) 1c�o H (s)

c�o was introduced in (2.18).


�

�

�

�

�

�

�

�


Proof. Let η(t) � θ�(t)x(t)+σ (t). It follows from (2.35) that

u(s) = −kD(s)(ωu(s)+η(s)−kgr(s)+ η(s)) .

Consequently

u(s) = − kD(s)

1+ωkD(s)(η(s)−kgr(s)+ η(s)) .

Using the definition of C(s) in (2.36), we can write

u(s) = −C(s)

ω(η(s)−kgr(s)+ η(s)) , (2.52)

and the system in (2.31) takes the form

x(s) =G(s)η(s)+H (s)C(s)kgr(s)−H (s)C(s)η(s)+xin(s) .

Similarly, it follows from (2.40)–(2.42) that

xref (s) =G(s)ηref (s)+H (s)C(s)kgr(s)+xin(s) .

Then

xref (s)−x(s) =G(s)ηe(s)+H (s)C(s)η(s) , ηe(t) � θ�(t)(xref (t)−x(t)) . (2.53)

Using (2.44), we can rewrite

xref (s)−x(s) =G(s)ηe(s)+C(s)x(s) .

Lemma A.7.1 gives the following upper bound:

‖(xref −x)τ‖L∞ ≤ ‖G(s)‖L1‖ηeτ‖L∞ +‖C(s)‖L1‖xτ‖L∞ . (2.54)

From the definition ofL in (2.37) it follows that ‖ηeτ‖L∞ ≤L‖(xref −x)τ‖L∞ . Substitutingthis back into (2.54), and solving for ‖(xref −x)τ‖L∞ , with account of the upper bound fromLemma 2.2.3 and the condition in (2.38), one gets


1−‖G(s)‖L1L

√θm

λmin(P )�,

which holds uniformly for all τ ≥ 0, leading to the first bound in (2.51).To prove the second bound in (2.51), we notice that from (2.41) and (2.52) one can

derive

uref (s)−u(s) = −C(s)

ωηe(s)+ C(s)

ωη(s) . (2.55)

Similar to the proof of (2.19) in Theorem 2.1.1, we refer to Lemma A.12.1 and rewrite

C(s)

ωη(s) = 1

ωH1(s)x(s) .


�

�

�

�

�

�

�

�


SinceC(s) is strictly proper and stable, the systemH1(s) is proper and stable. Then it followsfrom Lemma A.7.1 that the difference in (2.55) can be upper bounded as

‖(uref −u)τ‖L∞ ≤∥∥∥∥C(s)

ω

∥∥∥∥L1

L‖(xref −x)τ‖L∞ +∥∥∥∥H1(s)

ω

∥∥∥∥L1

‖xτ‖L∞ .

Using the first bound in (2.51) and the upper bound from Lemma 2.2.3 in the above expres-sion leads to the second bound in (2.51). �

Remark 2.2.1 Notice that letting k → ∞ leads to C(s) → 1, and thus the reference con-troller in (2.41), in the limit, leads to perfect cancelation of uncertainties and recovers theperformance of the ideal desired system. In this case, the uniform bound for the control sig-nal is lost, as C(s) = 1 corresponds to anH1(s), which is improper and, hence, its L1-normdoes not exist.

Theorem 2.2.2 For the closed-loop system in (2.31) with the L1 adaptive controller definedvia (2.32), (2.33), and (2.35), ifAg in (2.39) is Hurwitz, θ (t) ≡ θ is constant andD(s) = 1/s,we have

‖xref −x‖L∞ ≤ γ3√�

, ‖uref −u‖L∞ ≤ γ4√�

, (2.56)

where

γ3 �∥∥Hg(s)H1(s)

∥∥L1

√θm

λmin(P ), Hg(s) � (sI−Ag)−1

[b

0

],

γ4 �∥∥∥∥C(s)

ωθ�∥∥∥∥

L1

γ3 +∥∥∥∥H1(s)

ω

∥∥∥∥L1

√θm

λmin(P ).

Proof. Let e(t) � xref (t)−x(t) and ς (s) � −C(s)ωθ�e(s). With this notation, (2.53) can be

written as

e(s) =H (s)(θ�e(s)+ως (s)+C(s)η(s))

and further expressed in state-space form as[e(t)ς (t)

]= Ag

[e(t)ς (t)

]+[b

0

]ηC(t) ,

e(0) = 0,ς (0) = 0,

ηC(s) � C(s)η(s) .

Let xς (t) � [e�(t) ς (t)]�. Given a Hurwitz matrix Ag , the system Hg(s) is BIBO stableand strictly proper. Since

xς (s) =Hg(s)ηC(s) =Hg(s)H1(s)x(s) ,

we can follow similar arguments as in the proof of Theorem (2.2.1) and derive the twobounds in (2.56). �

Thus, the tracking errors between xref (t) and x(t) and between uref (t) and u(t) areuniformly bounded by a constant inverse proportional to

√�, implying that one can arbi-

trarily improve the tracking performance for both signals simultaneously by increasing �.In the case of constant θ and σ , one can prove in addition the following asymptotic result.


�

�

�

�

�

�

�

�


Lemma 2.2.4 If in the system (2.31) the parameters θ (t) ≡ θ and σ (t) ≡ σ are constant,Agin (2.39) is Hurwitz, andD(s) = 1/s, the L1 adaptive controller, defined via (2.32), (2.33),and (2.35), leads to the following asymptotic result:

limt→∞ x(t) = 0.

Proof. In the case of constant θ (t) and σ (t), the derivative of the Lyapunov function in(2.48) takes the form V (t) = −x�(t)Qx(t) ≤ 0, implying boundedness of x(t) and θ (t),ω(t), σ (t). It follows from Theorem 2.2.2 that x(t) is bounded, and consequently x(t) isbounded. Also, the adaptive laws in (2.33) ensure that θ (t), ω(t), σ (t) are bounded. Hence, itcan be checked straightforwardly that ˙x(t) is bounded, which leads to uniform boundednessof V (t), and hence uniform continuity of V (t). It then follows from Barbalat’s lemma thatV (t) → 0 as t → ∞, leading to limt→∞ x(t) = 0. �

2.2.4 Performance in the Presence of Nonzero TrajectoryInitialization Error

In this section we prove that in the case of constant θ , arbitrary nonzero trajectory initializa-tion errors lead to exponentially decaying transient errors in the input and output signals ofthe system. Thus, we consider the system in (2.31) with constant unknown θ while retainingthe time-varying disturbance σ (t):

x(t) = Amx(t)+b(ωu(t)+ θ�x(t)+σ (t)) , x(0) = x0 ,

y(t) = c�x(t) .(2.57)

We consider the same state predictor as in the previous section,


), x(0) = x0 ,

y(t) = c�x(t) ,(2.58)

where, however, x0 might not be equal to x0 in general. Since x0 = x0, then the prediction-error dynamics take the form

˙x(t) = Amx(t)+b(ω(t)u(t)+ θ�(t)x(t)+ σ (t)) , x(0) = x0 −x0 . (2.59)

Lemma 2.2.5 For the prediction-error dynamics in (2.59), the following upper bound holds:

‖x(t)‖ ≤ ρ(t) , ∀ t ≥ 0,

where

ρ(t) �

√√√√(V (0)− θn

�

)e−αt

λmin(P )+ θn

λmin(P )�, α � λmin(Q)

λmax(P ),

θn � 4maxθ∈� ‖θ‖2 +4�2 + (ωu−ωl)2 + 4�dσ

α

(2.60)

with V (t) being the Lyapunov function in (2.47).


�

�

�

�

�

�

�

�


Proof. Consider the Lyapunov function candidate in (2.47). Since ω and θ are constant, itfollows from (2.48) that

V (t) ≤ −x�(t)Qx(t)+2�−1c |σ (t)σ (t)| . (2.61)

Projection ensures that θ (t) ∈�, |σ (t)| ≤�, ω(t) ∈�, and therefore

1

�maxt≥0

(ω2(t)+ θ�(t)θ (t)+ σ 2(t)) ≤ 1

�

(4maxθ∈� ‖θ‖2 +4�2 + (ωu−ωl)2

).

Hence

V (t) ≤ x�(t)P x(t)+ θn

�− 4�dσλmax(P )

λmin(Q)�,

where θn is given in (2.60). Further, since

x�(t)Qx(t) ≥ λmin(Q)

λmax(P )x�(t)P x(t) ≥ λmin(Q)

λmax(P )

(V (t)− θn

�

)+ 4�dσ

�,

the upper bound in (2.61) can be used to obtain

V (t) ≤ −λmin(Q)

λmax(P )V (t)+ λmin(Q)

λmax(P )

θn

�,

which leads to

V (t) ≤(V (0)− θn

�

)e−αt + θn

�

with α being given in (2.60). Substituting this expression into

‖x(t)‖2 ≤ x�(t)P x(t)

λmin(P )≤ V (t)

λmin(P )

completes the proof. �

To derive the performance bounds for the closed-loop adaptive system with theL1 adaptive controller, we first need to introduce the following definition and prove apreliminary result (Lemma 2.2.6). For an m-input n-output stable proper transfer functionF (s) with impulse response f (t), let

�F (t) = maxi=1,...,n

√√√√ m∑j=1

f 2ij (t) , (2.62)

where fij (t) is the ith row, j th column of the impulse response matrix of F (s).

Lemma 2.2.6 Consider an m-input n-output stable proper transfer function F (s) and letp(s) = F (s)q(s). If ‖q(t)‖ ≤µ(t), then ‖p(t)‖∞ ≤�F (t)∗µ(t), where ∗ is the convolutionoperation.


�

�

�

�

�

�

�

�


Proof. Since p(s) = F (s)q(s), then

pi(t) =∫ t

0fi(t− τ )q(τ )dτ , ∀ i = 1, . . . ,n ,

wherefi corresponds to the ith row of the impulse response matrix forF (s). Upper boundingpi(t) gives

|pi(t)| ≤∫ t

0‖fi(t− τ )‖‖q(τ )‖dτ ≤

∫ t

0�F (t− τ )µ(τ )dτ , ∀ i = 1, . . . ,n ,

which completes the proof. �

Let

H2(s) � (I−G(s)θ�)−1C(s) , H3(s) � −C(s)θ�

ω.

Theorem 2.2.3 Given the system in (2.57) and theL1 adaptive controller, defined via (2.58),(2.33), and (2.35), subject to the L1-norm condition in (2.38), the following bounds holdfor all t ≥ 0:

‖xref (t)−x(t)‖∞ ≤ γ (t) , (2.63)

‖uref (t)−u(t)‖∞ ≤ 1

ω�H1 (t)∗ (ρ(t)+‖xin(t)‖)+�H3 (t)∗γ (t) , (2.64)

where

γ (t) ��H2 (t)∗ (ρ(t)+‖xin(t)‖) , (2.65)

and xin(t) is the inverse Laplace transform of xin(s) � (sI−Am)−1(x0 −x0).

Proof. It follows from (2.52) that

u(s) = C(s)

ω(−θ�x(s)−σ (s)+kgr(s)− η(s)) , (2.66)

and the system in (2.57) consequently takes the form

x(s) =H (s)(C(s)kgr(s)+ (1−C(s))

(θ�x(s)+σ (s)

)−C(s)η(s))

+xin(s) .

In the case of constant θ , the reference system in (2.40)–(2.42) is given by

xref (s) =H (s)(C(s)kgr(s)+ (1−C(s))(θ�xref (s)+σ (s))

)+xin(s) ,

uref (s) = C(s)

ω

(−θ�xref (s)−σ (s)+kgr(s)

).

(2.67)

Thus

xref (s)−x(s) =H (s)(

(1−C(s))θ�(xref (s)−x(s))−C(s)η(s))

=H2(s)H (s)η(s) .


�

�

�

�

�

�

�

�


It follows from (2.57) and (2.58) that

x(s) =H (s)η(s)+ xin(s) ,

and consequently

xref (s)−x(s) =H2(s)(x(s)− xin(s)) .

Using the upper bound in Lemma 2.2.5, we have

‖x(t)− xin(t)‖ ≤ ‖x(t)‖+‖xin(t)‖ ≤ ρ(t)+‖xin(t)‖ ,

and, hence, Lemma 2.2.6 leads to the upper bound in (2.63).To prove the bound in (2.64), we notice that from (2.66) and (2.67) one has

uref (s)−u(s) = −C(s)θ�

ω(xref (s)−x(s))+ C(s)

ωη(s)

= −C(s)θ�

ω(xref (s)−x(s))+ C(s)

ω

1

c�o H (s)c�o H (s)η(s)

=H3(s)(xref (s)−x(s))+ H1(s)

ω(x(s)− xin(s)) ,

and, therefore, application of Lemma 2.2.6 leads to the bound in (2.64). �

Remark 2.2.2 We notice that the above bounds are derived using a conservative estimation‖x(t)− xin(t)‖ ≤ ‖x(t)‖+‖xin(t)‖, which leads to a conservative upper bound for ‖x(t)−xin(t)‖. In fact, x(t) and xin(t) tend to cancel each other in certain cases, leading to verysmall transient deviation, as observed in simulations.

2.2.5 Time-Delay Margin Analysis

L1 Adaptive Controller in the Presence of Time Delay

In this section, we derive the time-delay margin of the L1 adaptive controller for the systemin (2.57) with the predictor given in (2.58), subject to x0 = x0. For simplicity, we chooseD(s) = 1/s, although arbitraryD(s) satisfying (2.38) can be accommodated in the analysisbelow.

We proceed by considering the following three systems.

System 1. Let τ > 0 denote the unknown constant time delay in the control channel. Thesystem in (2.57), when closed with the delayed L1 adaptive controller, takes the form

x(t) = Amx(t)+b(ωud (t)+ θ�x(t)+σ (t)) , x(0) = x0 , (2.68)

where

ud (t) ={

0, 0 ≤ t < τ ,u(t− τ ) , t ≥ τ ,

(2.69)


�

�

�

�

�

�

�

�


with u(t) being computed via (2.33), (2.35), (2.58) and ud (t) being the delayed controlsignal. We further notice that this closed-loop system has a unique solution. It is the stabilityof this closed-loop system that we are trying to determine dependent upon τ . We furthernotice that due to the delay in the control channel, one could not derive the dynamicsof the error between the system and the predictor, the boundedness of which is stated inLemmas 2.2.3 and 2.2.5. Theorems 2.2.1, 2.2.2, and 2.2.3 also do not hold.

System 2. Next, we consider the following closed-loop system without delay but withadditional disturbance:

xq (t) = Amxq (t)+b(ωuq (t)+ θ�xq (t)+σ (t)+ν(t)

), xq (0) = x0 , (2.70)

and let uq (t) be given via (2.33), (2.35), and (2.58) with x(t) being replaced by xq (t). We no-tice that compared to the system in (2.57), the system in (2.70) has an additional disturbancesignal ν(t). If ν(t) is continuously differentiable with uniformly bounded derivative and

|σ (t)+ν(t)| ≤� , ∀ t ≥ 0, (2.71)

where � was introduced in (2.34), then application of the L1 adaptive controller to thesystem in (2.70) is well defined, and hence the results of Lemma 2.2.5 and Theorem 2.2.3are valid.

System 3. Finally, we consider the open-loop system in (2.57) and apply uqd (t) to it withuqd (t) being the τ -delayed signal of uq (t):

uqd (t) ={

0, 0 ≤ t < τ ,uq (t− τ ) , t ≥ τ .

(2.72)

We denote the system’s response to this signal by xo(t), where the subindex o indicates theopen-loop nature of this system. It is important to notice that at this point we view uqd (t)as a time-varying input signal for (2.57), not as a feedback signal, so that (2.57) remains anopen-loop system in this context. These last two systems are illustrated in Figure 2.14.

In preparation for the proof of the main result in Theorem 2.2.4, we now derive somepreliminary results that will be used in future derivations.

Lemma 2.2.7 If the output of the open-loop System 3 has the same time history as theclosed-loop output of System 2, i.e.,

xo(t) ≡ xq (t) , ∀ t ≥ 0, (2.73)

then u(t) ≡ uq (t), x(t) ≡ xq (t), for all t ≥ 0.

Proof. Equation (2.73) implies that the open-loop time-delayed System 3, given by (2.57),(2.72), generates xq (t) in response to the input uq (t). When applied to (2.70), uq (t) leadsto xq (t). Hence, uq (t) and xq (t) are also solutions of the closed-loop adaptive System 1,defined via (2.33), (2.35), (2.58), (2.68), and (2.69). �


�

�

�

�

�

�

�

�


e−τs

Open-loop time-delayed System 3

ω

ω

H (s)

H (s)

θ�

θ�

uq (t)

uqd (t)

σ (t)

σ (t) ν(t)

xo(t)

xq (t)


ζ (t)

System 2 with additional disturbance

Figure 2.14: Block diagram of Systems 2 and 3.

Lemma 2.2.7 consequently implies that to ensure the stability of System 1 in thepresence of a given time delay τ , it is sufficient to prove the existence of ν(t) in System 2,satisfying (2.71) and verifying (2.73). We notice, however, that the closed-loop System 2 isa nonlinear system due to the nonlinear adaptive laws, so the proof of existence of such ν(t)for this system and the explicit construction of the bound� is not straightforward. Moreover,we note that the condition in (2.73) relates the time delay τ of System 1 (or System 3) tothe signal ν(t) implicitly. In the next section we introduce an LTI system that helps to provethe existence of such ν(t) and leads to explicit computation of �. The definition of thisLTI system is the key step in the overall time-delay margin analysis. The LTI system has anexogenous input that lumps the time trajectories of the nonlinear elements of the closed-loopSystem 2. For this LTI system, the time-delay margin can be computed via its loop transferfunction, which consequently defines a conservative, but guaranteed, lower bound for thetime-delay margin of the closed-loop adaptive system.

LTI System in the Presence of the Same Time Delay

We consider the following closed-loop LTI system in the presence of the same time delay τ :

xl(t) = (Am+bθ�)xl(t)+bζl(t) , xl(0) = x0 , (2.74)

ζl(s) = ωuld (s)+σ (s) , (2.75)

uld (t) ={

0, 0 ≤ t < τ ,ul(t− τ ) , t ≥ τ ,

(2.76)


�

�

�

�

�

�

�

�


H (s)kgC(s)ω

C(s)ω

e−τs

ω

ω

θ�

r(t)

εl(t)

ul(t) uld (t)

νl(t)

σ (t)

σ (t)

ζl(t)

xl(t)

ηl(t)

Figure 2.15: Equivalent LTI system in (2.74)–(2.79).

ul(s) = C(s)

ω

(kgr(s)− θ�xl(s)−σ (s)−νl(s)

)− εl(s) , (2.77)

εl(s) = C(s)

ωηl(s) , (2.78)

νl(s) = ωuld (s)−ωul(s) , (2.79)

where xl(t), ul(t), and εl(t) are the states of this LTI system; ηl(t) is an exogenous signal;r(t) and σ (t) are the same bounded reference and disturbance input as in (2.57) with |σ (t)| ≤�0; and C(s) is the same low-pass filter used in the definition of the reference system in(2.40)–(2.42). Further, let

H (s) � (sI−Am−bθ�)−1b .

The system (2.74)–(2.79) is highly coupled, as shown in Figure 2.15.

Remark 2.2.3 In the absence of time delay, i.e., when τ = 0, we have uld (t) = ul(t), andthe system in (2.74)–(2.79) is reduced to

xl(t) = (Am+bθ�)xl(t)+b(ωul(t)+σ (t)) , xl(0) = x0 ,

ul(s) = 1

ωC(s)(kgr(s)− θ�xl(s)−σ (s))− εl(s) ,

εl(s) = C(s)

ωηl(s) .

We notice that the trajectories of this LTI system are uniquely defined, once ηl(t) and r(t)are specified. Omitting the nonzero initial condition x0, we notice the following relationshipbetween this LTI system and the original reference system (2.40)–(2.42) (with θ (t) ≡ θ ):

xl(s)

r(s)= kgC(s)

(I+C(s)H (s)θ�)−1

H (s) = xref (s)

r(s),

xl(s)

σ (s)= (

1−C(s))(

I+C(s)H (s)θ�)−1H (s) = xref (s)

σ (s).


�

�

�

�

�

�

�

�


11−C(s)

C(s)ω

(1+ θ�H (s)

)

ul(t)

uld (t)

ω

σ (t)

ηb(t)

ηf (t)

Figure 2.16: LTI system for time-delay margin prediction.

Time-Delay Margin of the LTI System

We notice that the phase margin of the LTI system in (2.74)–(2.79) can be determined fromits open-loop transfer function from uld (t) to ul(t), and it does not depend upon its initialcondition. Omitting the initial condition xl(0) for the time being, it follows from (2.74)–(2.79) that

ul(s) = 1

1−C(s)

(C(s)

ωkgr(s)− C(s)

ω

(1+ θ�H (s)

)(ωuld (s)+σ (s)

)− εl(s)) .

Therefore, it can be equivalently presented as in Figure 2.16 with the following signals:

ul(s) = ηb(s)−ηf (s)

1−C(s),

ηf (s) = C(s)

ω

(1+ θ�H (s)

)(ωuld (s)+σ (s)

),

ηb(s) = C(s)

ωkgr(s)− εl(s) .

Assume ηl(t) is bounded. Then, the resulting εl(t) is bounded. Also, the loop transferfunction of the system in Figure 2.16 is

Lo(s) = C(s)

1−C(s)(1+ θ�H (s)) . (2.80)

Since σ (t) and r(t) are bounded, the phase margin φm of (2.80) can be derived from itsBode plot. Its time-delay margin is given by

T = φm

ωgc, (2.81)

where ωgc is the gain crossover frequency of Lo(s). The next lemma states a sufficientcondition for boundedness of all the states in the system (2.74)–(2.79), including the internalstates.

Lemma 2.2.8 For arbitrary bounded signal εl(t), if

τ < T , (2.82)

the signals xl(t), ul(t), νl(t) are bounded.


�

�

�

�

�

�

�

�


Proof. From the definition of the time-delay margin of LTI systems, it follows that for allτ < T , if εl(t) is bounded, ul(t) is bounded. Hence, it follows from (2.76) that uld (t) isbounded, and the relationship in (2.79) implies that νl(t) is bounded. Since ul(t), εl(t), νl(t),r(t), and σ (t) are bounded, it follows from (2.77) that θ�xl(t) is bounded. From (2.74) wenotice that

xl(s) =H (s)(θ�xl(s)+ ζl(s))+xin(s) ,

which leads to boundedness of xl(t). �

Lemma 2.2.9 Let εl(t) be bounded, and let τ comply with (2.82). If ηl(t) is bounded, thenνl(t) is bounded.

Proof. Using Lemma 2.2.8, we immediately conclude that xl(t) and ul(t) are bounded.Since C(s) is strictly proper and stable, bounded ηl(t) ensures that εl(t) is continuously dif-ferentiable with uniformly bounded derivative. Using similar arguments, one can concludethat both ul(t) and uld (t) have bounded derivatives. Hence, it follows from (2.79) that νl(t)is bounded. �

Time-Delay Margin of the Closed-Loop Adaptive System

Let

εb(εα , t) ��H1 (t)∗ (κα(εα , t)+‖xin(t)‖)+ εβ ,

κα(εα , t) �√

(V (0)− εα)e−αt + εαλmin(P )

,

where �H1 (t) is determined from (2.62) and (2.18), ‖xin(t)‖ is introduced in (2.65), andV (t) is the Lyapunov function in (2.47), while εα > 0 and εβ > 0 are arbitrary positiveconstants. We notice that εb(εα , t) is positive for all t ≥ 0 and is uniformly bounded. Furtherassume that εl(t) verifies the upper bound

|εl(t)| ≤ εb(εα , t) , ∀ t ≥ 0. (2.83)

For all τ < T , where T is the time-delay margin of Lo(s), Lemma 2.2.8 guarantees that themap �n : R

+ × [0,T ) → R+, given by

�n(εα ,τ ) � max|εl (t)|≤εb(εα ,t)

‖σ +νl‖L∞ , (2.84)

is well defined. Finally, let δ1 > 0 be an arbitrary constant, and let

��n(εα ,τ )+ δ1 . (2.85)

Remark 2.2.4 Notice that the computation of�n in (2.84) depends upon σ (t), the controlsignal ul(t), and the delay τ , involved in definition of νl(t). While �0 is independent ofεl(t), maximization of νl(t) depends upon ul(t) and uld (t) and hence upon εl(t). If theexternal signals in the LTI system (2.74)–(2.79) are well defined, then one can compute aconservative estimate for�n from (2.84). We also notice that the definition of εb(t) dependsupon the nonzero trajectory initialization error.


�

�

�

�

�

�

�

�


Similarly, we define

�u(εα ,τ ) � max|εl (t)|≤εb(εα ,t)

‖ul‖L∞ , �x(εα ,τ ) � max|εl (t)|≤εb(εα ,t)

‖xl‖L∞ .

If ηl(t) verifies the upper bound

‖ηl‖L∞ ≤ (ωu−ωl)�u(εα ,τ )+2L�x(εα ,τ )+2� , (2.86)

and the resulting εl(t) complies with (2.83), Lemma 2.2.9 guarantees that for all τ < T

�d � maxηl (t)

‖σ + νl‖L∞ (2.87)

is a computable (possibly unknown) constant. Further, let

εc(t) ��H1 (t)∗ (κβ (θq , t)+‖xin(t)‖) ,

κβ (θq , t) �

√√√√(V (0)− θq

�

)e−αt + θq

�

λmin(P ),

θq � 4maxθ∈� ‖θ‖2 +4�2 + (ωu−ωl)2 +4

λmax(P )�d�

λmin(Q), (2.88)

where V (t) is the Lyapunov function in (2.47). We notice that for arbitrary εα > 0 andarbitrary τ > 0 verifying (2.82), if ηl(t) complies with (2.86) and the resulting εl(t) complieswith (2.83), one has bounded � and �d and bounded positive εc(t).

Theorem 2.2.4 Consider System 1 and the LTI system in (2.74)–(2.79) in the presence ofthe same time delay τ . For arbitrary εα ∈ R

+, choose� according to (2.85) and restrict theadaptive gain to the following lower bound:

� >θq

εα. (2.89)

Then, for every τ < T , there exists a bounded exogenous signal ηl(t) over [0,∞) verify-ing (2.86) such that the resulting εl(t) complies with (2.83), and xl(t) = x(t), ul(t) = u(t)∀ t ≥ 0.

Proof. To proceed with the proof, we introduce the following notations. Let xh(t) be thestate variable of a general LTI system Hx(s), and let xi(t) and xs(t) be the input and theoutput signals of it. We notice that for arbitrary time instant t1 and arbitrary fixed timeinterval [t1, t2], given a continuous input signal xi(t) over [t1, t2), xs(t) is uniquely definedfor t ∈ [t1, t2]. Let S be the map

xs(t)|t∈[t1, t2] = S(Hx(s),xh(t1),xi(t)|t∈[t1, t2)).

If xi(t) is continuous, then S is a continuous map. We further notice that xs(t) is defined overa closed interval [t1, t2], although xi(t) is defined over the corresponding open set [t1, t2).It follows from the definition of S that given

xs1 (t)|t∈[t1, t2] = S(Hx(s),xh1 ,xi1 (t)|t∈[t1, t2)) ,

xs2 (t)|t∈[t1, t2] = S(Hx(s),xh2 ,xi2 (t)|t∈[t1, t2)) ,

if xh1 = xh2 and xi1 (t) = xi2 (t) over [t1, t2), then xs1 (t) = xs2 (t) for arbitrary t ∈ [t1, t2].


�

�

�

�

�

�

�

�


In (2.70), we notice that if |σ (t) + ν(t)| ≤ �, for all t ∈ [0, t∗], for some t∗ > 0,and σ (t), ν(t) have bounded derivatives over [0, t∗], then application of the L1 adaptivecontroller is well defined over [0, t∗]. Letting

dt∗ � ‖(σ + ν)t∗‖L∞ ,

it follows from (2.58) and (2.70) that

˙xq (t) = Amxq (t)+bηq (t) , xq (0) = x(0)−x(0) ,

where xq (t) � x(t)−xq (t), and

ηq (t) � ω(t)uq (t)+ θ�(t)xq (t)+ σq (t) , σq (t) � σ (t)− (σ (t)+ν(t)) . (2.90)

The choice of D(s) = 1/s leads to a first-order low-pass filter,

C(s) = ωk

s+ωk .

The expression above, along with the definition of uq (t), implies that

uq (t)|t∈[0, t∗] = S(C(s)/ω,uq (0),

(kgr(t)− θ�xq (t)−σ (t)−ν(t)− ηq (t)

)|t∈[0, t∗)

),

xq (t)|t∈[0, t∗] = S(H (s), xq (0), ηq (t)|t∈[0,t∗)) .

We notice that uq (t)|t∈[0, t∗] can be equivalently presented as

uq (t)|t∈[0, t∗] = S(C(s)/ω,uq (0),

(kgr(t)− θ�xq (t)−σ (t)−ν(t)

)|t∈[0, t∗)

)− ε(t)|t∈[0, t∗] ,

(2.91)

where

ε(t)|t∈[0, t∗] = S(C(s)/ω,0, ηq (t)|t∈[0, t∗)) . (2.92)

Next, let

κ(θt∗ , t) �

√√√√(V (0)− θt∗

�

)e−αt + θt∗

�

λmin(P ), t ∈ [0, t∗] , α = λmin(Q)

λmax(P ),

θt∗ � 4maxθ∈� ‖θ‖2 +4�2 + (ωu−ωl)2 + 4�dt∗

α. (2.93)

From Lemma 2.2.5, it follows that ‖xq (t)‖ ≤ κ(θt∗ , t), for all t ∈ [0, t∗]. Since

ε(t)|t∈[0, t∗] = S

(C(s)

ωc�o H (s)c�o H (s),0, ηq (t)|t∈[0, t∗)

)

= S

(C(s)

ωc�o H (s)c�o ,0, (xq (t)− xin(t))|t∈[0, t∗)

),


�

�

�

�

�

�

�

�


then ε(t) can be upper bounded as

|ε(t)| ≤�H1 (t)∗ (κ(θt∗ , t)+‖xin(t)‖) , ∀ t ∈ [0, t∗] . (2.94)

Next, we prove the existence of a continuously differentiable ν(t), t ≥ 0, in the closed-loop adaptive System 2 and the existence of bounded ηl(t) in the time-delayed LTI systemsuch that for all t ≥ 0,

|σ (t)+ν(t)|<� , |σ (t)+ ν(t)|<�d , xo(t) = xq (t) , (2.95)

ul(t) = uq (t) , xl(t) = xq (t) , εl(t) = ε(t) , |εl(t)|< εb(εα , t) . (2.96)

We notice that these two systems are well defined if the external inputs ν(t) and ηl(t) arespecified. With (2.95), Lemma 2.2.7 implies that x(t) = xq (t) and u(t) = uq (t) for all t ≥ 0,while (2.96) proves Theorem 2.2.4.

Let ζ (t) � ωuqd (t)+σ (t), t ≥ 0. It follows from the definition of the map S and thedefinition of the time-delayed open-loop System 3 that, for all i ≥ 0, one can write

xo(t)|t∈[iτ , (i+1)τ ] = S(H (s),xo(iτ ),ζ (t)|t∈[iτ , (i+1)τ )

). (2.97)

In the three steps below, we prove by iteration that for all t ∈ [0, iτ ):

uq (t) = ul(t) , ε(t) = εl(t) , xo(t) = xq (t) = xl(t) , (2.98)

uqd (t) = uld (t) , |ε(t)|< εb(εα , t) ,

for all i ≥ 0. In addition, we prove that over t ∈ [0, iτ ) there exist bounded ηl(t) andcontinuously differentiable ν(t) such that for all i ≥ 0

ν(t) = νl(t) , |σ (t)+ν(t)|<� , |σ (t)+ ν(t)|<�d . (2.99)

Step 1: In this step, we prove that (2.98)–(2.99) hold for i = 0. It follows from (2.92)that ε(0) = 0. Due to zero initialization of C(s), one has uq (0) = ul(0) = 0. Recall thatxo(0) = xl(0) = x0. The expressions in (2.72) and (2.76) imply that uqd (t) = uld (t) = 0,t ∈ [0, τ ). Hence, it follows that for i = 0, we have

uq (iτ ) = ul(iτ ) , ε(iτ ) = εl(iτ ) , xo(iτ ) = xq (iτ ) = xl(iτ ) ,

uqd (t) = uld (t) , ∀ t ∈ [iτ , (i+1)τ ) ,

|ε(iτ )|< εb(εα , iτ ) .

For i = 0, the existence of ν(t) satisfying (2.99) is trivial.

Step 2: Assume that, for some i ≥ 0, the following conditions hold:

uq (t) = ul(t) , ∀ t ∈ [0, iτ ] , (2.100)

ε(t) = εl(t) , ∀ t ∈ [0, iτ ] , (2.101)

xo(t) = xq (t) = xl(t) , ∀ t ∈ [0, iτ ] , (2.102)

uqd (t) = uld (t) , ∀ t ∈ [iτ , (i+1)τ ) , (2.103)

|ε(t)| < εb(εα , t) , ∀ t ∈ [0, iτ ] , (2.104)


�

�

�

�

�

�

�

�


and there exist bounded ηl(t) and continuously differentiable ν(t) such that

ν(t) = νl(t) , |σ (t)+ν(t)|<� , |σ (t)+ ν(t)|<�d , t ∈ [0, iτ ) . (2.105)

We prove below that there exist bounded ηl(t) and continuously differentiable ν(t) withbounded derivative such that (2.100)–(2.105) hold for (i+1).

We notice that the relationship in (2.74) implies

xl(t)|t∈[iτ , (i+1)τ ] = S(H (s),xl(iτ ),ζl(t)|t∈[iτ , (i+1)τ )

). (2.106)

It follows from (2.103) that ζl(t) = ζ (t), for all t ∈ [iτ , (i+1)τ ). Using (2.102), it followsfrom (2.97) and (2.106) that

xo(t) = xl(t) , ∀ t ∈ [0, (i+1)τ ] . (2.107)

We now define ν(t) over [iτ , (i+1)τ ) as

ν(t) = ωuqd (t)−ωuq (t) , t ∈ [iτ , (i+1)τ ) . (2.108)

Since (2.70) implies that

xq (t)|t∈[iτ , (i+1)τ ] = S(H (s),xq (iτ ), (ωuq (t)+σ (t)+ν(t))|t∈[iτ , (i+1)τ )) ,

it follows from (2.108) that

xq (t)|t∈[iτ , (i+1)τ ] = S(H (s),xq (iτ ),ζ (t)t∈[iτ , (i+1)τ )) .

Along with (2.97) and (2.102), the expression above ensures that

xq (t) = xo(t) , ∀ t ∈ [0, (i+1)τ ] . (2.109)

We need to prove that the definition in (2.108) guarantees

|σ (t)+ν(t)|<� , t ∈ [iτ , (i+1)τ ) , (2.110)

which is required for application of the L1 adaptive controller. Assume that it is not true.Since (2.105) holds for all t ∈ [0, iτ ) and ν(t) is continuous over [iτ , (i+1)τ ), there mustexist t ′ ∈ [iτ , (i+1)τ ) such that |σ (t)+ν(t)|<� for all t < t ′ and

|σ (t ′)+ν(t ′)| =� . (2.111)


xo(t)t∈[iτ , t ′] = S(H (s),xo(iτ ), (ωuq (t)+σ (t)+ν(t))|t∈[iτ , t ′)

),

while the relationships in (2.91) and (2.92) imply that

uq (t)|t∈[iτ , t ′] = S

(C(s)

ω,uq (iτ )+ ε(iτ ), (kgr(t)− θ�xq (t)−σ (t)−ν(t))|t∈[iτ , t ′)

)−ε(t)|t∈[iτ , t ′] , (2.112)


�

�

�

�

�

�

�

�


whereε(t)|t∈[iτ , t ′] = S

(C(s)/ω,ε(iτ ), ηq (t)|t∈[iτ , t ′)

). (2.113)

Letηl(t) = ηq (t) , t ∈ [iτ , t ′) . (2.114)

Then, we have

εl(t)|t∈[iτ , t ′] = S(C(s)/ω,εl(iτ ), ηq (t)|t∈[iτ , t ′)

),

which along with (2.101) and (2.113) implies

εl(t) = ε(t) , ∀ t ∈ [iτ , t ′] . (2.115)

Substituting (2.100), (2.107), (2.109), and (2.115) into (2.112) yields

uq (t)|t∈[iτ , t ′] = S(C(s)/ω,ul(iτ )+ εl(iτ ), (kgr(t)− θ�xl(t)−σ (t)−ν(t))|t∈[iτ , t ′)

)−εl(t)|t∈[iτ , t ′] . (2.116)

The relationships in (2.79) and (2.103) imply that

νl(t) = ωuqd (t)−ωul(t) , t ∈ [iτ , t ′] . (2.117)

Since (2.77) implies

ul(t)|t∈[iτ , t ′] = S(C(s)/ω,ul(iτ )+ εl(iτ ), (kgr(t)− θ�xl(t)−σ (t)−νl(t))|t∈[iτ , t ′)

)−εl(t)|t∈[iτ , t ′] ,

it follows from (2.108), (2.116), and (2.117) that

uq (t) = ul(t), ∀ t ∈ [iτ , t ′] , (2.118)

ν(t) = νl(t), ∀ t ∈ [iτ , t ′) . (2.119)


ν(t) = νl(t) , ∀ t ∈ [0, t ′) . (2.120)

We now prove by contradiction that

|ε(t)|< εb(εα , t) , ∀ t ∈ [iτ , t ′] . (2.121)

If (2.121) is not true, then since ε(t) is continuous, there exists t ∈ [iτ , t ′] such that |ε(t)|<εb(εα , t), for all t ∈ [iτ , t), and

|ε(t)| = εb(εα , t) . (2.122)

It follows from (2.104) that

|ε(t)| ≤ εb(εα , t) , ∀ t ∈ [0, t] . (2.123)

The relationships in (2.100), (2.102), (2.107), (2.109), and (2.118) imply that uq (t) = ul(t),xq (t) = xl(t) for all t ∈ [0, t]. Therefore, (2.90) and (2.114) imply that

ηl(t) = ω(t)ul(t)+ θ�(t)xl(t)+ σq (t) ,


�

�

�

�

�

�

�

�


and consequently

‖ηlt ‖L∞ ≤ (ωu−ωl)‖ult ‖L∞ +2L‖xlt ‖L∞ +2� . (2.124)

It follows from (2.101) and (2.115) that εl(t) = ε(t), for all t ∈ [0, t ′], and hence (2.123)implies

|εl(t)| ≤ εb(εα , t) , ∀ t ∈ [0, t] . (2.125)

Using (2.125), Lemma 2.2.8 implies that νl(t), ul(t), xl(t) are bounded subject to

|σ (t)+νl(t)| ≤�n(εα ,τ )<� , ‖ult ‖L∞ ≤�u(εα ,τ ) , ‖xlt ‖L∞ <�x(εα ,τ ) . (2.126)


‖ηlt ‖L∞ ≤ (ωu−ωl)�u(εα ,τ )+2L�x(εα ,τ )+2� ,

which along with (2.125) verifies (2.83) and (2.86). Hence, it follows from (2.87) that|σ (t) + νl(t)| ≤�d for all t ∈ [0, t]. Since (2.120) holds for all t ∈ [0, t ′), ν(t) is boundedand differentiable such that

|σ (t)+ν(t)| ≤�n(εα ,τ )<� , |σ (t)+ ν(t)| ≤�d , ∀ t ∈ [0, t] . (2.127)


|ε(t)| ≤�H1 (t)∗ (κβ (θt , t)+‖xin(t)‖)

for all t ∈ [0, t]. The relationships in (2.88), (2.93), and (2.127) imply that θt ≤ θq , and usingthe upper bound from (2.94) we have

|ε(t)| ≤�H3 (t)∗ (κ(θq , t)+‖xin(t)‖) = εc(t) .

From (2.89) we have θq�< εα and hence κβ (θq , t) ≤ κα(εα , t), for all t ≥ 0. From the definition

of �H1 (t) and the properties of convolution, it follows that εc(t) < εb(εα , t) and hence|ε(t)|< εb(εα , t) for all t ∈ [0, t], which contradicts (2.122). Therefore, (2.121) holds.

Since (2.121) is true, it follows from (2.127) that |σ (t) + ν(t)| ≤ �n < �, for allt ∈ [0, t ′], which contradicts (2.111). Thus, the upper bound in (2.110) holds. Therefore,the relationships in (2.107), (2.109), (2.110), (2.115), (2.118), (2.119), (2.121), and (2.127)imply that there exist bounded ηl(t) and continuously differentiable ν(t) in [0, (i+1)τ ) suchthat

xo(t) = xq (t) = xl(t) , uq (t) = ul(t) , ε(t) = εl(t) ,

|ε(t)|< εb(εα , t) , ∀ t ∈ [0, (i+1)τ ] ,

ν(t) = νl(t) , |σ (t)+ν(t)|<� , (2.128)

|σ (t)+ ν(t)| ≤�d , ∀ t ∈ [0, (i+1)τ ) .

It follows from (2.72), (2.76), and the fact that uq (t) = ul(t) for all t ∈ [iτ , (i+ 1)τ ] thatuqd (t) = uld (t) for all t ∈ [(i+ 1)τ , (i+ 2)τ ), which along with (2.128) proves (2.100)–(2.104) for i+1.


�

�

�

�

�

�

�

�


Step 3: By iterating the results from Step 2, we prove (2.98)–(2.99) for arbitrary i ≥ 0.We note that the relationships in (2.98)–(2.99) lead to (2.95)–(2.96) directly, which

completes the proof. �

Theorem 2.2.4 establishes the equivalence of state and control trajectories of theclosed-loop adaptive system and the LTI system in (2.74)–(2.79) in the presence of thesame time delay. Therefore the time-delay margin of the system in (2.74)–(2.79) can beused as a conservative lower bound for the time-delay margin of the closed-loop adaptivesystem. The proof of the following result follows from Lemma 2.2.8 and Theorem 2.2.4directly.

Corollary 2.2.1 Subject to (2.38) and (2.82), if� and� are selected appropriately large, theclosed-loop system in (2.68) with the τ -delayed controller, given by (2.32), (2.33), (2.35),and (2.69), is stable.

Remark 2.2.5 Notice that the bound � used in the projection-based adaptive law for σ (t)depends upon the initial condition x0 via ηl(t). This implies that the result is semiglobal,in a sense that larger x0 may require larger � to ensure the same time-delay margin givenby (2.81).

2.2.6 Gain-Margin Analysis

We now analyze the gain margin of the system in (2.57) with the L1 adaptive controller.Consider the system

x(t) = Amx(t)+b(ωgu(t)+ θ�(t)x(t)+σ (t)

), x(0) = x0 ,

where ωg � gω, with g being a positive constant. We note that this transformation impliesthat the set� in the application of the projection operator for adaptive laws needs to increaseaccordingly. However, increasing � will not violate the condition in (2.38) required forstability, as the latter depends only on the bounds of θ (t). Thus, it follows from (2.34) thatthe gain margin of the L1 adaptive controller is determined by

gm = [ωl/ωl0 , ωu/ωu0 ] .

We note that the lower bound of gm is greater than zero, while its definition implies thatarbitrary gain margin can be obtained through appropriate choice of �.

2.2.7 Simulation Example: Robotic Arm

System Dynamics and Assumptions

Consider the dynamics of a single-link robot arm rotating on a vertical plane:

I q(t) = u(t)+ Mglcos(q(t))

2+ σ (t)+F1(t)q(t)+F (t)q(t) ,

where q(t) and q(t) are the angular position and velocity, respectively, u(t) is the inputtorque, I is the unknown moment of inertia, M is the unknown mass, l is the unknown


�

�

�

�

�

�

�

�


length,F (t) is an unknown time-varying friction coefficient,F1(t) is the position-dependentexternal torque coefficient, and σ (t) is an unknown uniformly bounded disturbance. Theequations of motion can be cast into the following normal form:

x(t) = Amx(t)+b(ωu(t)+ θ�(t)x(t)+σ (t)),

y(t) = c�x(t) ,

where ω = 1/I is the unknown control effectiveness, and

Am =[

0 1−1 −1.4

], b =

[01

], θ (t) =

[1+ F1(t)

I, 1.4+ F (t)

I

]�,

σ (t) = Mglcos(x1(t))

2I+ σ (t)

I.

Consider three cases of parametric uncertainties,

ω1 = 1, θ1(t) = [2+ cos(πt), 2+0.3sin(πt)+0.2cos(2t)]� ,

ω2 = 1.5 , θ2(t) = [sin(0.5πt)+ cos(πt), −1+0.1sin(3πt)]� ,

ω3 = 0.8 , θ3(t) = [4.5, 3− sin(t)]� ,

and the following three cases of disturbances:

σ1(t) = sin(π

2t)

,

σ2(t) = cos(x1(t))+2sin(πt)+ cos

(7π

5t

),

σ3(t) = cos(x1(t))+2sin(2πt)+ cos

(16π

5t

).

The compact sets can be conservatively set to �0 = [0.5, 1.8], �= {ϑ = [ϑ1, ϑ2]� ∈ R2 :

ϑi ∈ [−5, 5] , i = 1, 2}, and �0 = 10.

Design of the L1 Adaptive Controller

We implement the L1 adaptive controller according to (2.32), (2.33), and (2.35) subject tothe L1-norm condition in (2.38). Letting D(s) = 1/s, we have

G(s) = s

s+ωkH (s), H (s) =[

1

s2 +1.4s+1,

s

s2 +1.4s+1

]�.

It is straightforward to verify numerically that for ωk > 30, one has ‖G(s)‖L1L< 1. Sinceω > 0.5, we set k = 60. For implementation of the adaptation laws we select the followinglarger projection bounds �= [0.1, 2],�= 50 for ω and σ (t), and retain the original� forθ (t). Finally, let � = 100000.


�

�

�

�

�

�

�

�


0 5 10 15 20 25�1

�0.5

0

0.5

1

time [s]

x1(t)x1(t)r(t)

(a) x1(t), x1(t), and r(t)

0 5 10 15 20 25−4

−2

0

2

4

time [s]


Figure 2.17: Performance of the L1 adaptive controller for ω1, θ1(t) and σ1(t).

0 5 10 15 20 25�1

�0.5

0

0.5

1

time [s]

x1(t)x1(t)r(t)

(a) x1(t), x1(t), and r(t)

0 5 10 15 20 25−6

−4

−2

0

2

4

time [s]


Figure 2.18: Performance of the L1 adaptive controller for ω1, θ1(t) and σ2(t).

Performance Verification for Different Uncertainties and Disturbances

Let r(t) = cos(2t/π ) be the reference trajectory, and let the uncertainties be given by ω1and θ1(t). The simulation results obtained with the L1 adaptive controller for differentdisturbances σi(t), i = 1,2,3, without any retuning, are shown in Figures 2.17 through 2.19.We note that the fast adaptation of the L1 adaptive controller guarantees smooth and uniformtransient performance in the presence of different unknown time-varying disturbances. Thefrequencies in the control signal match the frequencies of the disturbance that the controlleris supposed to compensate for. Notice that x1(t) and x1(t) are almost the same in all thefigures.

Further, let the disturbance be given by σ1(t). The simulation results with the L1 adap-tive controller (without any retuning) for different uncertainties ωi and θi(t), i = 1,2,3 areshown in Figures 2.17, 2.20, and 2.21. One can see that the L1 adaptive controller retainsits uniform performance.

Next, we simulate the response of the closed-loop adaptive system with the L1 adap-tive system for various step reference inputs. For this example, let the disturbance be given byσ1(t) and let the uncertainties be given by ω1 and θ1(t). The results are given in Figure 2.22.One can see from the plot that the system response scales uniformly. This scaling property istypical of linear systems and is consistent with the claims in Section 2.2.3, which state thatin the presence of fast adaptation, the input-output signals of the closed-loop L1 adaptivesystem remain close to the same signals of a bounded linear reference system.


�

�

�

�

�

�

�

�


0 5 10 15 20 25�1

�0.5

0

0.5

1

time [s]

x1(t)x1(t)r(t)

(a) x1(t), x1(t), and r(t)

0 5 10 15 20 25−6

−4

−2

0

2

4

time [s]


Figure 2.19: Performance of the L1 adaptive controller for ω1, θ1(t), and σ3(t).

0 5 10 15 20 25�1

�0.5

0

0.5

1

time [s]

x1(t)x1(t)r(t)

(a) x1(t), x1(t), and r(t)

0 5 10 15 20 25−2

−1

0

1

2

3

time [s]



Performance Verification for Nonzero Trajectory Initialization Error

To check the performance in the presence of nonzero initialization errors, we set ω = 1,θ (t) = θ = [2, 2]�, and let σ (t) = σ1(t). We use (2.58), along with (2.33), and (2.35)subject to (2.38), setting x(0) = [1, 1]� and x(0) = [0, 0]�. The simulation results inFigure 2.23 verify the performance of the L1 adaptive controller in the presence of nonzeroinitialization errors. We emphasize that there is no retuning of the L1 adaptive controllerfrom the previous case.

Next we test the performance of L1 adaptive controller in the case of time-varyinguncertainties. We set ω = 1, θ (t) = θ1(t), σ (t) = σ1(t), and use the same L1 adaptivecontroller without retuning. Figure 2.24 demonstrates no degradation in performance.

Stability Margins

Next, we verify the stability margins of the L1 adaptive controller for this system. For thispurpose, we use the same constants ω = 1, θ = [2, 2]�, and let σ (t) = σ1(t). We deriveLo(s) in (2.80) and compute the worst-case time-delay margin for all ω ∈ �0 and θ ∈ �from the open-loop system’s Bode plot. The worst-case values for the unknown parametersare ω= 1.625 and θ = [−5, 2.5]�, which lead to the phase margin φm = 88.53 deg and thetime-delay margin T = 0.0158 s.


�

�

�

�

�

�

�

�


0 5 10 15 20 25�1

�0.5

0

0.5

1

time [s]

x1(t)x1(t)r(t)

(a) x1(t), x1(t), and r(t)

0 5 10 15 20 25−10

−5

0

5

10

time [s]



0 5 10 15 20 250

5

10

15

time [s]

(a) x1(t) and r(t)

0 5 10 15 20 25−30

−20

−10

0

10

time [s]

r(t) = 1r(t) = 5r(t) = 10


Figure 2.22: Performance of the L1 adaptive controller for step reference inputs.

From (2.80) it follows that in the structure of Lo(s) the system H (s) is decoupledfrom C(s), and therefore the time-delay margin can be tuned by selection of C(s). Thus,choosing k = 13 and

D1(s) = s+0.1

s(s+0.9)

leads to C1(s) = (13s+1.3)/(s2 +13.9s+1.3), which has a lower bandwidth compared tothe previous filter and consequently leads to improved time-delay margins (see Table 2.2).The Bode plots for both C(s) and C1(s) are given in Figure 2.25. Notice that the simula-tion results for the closed-loop adaptive system without time delay in the loop, given inFigure 2.26, show that there is some degradation in the tracking performance, as expected.Thus, improving the time-delay margin hurts the transient performance, which is consistentwith the conventional claims in classical and robust control.

The example above clearly illustrates the main advantage of the L1 adaptive controller,the tuning of which is limited to the design of a linear low-pass filter, as opposed to theselection of the rate of the nonlinear gradient minimization scheme in the adaptation laws.From this perspective, the L1 adaptive control paradigm achieves clear separation betweenadaptation and robustness: the adaptation can be as fast as the CPU permits, while robustnesscan be resolved via conventional methods from linear feedback control.


�

�

�

�

�

�

�

�


0 5 10 150

0.5

1

1.5

time [s]

x1(t)x1(t)x1(t)r(t)

(a) x1(t), x1(t), x(t) and r(t)

0 5 10 15−20

0

20

40

60

time [s]


Figure 2.23: Performance of the L1 adaptive controller in the presence of nonzero initial-ization error for constant parametric uncertainties.

0 5 10 150

0.5

1

1.5

time [s]

x1(t)x1(t)x1(t)r(t)

(a) x1(t), x1(t), x(t) and r(t)

0 5 10 15−20

0

20

40

60

time [s]


Figure 2.24: Performance of the L1 adaptive controller in the presence of nonzero initial-ization error for time-varying uncertainties.

As stated in Corollary 2.2.1, the time-delay margin of the LTI system in (2.80), com-puted for the worst-case parameters, provides a conservative lower bound for the time-delaymargin of the closed-loop adaptive system. So, we set � = 109, �= [0.1, 109], � = 108,and simulate the L1 adaptive controller and obtain the values for the time-delay margin forboth filters numerically. These are given in Table 2.2. Notice that one can reduce the levelof conservatism in the estimate of the lower bound of the time-delay margin if more accu-rate information about the unknown parameters is available, which can be used for settingtighter bounds in the implementation of the projection operator. For example, consideringD(s) and letting�0 = [0.5, 1.5], and�= {ϑ = [ϑ1, ϑ2]� ∈ R

2 : ϑi ∈ [0, 3], for all i = 1, 2}leads to T = 0.021 s. This value is much closer to the real time-delay margin observed insimulations. This implies that our conservative knowledge of uncertainty hurts our abilityto predict the margin accurately, but not the margin itself.

Further, notice that a smaller value of � is preferable from an implementation point ofview. Simulations show that for the closed-loop adaptive system with D(s), if � decreasesfrom 108 to 105, the time-delay margin decreases from 0.0258 s to 0.0236 s (i.e., about 9%).However, for the case with D1(s) it decreases from 0.115 s to 0.114 s (i.e., less than 1%),which is related to the fact thatD1(s) results in C1(s) with smaller bandwidth as comparedto C(s) (see Figure 2.25). Thus, with smaller adaptation gain �, the design with D1(s) ismore suitable in terms of robustness as compared toD(s). A similar observation was made


�

�

�

�

�

�

�

�


−40

−30

−20

−10

0

Mag

nitu

de (

dB)

10−2

10−1

100

101

102

103

−90

−45

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

C(s)C1(s)

Figure 2.25: Bode plot for the low-pass filters.

Table 2.2: Worst-case phase and time-delay margins (TDM) of Lo(s) and the actual closed-loop adaptive system.

D(s) D1(s)

Phase margin of Lo(s) 88.53 deg 85.40 deg

Gain crossover frequency of Lo(s) 97.6 rad/s 21.4 rad/s

Predicted time-delay margin 0.0158 s 0.0698 s

Numerically verified TDM (� = 108) 0.0258 s 0.115 s

Numerically verified TDM (� = 105) 0.0236 s 0.114 s

in Section 2.1.5, where increasing the order of the low-pass filter helped to achieve a similarperformance level with a smaller adaptive gain �. Table 2.2 summarizes this results.

Next, keeping the same projection bounds (� = 109, � = [0.1, 109]) and settingthe adaptation gain � = 106, in Figure 2.27 we give the simulation results for the closed-loop system with D(s) in the presence of time delay of 0.015 s. The simulations verifyCorollary 2.2.1. Moreover, notice that a moderate level of the time delay does not influencethe transient tracking significantly.

Finally, we test the performance of the same L1 adaptive controller for the systemwith time-varying uncertainties without any retuning. Figure 2.28 shows the simulation


�

�

�

�

�

�

�

�


0 5 10 15 20 25 30�2

�1

0

1

2

time [s]

x1(t)x1(t)r(t)

(a) x1(t), x1(t), and r(t)

0 5 10 15 20 25 30−4

−2

0

2

4

time [s]


Figure 2.26: Performance of the L1 adaptive controller with filter D1(s) for σ1(t).

0 5 10 150

0.5

1

1.5

time [s]

x1(t)x1(t)r(t)

(a) x1(t), x1(t), and r(t)

0 5 10 15−3

−2

−1

0

1

2

time [s]


Figure 2.27: Performance of the L1 adaptive controller with time delay of 15 ms for constantparametric uncertainties.

results for ω = 1, θ (t) = θ1(t), σ (t) = σ1(t). We notice that the simulation results are verysimilar to the case of constant parametric uncertainties, given in Figure 2.27.

Remark 2.2.6 Recall the scalar system with the L1 controller considered in Section 1.3:

x(t) = −x(t)+ θ +u(t) , x(0) = 0,

˙x(t) = −x(t)+ θ (t)+u(t) , x(0) = 0,

u(s) = −C(s)θ (s) , C(s) = ωc

s+ωc ,

˙θ (t) = −�x(t) , θ (0) = 0,

x(t) = x(t)−x(t) .

The block diagram of this linear system is shown in Figure 1.5. The key feature of thissystem is that the system Lo(s) from (2.80), giving the lower bound for the time-delaymargin, depends only upon C(s) and is free of uncertainties:

Lo(s) = C(s)

1−C(s)= ωc

s.


�

�

�

�

�

�

�

�

2.3. Extension to Systems with Unmodeled Actuator Dynamics 67

0 5 10 150

0.5

1

1.5

time [s]

x1(t)x1(t)r(t)

(a) x1(t), x1(t), and r(t)

0 5 10 15−3

−2

−1

0

1

2

time [s]


Figure 2.28: Performance of the L1 adaptive controller with time delay of 15 ms for time-varying uncertainties.

Recall that the loop transfer function of the system in Figure 1.5 for its phase-margin analysisis given by

L2(s) = �C(s)

s(s+1)+�(1−C(s)).

Notice that as �→ ∞ we obtain exactly Lo(s) in the limit:

L2l(s) � lim�→∞L2l(s) = C(s)

1−C(s).

This verifies that the guaranteed lower bound for the time-delay margin of the L1 adaptivecontroller, provided by Theorem 2.2.4 and Corollary 2.2.1, is achievable for at least one LTIsystem; i.e., the result is not conservative.

2.3 Extension to Systems with Unmodeled ActuatorDynamics

This section presents the L1 adaptive control architecture for the class of uncertain lineartime-varying systems with unmodeled actuator dynamics. We prove that subject to a set ofmild assumptions, the control architecture from Section 2.2 can be used for compensation ofuncertainties within the bandwidth of the control channel, provided the selected adaptationgain is sufficiently large [28].



x(t) = Amx(t)+b(µ(t)+ θ�(t)x(t)+σ0(t))

, x(0) = x0 ,

y(t) = c�x(t) ,(2.129)

where x(t) ∈ Rn is the system state vector (measured); y(t) ∈ R is the system regulated

output;Am ∈ Rn×n is a known Hurwitz matrix specifying the desired closed-loop dynamics;


�

�

�

�

�

�

�

�


b, c ∈ Rn are known constant vectors; θ (t) ∈ R

n is a vector of time-varying unknownparameters; σ0(t) ∈ R is a time-varying disturbance; and µ(t) ∈ R is the output of thesystem

µ(s) = F (s)u(s) ,

where u(t) ∈ R is the control input and F (s) is an unknown BIBO-stable transfer functionwith known sign of its DC gain.

We repeat the set of assumptions from Section 2.2 and impose one additional assump-tion on the stability of the unmodeled actuator dynamics.


θ (t) ∈� , |σ0(t)| ≤�0 , ∀ t ≥ 0,

where � is a known convex compact set and �0 ∈ R+ is a known (conservative) bound

of σ0(t).

Assumption 2.3.2 (Uniform boundedness of the rate of variation of parameters) Letθ (t) and σ0(t) be continuously differentiable with uniformly bounded derivatives:

‖θ (t)‖ ≤ dθ <∞ , |σ0(t)| ≤ dσ0 <∞ , ∀ t ≥ 0.

Assumption 2.3.3 (Partial knowledge of actuator dynamics) There exists LF > 0 ver-ifying ‖F (s)‖L1 ≤ LF . Also, we assume that there exist known constants ωl , ωu ∈ R

satisfying

0< ωl ≤ F (0) ≤ ωu ,

where, without loss of generality, we have assumedF (0)> 0. Finally, we assume (for designpurposes) that we know a set F� of all admissible actuator dynamics.

The control objective is to design a full-state feedback-adaptive controller to ensurethat y(t) tracks a given bounded piecewise-continuous reference signal r(t) with quantifiableperformance bounds.


Definitions and L1-Norm Sufficient Condition for Stability

As in Section 2.2.2, the design of the L1 adaptive controller proceeds by considering apositive-feedback gain k > 0 and a strictly proper stable transfer function D(s), whichimply that

C(s) � kF (s)D(s)

1+kF (s)D(s)(2.130)

is a strictly proper stable transfer function with DC gain C(0) = 1 for all F (s) ∈ F�. Next,similar to the previous development, let xin(s) � (sI−Am)−1x0. Notice that from the factthatAm is Hurwitz, it follows that ‖xin‖L∞ is bounded, and xin(t) is exponentially decaying.


�

�

�

�

�

�

�

�


For the proofs of stability and performance bounds, the choice of k and D(s) needsto ensure that the following L1-norm condition holds:

‖G(s)‖L1L < 1, (2.131)

whereG(s) � (1−C(s))H (s) , H (s) � (sI−Am)−1b , L� max

θ∈� ‖θ‖1 .

To streamline the subsequent analysis of stability and performance bounds, we intro-duce the following notation. LetCu(s) �C(s)/F (s). Notice that from the definition ofC(s)given in (2.130) and the fact thatD(s) is strictly proper and stable, while F (s) is proper andstable, it follows that Cu(s) is a strictly proper and stable transfer function. Next, let

H1(s) � Cu(s)1

c�o H (s)c�o , (2.132)

where co ∈ Rn is a vector that rendersH1(s) BIBO stable. Existence of such co is proved in

Lemma A.12.1. Further, let

ρr � ‖G(s)‖L1�0 +‖H (s)C(s)kg‖L1‖r‖L∞ +‖xin‖L∞1−‖G(s)‖L1L

,

ρur � ‖Cu(s)‖L1 (|kg|‖r‖L∞ +Lρr +�0) ,

ρ � ρr +γ1 ,

ρu � ρur +γ2 ,

where kg � −1/(c�A−1m b), and

γ1 � ‖C(s)‖L1

1−‖G(s)‖L1Lγ0 +β , (2.133)

γ2 � ‖Cu(s)‖L1Lγ1 +‖H1(s)‖L1γ0 , (2.134)

with β > 0 and γ0 > 0 being arbitrarily small positive constants.Finally, using the conservative knowledge of F (0), let

�µ � maxF (s)∈F�

‖F (s)− (ωl+ωu)/2‖L1ρu , (2.135)

��0 +�µ , (2.136)

ρu � ‖ksD(s)‖L1 (ρuωu+Lρ+�+|kg|‖r‖L∞ ) .

Remark 2.3.1 Notice that if F (s) = F is an unknown constant, the system in (2.129)degenerates into the system in (2.31). Consequently, the L1-norm condition in (2.131)reduces to the one in (2.38).

The elements of the L1 adaptive control architecture are introduced next.

State Predictor



), x(0) = x0 ,

y(t) = c�x(t) ,(2.137)


�

�

�

�

�

�

�

�


where x(t) ∈ Rn is the predictor state, while ω(t), σ (t) ∈ R, and θ (t) ∈ R

n are the adaptiveestimates.

Adaptation Laws

The adaptive laws are defined using the projection operator:

˙ω(t) = �Proj(ω(t),−x�(t)Pbu(t)

), ω(0) = ω0 ,

˙θ (t) = �Proj

(θ (t),−x�(t)Pbx(t)

), θ (0) = θ0 ,

˙σ (t) = �Proj(σ (t),−x�(t)Pb

), σ (0) = σ0 ,

(2.138)

where x(t) � x(t) − x(t), � ∈ R+ is the adaptation gain, Proj(·, ·) denotes the projection

operator, and the symmetric positive definite matrix P = P� > 0 solves the Lyapunovequation A�

mP +PAm = −Q for arbitrary Q =Q� > 0. The projection operator ensuresthat ω ∈�= [ωl , ωu], θ ∈�, |σ | ≤�.

Control Law


u(s) = −kD(s)(η(s)−kgr(s)) , (2.139)

where r(s) and η(s) are the Laplace transforms of r(t) and η(t) � ω(t)u(t)+ θ�(t)x(t)+ σ (t),while k and D(s) were introduced in (2.130).

The complete L1 adaptive controller is defined via (2.137), (2.138), and (2.139),subject to the L1-norm condition in (2.131).

2.3.3 Analysis of the L1 Adaptive Controller


Consider the following closed-loop reference system:

xref (t) = Amxref (t)+b(µref (t)+ θ�(t)xref (t)+σ0(t))

, xref (0) = x0 ,

µref (s) = F (s)uref (s) ,

uref (s) = Cu(s)(kgr(s)−ηref (s)) ,

yref (t) = c�xref (t) ,

(2.140)

where xref (t) ∈ Rn is the reference system state vector and ηref (s) is the Laplace transform

of ηref (t) � θ�(t)xref (t) + σ0(t). Notice that this reference system is not implementable,as it depends upon the unknown θ (t), σ0(t), and F (s). Similar to previous sections, thisreference system is used only for analysis purposes. The next lemma proves the stability ofthis closed-loop reference system.


�

�

�

�

�

�

�

�


Lemma 2.3.1 For the closed-loop reference system given in (2.140), subject to the L1-normcondition in (2.131), the following bounds hold:

‖xref ‖L∞ ≤ ρr , ‖uref ‖L∞ ≤ ρur . (2.141)

Proof. The closed-loop reference system in (2.140) can be rewritten in the frequency do-main as follows:

xref (s) =G(s)ηref (s)+H (s)C(s)kgr(s)+xin(s) . (2.142)

Lemma A.7.1, along with the fact that for bounded signals ‖(·)τ‖L∞ ≤ ‖ ·‖L∞ , implies

‖xref τ‖L∞ ≤ ‖G(s)‖L1‖ηref τ‖L∞ +‖H (s)C(s)kg‖L1‖r‖L∞ +‖xin‖L∞ .

It follows from Assumption 2.3.1 that

‖ηref τ‖L∞ ≤ L‖xref τ‖L∞ +�0 , (2.143)

which leads to

‖xref τ‖L∞ ≤ ‖G(s)‖L1 (L‖xref τ‖L∞ +�0)+‖H (s)C(s)kg‖L1‖r‖L∞ +‖xin‖L∞ .

Keeping in mind the L1-norm condition in (2.131), we solve for ‖xref τ‖L∞ in the expressionabove to obtain the following upper bound:

‖xref τ‖L∞ ≤ ‖G(s)‖L1�0 +‖H (s)C(s)kg‖L1‖r‖L∞ +‖xin‖L∞1−‖G(s)‖L1L

= ρr .

Notice that this upper bound holds uniformly for all τ ≥ 0 and therefore

‖xref ‖L∞ ≤ ρr . (2.144)

To prove the second bound in (2.141), notice that from (2.143) and (2.144) we have

‖ηref τ‖L∞ ≤ Lρr +�0 .

Using

uref (s) = Cu(s)(kgr(s)−ηref (s)),

along with Lemma A.7.1, gives

‖uref τ‖L∞ ≤ ‖Cu(s)‖L1 (|kg|‖r‖L∞ +‖ηref τ‖L∞ )

≤ ‖Cu(s)‖L1 (|kg|‖r‖L∞ +Lρr +�0) = ρur ,

which holds uniformly for all τ ≥ 0 and proves the second bound in (2.141). �


�

�

�

�

�

�

�

�


Equivalent Linear Time-Varying System

Following Lemma A.10.1, on every finite time interval the system with unmodeled multi-plicative dynamics can be transformed into an equivalent linear time-varying system withuncertain system input gain and with an additional disturbance. Thus, we transform theoriginal system with unmodeled dynamics in (2.129) into an equivalent linear time-varyingsystem with unknown time-varying parameters. This transformation requires us to imposethe following assumptions on the signals of the system: the control signal u(t) is continuous,and moreover the following bounds hold:

‖uτ‖L∞ ≤ ρu , ‖uτ‖L∞ ≤ ρu , ∀ τ ≥ 0. (2.145)

These assumptions will be verified later in the proof of Theorem 2.3.1.Consider the system in (2.129) with u(t) subject to (2.145). Then, Lemma A.10.1

implies that µ(t) can be rewritten as

µ(t) = ωu(t)+σµ(t) ,

where ω is an unknown constant, ω ∈ (ωl , ωu), and σµ(t) is a continuous signal with(piecewise)-continuous derivative defined over t ∈ [0, τ ], such that

|σµ(t)| ≤�µ , |σµ(t)| ≤ dσµ ,

with �µ being defined in (2.135), and dσµ � ‖F (s)− (ωl +ωu)/2‖L1ρu. This implies thatone can rewrite the system in (2.129) over t ∈ [0, τ ] as follows:

x(t) = Amx(t)+b(ωu(t)+ θ�(t)x(t)+σ (t))

, x(0) = x0 ,

y(t) = c�x(t) ,(2.146)

where σ (t) � σ0(t) + σµ(t) is an unknown continuous time-varying signal, satisfying|σ (t)|<�, with � being introduced in (2.136), and |σ (t)|< dσ with dσ � dσ0 +dσµ .


Using (2.146), one can write the error dynamics over t ∈ [0, τ ] as

˙x(t) = Amx(t)+b(ω(t)u(t)+ θ�(t)x(t)+ σ (t)) , x(0) = 0, (2.147)

where ω(t) � ω(t)−ω, θ (t) � θ (t)− θ (t), and σ (t) � σ (t)−σ (t).

Lemma 2.3.2 For the error dynamics in (2.147), if u(t) is continuous, and moreover thefollowing bounds hold:

‖uτ‖L∞ ≤ ρu , ‖uτ‖L∞ ≤ ρu ,

then

‖xτ‖L∞ ≤√θm(ρu,ρu)

λmin(P )�, (2.148)

where

θm(ρu,ρu) � (ωu−ωl)2 +4maxθ∈� ‖θ‖2 +4�2 +4

λmax(P )

λmin(Q)(maxθ∈� ‖θ‖dθ +�dσ ) .


�

�

�

�

�

�

�

�



V (x(t), ω(t), θ (t), σ (t)) = x�(t)P x(t)+ 1

�(ω2(t)+ θ�(t)θ (t)+ σ 2(t)) .

Using the adaptation laws in (2.138) and Property B.2 of the projection operator, we com-pute the upper bound on the derivative of the Lyapunov function similar to the proof ofLemma 2.2.3:

V (t) ≤ −x�(t)Qx(t)+ 2

�

(|θ�(t)θ (t)|+ |σ (t)σ (t)|

).

Further, following steps similar to those of the proof of Lemma 2.2.3, we obtain the followinguniform upper bound on the prediction error, which leads to the bound in (2.148):

‖x(t)‖2 ≤ θm(ρu,ρu)

λmin(P )�, ∀ t ∈ [0, τ ]. �

Theorem 2.3.1 If the adaptive gain satisfies the design constraint

� ≥ θm(ρu,ρu)

λmin(P )γ 20

, (2.149)

where γ0 > 0 is an arbitrary constant introduced in (2.133), we have

‖x‖L∞ ≤ γ0 , (2.150)

‖xref −x‖L∞ ≤ γ1 , (2.151)

‖uref −u‖L∞ ≤ γ2 . (2.152)

Proof. We prove the bounds in (2.151) and (2.152) following a contradicting argument.Assume that (2.151) and (2.152) do not hold. Then, since

‖xref (0)−x(0)‖∞ = 0, ‖uref (0)−u(0)‖∞ = 0,

continuity of xref (t), x(t), uref (t), u(t) implies that there exists time τ > 0 for which

‖xref (t)−x(t)‖∞ < γ1 , ‖uref (t)−u(t)‖∞ < γ2, ∀t ∈ [0,τ )

and

‖xref (τ )−x(τ )‖∞ = γ1 or ‖uref (τ )−u(τ )‖∞ = γ2 .

This implies that at least one of the following equalities holds:

‖(xref −x)τ‖L∞ = γ1 , ‖(uref −u)τ‖L∞ = γ2 . (2.153)

Moreover, Lemma 2.3.1 yields the bounds

‖xref τ‖L∞ ≤ ρr , ‖uref τ‖L∞ ≤ ρur ,

which, together with the bounds in (2.153), lead to

‖xτ‖L∞ ≤ ρr +γ1 = ρ , ‖uτ‖L∞ ≤ ρur +γ2 = ρu . (2.154)


�

�

�

�

�

�

�

�


Further, consider the control law in (2.139), whose derivative can be written in thefrequency domain as

su(s) = −ksD(s)(η(s)−kgr(s)) .

From the properties of the projection operator and the upper bounds in (2.154), we have

‖ητ‖L∞ ≤ ωuρu+Lρ+� ,

and consequently

‖uτ‖ ≤ ‖ksD(s)‖L1

(ωuρu+Lρ+�+|kg|‖r‖L∞

)= ρu .

These bounds imply that the conditions of Lemma 2.3.2 hold. Then, selecting the adaptivegain � according to the design constraint in (2.149), it follows that

‖xτ‖L∞ ≤ γ0 . (2.155)

Consider next the system in (2.129). Notice that for t ∈ [0, τ ], we have

η(t) = µ(t)+ η(t)+η(t) ,

where η(t) � θ�(t)x(t)+σ0(t), η(t) � ω(t)u(t)+ θ�(t)x(t)+ σ (t). Substituting the Laplacetransform of η(t) into (2.139) yields

u(s) = −Cu(s)(η(s)+η(s)−kgr(s)) , (2.156)

which implies that the closed-loop response of the system in (2.129) with the L1 adaptivecontroller can be rewritten (in the frequency domain) as

x(s) =G(s)η(s)−H (s)C(s)(η(s)−kgr(s))+xin(s) .

Recall that, in (2.142), the response of the reference system was presented as


Then, the two expressions above lead to

xref (s)−x(s) =G(s)(ηref (s)−η(s))+H (s)C(s)η(s) . (2.157)

Since

ηref (t)−η(t) � θ�(t)(xref (t)−x(t)) ,

the following upper bound holds:

‖(ηref −η)τ‖L∞ ≤ L‖(xref −x)τ‖L∞ . (2.158)

Moreover, it follows from (2.147) that

x(s) =H (s)η(s) ,

which, along with (2.157) and (2.158), leads to

‖(xref −x)τ‖L∞ ≤ ‖G(s)‖L1L‖(xref −x)τ‖L∞ +‖C(s)‖L1‖xτ‖L∞ .


�

�

�

�

�

�

�

�


The bound in (2.155) and the definition of γ1 in (2.133) yield


1−‖G(s)‖L1L‖xτ‖L∞ ≤ ‖C(s)‖L1

1−‖G(s)‖L1Lγ0 < γ1 , (2.159)

and, thus, we obtain a contradiction to the first equality in (2.153).To show that the second equation in (2.153) also cannot hold, consider (2.140)

and (2.156), which lead to

uref (s)−u(s) = Cu(s)(η(s)−ηref (s))+Cu(s)η(s) .

Using Lemma A.12.1 and the definition of H1(s) from (2.132), we can write

Cu(s)η(s) =H1(s)x(s) ,

which, along with the result of Lemma A.7.1, yields the following upper bound:

‖(uref −u)τ‖L∞ ≤ L‖Cu(s)‖L1‖(xref −x)τ‖L∞ +‖H1(s)‖L1‖xτ‖L∞ .

Then, from the the upper bounds on ‖xτ‖L∞ and ‖(xref −x)τ‖L∞ in (2.155) and (2.159),and the definition of γ2 in (2.134), it follows that

‖(uref −u)τ‖L∞ ≤ ‖Cu(s)‖L1L(γ1 −β)+‖H1(s)‖L1γ0 < γ2 , (2.160)

which contradicts the second equality in (2.153). Notice that the upper bounds in (2.159)and (2.160) hold uniformly, which proves (2.151)–(2.152). Then, the upper bound in (2.150)follows from these two bounds and (2.155) directly. �

Remark 2.3.2 It follows from (2.149) that one can prescribe arbitrarily small γ0 by increas-ing the adaptive gain, which further implies that the performance bounds γ1 and γ2 for thesystem’s signals, both input and output, can be rendered arbitrarily small simultaneously.

Remark 2.3.3 Notice that letting k → ∞ leads to C(s) → 1, and thus the reference con-troller in the definition of the closed-loop reference system in (2.140) leads, in the limit,to perfect cancelation of uncertainties and recovers the performance of the ideal desiredsystem. Notice that if we set C(s) = 1, the transfer function

Cu(s) = C(s)

F (s)= 1

F (s)

becomes improper. Moreover, the norm of H1(s), given by

‖H1(s)‖L1 =∥∥∥∥Cu(s)

1

c�o H (s)c�o∥∥∥∥

L1

,

is not bounded since c�o H (s) is strictly proper and Cu(s) is improper. Therefore, in theabsence of C(s), one cannot obtain a uniform performance bound for the control signalsimilar to Theorem 2.3.1.

Remark 2.3.4 We notice that the ideal system input signal

µid(t) = kgr(t)− θ�(t)x(t)−σ0(t) (2.161)


�

�

�

�

�

�

�

�


is the one that leads to desired system response,

xid(t) = Amxid(t)+bkgr(t) , x(0) = x0 ,

yid(t) = c�xid(t) ,(2.162)

by canceling the uncertainties exactly. In the closed-loop reference system (2.140), µid(t) isfurther low-pass filtered by C(s) to have guaranteed low-frequency range. Thus, the refer-ence system in (2.140) has a different response as compared to (2.162) achieved with (2.161).Similar to the previous sections, the response of xref (t) and uref (t) can be made arbitrarilyclose to (2.162) by reducing ‖G(s)‖L1 . If F (s) is a relative degree one and minimum phasesystem, then ‖G(s)‖L1 can be made arbitrarily small by appropriately choosing the designparameters k andD(s). However, for the general case of unknown F (s), the design of k andD(s) which satisfy (2.131), is an open question.

2.3.4 Simulation Example: Rohrs’ Example

In this section we analyze Rohrs’ example from [146, 147], which was constructed with aparticular objective of analyzing the robustness properties of MRAC architectures. The sys-tem under consideration in [146] is a first-order-stable system with unknown time constantand DC gain, and with two highly damped unmodeled poles:

y(s) = 2

s+1µ(s) ,

µ(s) = 229

s2 +30s+229u(s) .

The system has a gain crossover frequency of ωgc = 1.70 rad/s and a phase margin of φm =107.67 deg. Its phase crossover frequency and the gain margin are ωφc = 16.09 rad/s andgm = 24.62 dB, respectively. The control objective in [146] is given via the followingfirst-order-stable reference model:

ym(s) = 3

s+3r(s) .

Model Reference Adaptive Control: Parameter Drift

The conventional MRAC controller for this system takes the form

u(t) = ky(t)y(t)+ kr (t)r(t) ,˙ky(t) = −e(t)y(t) , ky(0) = ky0 ,˙kr (t) = −e(t)r(t) , kr (0) = kr0 ,

where e(t) = y(t) −ym(t). The corresponding feedback loop of the MRAC architecture isshown in Figure 2.29.

For simulations we consider the same reference inputs as in [146]. The first referenceinput has the exact phase crossover frequency

r1(t) = 0.3+1.85sin(16.1t) ,


�

�

�

�

�

�

�

�


2s+1

229s2+30s+229

r u y

ky

kr

Plant with unmodeled dynamics

Figure 2.29: Rohrs’ example. Closed-loop MRAC system.

0 5 10 15 20−4

−2

0

2

4

time [s]

y(t)ym(t)

(a) System output y(t), ym(t)

0 5 10 15 20�20

�10

0

ky(t

)

time [s]

0 5 10 15 200

10

20

kr(t

)

(b) Controller parameters kr (t), ky (t)

Figure 2.30: MRAC: Closed-loop system’s response to r1(t).

0 5 10 15 20−4

−2

0

2

4

time [s]

y(t)ym(t)


0 5 10 15 20�10

�5

0

ky(t

)

0 5 10 15 200

5

time [s]

kr(t

)

(b) Controller parameters kr (t), ky (t)

Figure 2.31: MRAC: Closed-loop system’s response to r2(t).

while the second one is also a sinusoidal reference signal, but at a frequency which isapproximately half the phase crossover frequency,

r2(t) = 0.3+2sin(8t) .

We use the same initial conditions as in [146]:y(0) = 0, kr (0) = 1.14, and ky(0) = −0.65. Thesimulation results from [146] are reproduced here in Figures 2.30 and 2.31. In Figure 2.30,one can see that, while tracking r1(t), the closed-loop system is unstable due to parameterdrift. In Figure 2.31, bursting takes place in the response of the closed-loop adaptive systemto the reference signal r2(t).


�

�

�

�

�

�

�

�


2s+1

229s2+30s+229

ru x

− ks

kg

ω

θ

σ Plant with unmodeled dynamics

Figure 2.32: Rohrs’ example. Closed-loop L1 control system.


It is straightforward to see that Rohrs’ example can be cast into the framework of theL1 adaptive controller of this section. In fact, we can rewrite Rohrs’ example in state-spaceform as

x(t) = −3x(t)+2(µ(t)+x(t)) , x(0) = x0 ,

y(t) = x(t) ,

where

µ(s) = 229

s2 +30s+229u(s) .

Then, the state predictor takes the form

˙x(t) = −3x(t)+2(ω(t)u(t)+ θ (t)x(t)+ σ (t)

), x(0) = x0 ,

y(t) = x(t) ,

with ω(t), θ (t), and σ (t) being governed by

˙ω(t) = �Proj (ω(t),−x(t)u(t)) , ω(0) = ω0 ,˙θ (t) = �Proj

(θ (t),−x(t)x(t)

), θ (0) = θ0 ,

˙σ (t) = �Proj (σ (t),−x(t)) , σ (0) = σ0 .

The control law is given by

u(s) = −kD(s)(η(s)−kgr(s)) ,

where η(t) � ω(t)u(t)+ θ (t)x(t)+ σ (t) and kg = 32 . The block diagram of the L1 adaptive

control system is given in Figure 2.32.The simulation plots for the inputs r1(t) and r2(t), using the L1 adaptive controller

with k = 5, D(s) = 1/s, and � = 1000, are given in Figures 2.33 through 2.35. The initialconditions have been set to x0 = 0, ω0 = 1.14, θ0 = 0.65, and σ0 = 0. The projection


�

�

�

�

�

�

�

�


0 1 2 3 4 5−0.2

0

0.2

0.4

0.6

0.8

time [s]

y(t)ym(t)


0 2 4 6 8 100

5

10

ω(t

)

0 2 4 6 8 10�2

0

2

θ(t)

0 2 4 6 8 10�5

0

5

time [s]

σ(t

)

(b) Controller parameters ω(t), θ (t), σ (t)

Figure 2.33: L1 adaptive control: Closed-loop system response to r1(t).

0 1 2 3 4 5−1

−0.5

0

0.5

1

1.5

time [s]

y(t)ym(t)


0 2 4 6 8 100

5

ω(t

)

0 2 4 6 8 100

5

10θ(

t)

0 2 4 6 8 10�5

0

5

time [s]

σ(t

)

(b) Controller parameters ω(t), θ (t), σ (t)

Figure 2.34: L1 adaptive control: Closed-loop system response to r2(t).

0 1 2 3 4 5−4

−2

0

2

4

time [s]

(a) u(t) for r1(t)

0 1 2 3 4 5−4

−2

0

2

4

time [s]

(b) u(t) for r2(t)

Figure 2.35: L1 adaptive control: Control signal time history.

bounds are set to�= [−10, 10],�= 10, and �= [0.55.5]. We see from the plots that theL1 adaptive controller guarantees that both the system output and the parameters remainbounded, while achieving an expected level of performance (as both reference commandsare well beyond the bandwidth of the system).

We can get further insight into the L1 adaptive controller by analyzing the imple-mentation block diagram in Figure 2.32 from a classical control perspective. We note thatwhile the feedback gain θ (t) plays a similar role as the feedback gain ky(t) in MRAC,the adaptive parameter ω(t) appears in the feedforward path as a feedback gain around


�

�

�

�

�

�

�

�


the integrator ks

and thus has the ability to adjust the bandwidth of the low-pass filter, theoutput of which is the feedback signal of the closed-loop adaptive system. Recall that inMRAC the feedforward gain kr (t) did not play any role in the stabilization process andonly scaled the reference input. It is important to note also that the various modificationsof the adaptive laws, such as the σ -modification and the e-modification, are only meansto ensure boundedness of the parameter estimates (and thus avoid the parameter drift) butby no means affect the phase in the system, as ω(t) does in the L1 architecture. Instead,with the L1 adaptive controller, the open-loop system bandwidth changes as both ω(t) andθ (t) adapt, which leads to simultaneous adaptation on the loop gain and the phase of theclosed-loop system.

2.4 L1 Adaptive Controller for Nonlinear SystemsIn this section we present the L1 adaptive controller for systems with unknown state-and time-dependent nonlinearities. Under a mild set of assumptions, we prove that theL1 adaptive controller leads to uniform transient and steady-state performance boundsfor the system’s signals, both input and output, which can be systematically improved byincreasing the rate of adaptation. Moreover, the semiglobal positively invariant sets, towhich the state and the control signal are confined, are quantified dependent upon the filterparameters and the bounds on the partial derivatives of the unknown nonlinear function [30].


We consider the nonlinear system dynamics

x(t) = Amx(t)+b (ωu(t)+f (t ,x(t))) , x(0) = x0 ,

y(t) = c�x(t) ,(2.163)

where x(t) ∈ Rn is the system state (measured); Am ∈ R

n×n is a known Hurwitz matrixspecifying the desired closed-loop dynamics; b, c ∈ R

n are known constant vectors; u(t) ∈ R

is the control input; ω ∈ R is an unknown constant parameter with known sign, representinguncertainty in the system input gain; f (t ,x) : R×R

n → R is an unknown nonlinear mapcontinuous in its arguments; and y(t) ∈ R is the regulated output. The initial condition x0is assumed to be inside an arbitrarily large known set, i.e., ‖x0‖∞ ≤ ρ0 <∞ with knownρ0 > 0.


ω ∈�� [ωl , ωu] ,

where 0< ωl < ωu are known conservative bounds.

Assumption 2.4.2 (Uniform boundedness of f (t ,0)) There exists B > 0 such that

|f (t ,0)| ≤ B , ∀ t ≥ 0.

Assumption 2.4.3 (Semiglobal uniform boundedness of partial derivatives) For arbit-rary δ > 0, there exist dfx (δ) > 0 and dft (δ) > 0 independent of time, such that for


�

�

�

�

�

�

�

�

2.4. L1 Adaptive Controller for Nonlinear Systems 81

arbitrary ‖x‖∞ ≤ δ, the partial derivatives of f (t ,x) are piecewise-continuous and bounded,∥∥∥∥∂f (t ,x)

∂x

∥∥∥∥1≤ dfx (δ) ,

∣∣∣∣∂f (t ,x)

∂t

∣∣∣∣≤ dft (δ) .

The control objective is to design a full-state feedback adaptive controller to ensurethat y(t) tracks a given bounded piecewise-continuous reference signal r(t) with quantifiableperformance bounds.


Definitions and L1–Norm Sufficient Condition for Stability

As in Section 2.2, the design of theL1 adaptive controller proceeds by considering a feedbackgain k > 0 and a strictly proper transfer functionD(s), which lead, for allω ∈�, to a strictlyproper stable transfer function

C(s) � ωkD(s)

1+ωkD(s)

with DC gain C(0) = 1. As before, we let xin(t) be the signal with its Laplace transformbeing xin(s) � (sI−Am)−1x0. Since Am is Hurwitz, ‖xin‖L∞ ≤ ρin, where ρin � ‖s(sI−Am)−1‖L1ρ0. Further, let

Lδ � δ(δ)

δdfx (δ(δ)) , δ(δ) � δ+ γ1 , (2.164)

where dfx (·) was introduced inAssumption 2.4.3 and γ1> 0 is an arbitrary positive constant.For the proofs of stability and performance bounds, the choice of k andD(s) needs to

ensure that for a given ρ0, there exists ρr > ρin, such that the following L1-norm conditioncan be verified:

‖G(s)‖L1 <ρr −‖H (s)C(s)kg‖L1‖r‖L∞ −ρin

Lρr ρr +B, (2.165)

where G(s) �H (s)(1−C(s)), H (s) � (sI−Am)−1b, and kg � −1/(c�A−1m b) is the feed-

forward gain required for tracking of step reference command r(t) with zero steady-stateerror.

To streamline the subsequent analysis of stability and performance bounds, we needto introduce some notation. Let

H1(s) � C(s)1

c�o H (s)c�o ,

where co ∈ Rn is a vector that renders H1(s) BIBO stable and proper. We refer to Lemma

A.12.1 for the existence of such co.We also define

ρ � ρr + γ1 , (2.166)


�

�

�

�

�

�

�

�


and let γ1 be given by

γ1 � ‖C(s)‖L1

1−‖G(s)‖L1Lρrγ0 +β , (2.167)

where β and γ0 are arbitrary small positive constants such that γ1 ≤ γ1. Moreover, let

ρu � ρur +γ2 , (2.168)

where ρur and γ2 are defined as

ρur �∥∥∥∥C(s)

ω

∥∥∥∥L1

(|kg|‖r‖L∞ +Lρr ρr +B) , (2.169)

γ2 �∥∥∥∥C(s)

ω

∥∥∥∥L1

Lρr γ1 +∥∥∥∥H1(s)

ω

∥∥∥∥L1

γ0 . (2.170)

Finally, let

θb � dfx (ρ) , �� B+ ε , (2.171)

where ε > 0 is an arbitrary constant.

Remark 2.4.1 In the following analysis we demonstrate that ρr and ρ characterize thepositively invariant sets for the state of the closed-loop reference system (yet to be defined)and the state of the closed-loop adaptive system, respectively. We notice that, since γ1 canbe set arbitrarily small, ρ can approximate ρr arbitrarily closely.

Remark 2.4.2 Notice that the L1-norm condition in (2.165) is a consequence of thesemiglobal boundedness of the partial derivatives of f (t ,x), stated in Assumption 2.4.3.If f (t ,x) has a uniform bound for its derivative with respect to x, i.e., ‖ ∂f

∂x‖ ≤ dfx =L holds

uniformly for all x ∈ Rn, then

limρr→∞

ρr −‖H (s)C(s)kg‖L1‖r‖L∞ −ρin

Lρr +B = 1

L,

and the L1-norm condition in (2.165) degenerates into

‖G(s)‖L1L < 1,

which is the same condition as the one in (2.7), derived in Chapter 2 for systems with constantunknown parameters. We will prove that (2.165) is a sufficient condition for stability of theclosed-loop adaptive system.

The elements of L1 adaptive controller are introduced next.

State Predictor


˙x(t) = Amx(t)+b(ω(t)u(t)+ θ (t)‖x(t)‖∞ + σ (t)) , x(0) = x0 ,

y(t) = c�x(t) ,(2.172)

where ω(t) ∈ R, θ (t) ∈ R, and σ (t) ∈ R are the adaptive estimates.


�

�

�

�

�

�

�

�


Adaptation Laws

The adaptive estimates ω(t), θ (t), and σ (t) are governed by the following adaptation laws:

˙θ (t) = �Proj(θ (t),−x�(t)Pb‖x(t)‖∞) , θ (0) = θ0 ,

˙σ (t) = �Proj(σ (t),−x�(t)Pb) , σ (0) = σ0 ,


(2.173)

where x(t) � x(t)−x(t),� ∈ R+ is the adaptation gain, whileP =P� > 0 is the solution of

the algebraic Lyapunov equationA�mP +PAm = −Q, for arbitrary symmetricQ=Q�> 0.

The projection operator ensures that ω(t) ∈ �, θ (t) ∈ � � [−θb, θb], |σ (t)| ≤ �, with θband � being defined in (2.171).

Control Law


u(s) = −kD(s)(η(s)−kgr(s)) , (2.174)

where η(s) is the Laplace transform of η(t) � ω(t)u(t)+ θ (t)‖x(t)‖∞ + σ (t).The L1 adaptive controller is defined via (2.172), (2.173), and (2.174) subject to the

L1-norm condition in (2.165).



We consider the following closed-loop reference system, in which the control signal attemptsonly to compensate for the uncertainties within the bandwidth of the low-pass filter C(s):

xref (t) = Amxref (t)+b (ωuref (t)+f (t ,xref (t))) ,

uref (s) = C(s)

ω(kgr(s)−ηref (s)) ,

yref (t) = c�xref (t) , xref (0) = x0 ,

(2.175)

where ηref (s) is the Laplace transform of the signal ηref (t) � f (t ,xref (t)). The next lemmaestablishes the stability of the closed-loop reference system in (2.175).

Lemma 2.4.1 For the closed-loop reference system in (2.175), subject to the L1-normcondition in (2.165), if

‖x0‖∞ ≤ ρ0 ,

then

‖xref ‖L∞ < ρr , (2.176)

‖uref ‖L∞ < ρur , (2.177)

where ρr and ρur were introduced in (2.165) and (2.169), respectively.


�

�

�

�

�

�

�

�


Proof. The proof is done by contradiction. First, we note that from (2.175), it follows that

xref (s) =G(s)ηref (s)+H (s)C(s)kgr(s)+xin(s) , (2.178)

and Lemma A.7.1 yields the following bound:

‖xref τ‖L∞ ≤ ‖G(s)‖L1‖ηref τ‖L∞ +‖H (s)C(s)kg‖L1‖r‖L∞ +‖xin‖L∞ . (2.179)

If (2.176) is not true, since ‖xref (0)‖∞ = ‖x0‖∞ ≤ ρ0 < ρr and xref (t) is continuous, thenthere exists τ > 0 such that

‖xref (t)‖∞ < ρr , ∀ t ∈ [0,τ ) ,

and

‖xref (τ )‖∞ = ρr ,

which implies that

‖xref τ‖L∞ = ρr . (2.180)

Recalling the definition of δ(δ) in (2.164), we have ρr < ρr (ρr ). Then, taking into consid-eration Assumptions 2.4.2 and 2.4.3, the equality in (2.180), together with the redefinitionin (2.164), yields the following upper bound:

‖ηref τ‖L∞ ≤ Lρr ρr +B . (2.181)

Substituting this upper bound into (2.179), and noticing that for uniformly bounded signals‖(·)τ‖L∞ ≤ ‖ ·‖L∞ , we obtain

‖xref τ‖L∞ ≤ ‖G(s)‖L1 (Lρr ρr +B)+‖H (s)C(s)kg‖L1‖r‖L∞ +ρin .

The condition in (2.165) can be solved for ρr to obtain the upper bound

‖G(s)‖L1 (Lρr ρr +B)+‖H (s)C(s)kg‖L1‖r‖L∞ +ρin < ρr ,

which implies that ‖xref τ‖L∞ < ρr , thus contradicting (2.180). This proves the boundin (2.176).

Using (2.181), it follows from the definition of the reference control signal in (2.175)that

‖uref ‖L∞ ≤∥∥∥∥C(s)

ω

∥∥∥∥L1

(|kg|‖r‖L∞ +Lρr ρr +B) ,

which proves the bound in (2.177). �

Equivalent (Semi-)Linear Time-Varying System

Next, we refer to Lemma A.8.1 to transform the nonlinear system in (2.163) into a(semi-)linear system with unknown time-varying parameters and disturbances.


�

�

�

�

�

�

�

�


Since

‖x0‖∞ ≤ ρ0 < ρ , u(0) = 0,

and x(t), u(t) are continuous, there always exists τ such that

‖xτ‖L∞ ≤ ρ , ‖uτ‖L∞ ≤ ρu . (2.182)

Then, it follows from the bounds in (2.182) and Lemma A.8.1 that the system in (2.163)can be rewritten over [0, τ ] as

x(t) = Amx(t)+b (ωu(t)+ θ (t)‖x(t)‖∞ +σ (t)) , x(0) = x0 ,

y(t) = c�x(t) ,(2.183)

where θ (t) and σ (t) are unknown signals satisfying

|θ (t)|< θb , |σ (t)|<� , ∀ t ∈ [0, τ ] . (2.184)

|θ (t)| ≤ dθ (ρ,ρu) , |σ (t)| ≤ dσ (ρ,ρu) , ∀ t ∈ [0, τ ] , (2.185)

with dθ (ρ,ρu)> 0 and dσ (ρ,ρu)> 0 being the bounds guaranteed by Lemma A.8.1.


It follows from (2.172) and (2.183) that over [0, τ ] the prediction error dynamics can bewritten as

˙x(t) = Amx(t)+b(ω(t)u(t)+ θ (t)‖x(t)‖∞ + σ (t))

, x(0) = 0, (2.186)

where

ω(t) � ω(t)−ω , θ (t) � θ (t)− θ (t) , σ (t) � σ (t)−σ (t) . (2.187)

Let η(t) � ω(t)u(t)+ θ�(t)‖x(t)‖∞ + σ (t) and let η(s) be the Laplace transform of it. Then,the error dynamics in (2.186) can be rewritten in the frequency domain as

x(s) =H (s)η(s) . (2.188)

Before the main theorem, we prove the following lemma.

Lemma 2.4.2 For the system in (2.186), if u(t) is continuous, and moreover the followingbounds hold:

‖xτ‖L∞ ≤ ρ , ‖uτ‖L∞ ≤ ρu , (2.189)

then

‖xτ‖L∞ ≤√θm(ρ,ρu)

λmin(P )�,

where

θm(ρ,ρu) � 4θ2b +4�2 + (ωu−ωl)2

+4λmax(P )

λmin(Q)(θbdθ (ρ,ρu)+�dσ (ρ,ρu)) .

(2.190)


�

�

�

�

�

�

�

�



V (x(t), ω(t), θ (t), σ (t)) = x�(t)P x(t)+ 1

�

(ω2(t)+ θ2(t)+ σ 2(t)

).

We can verify straightforwardly that

V (0) ≤ (ωu−ωl)2 +4θ2b +4�2

�≤ θm(ρ,ρu)

�.

Further, we need to prove that

V (t) ≤ θm(ρ,ρu)

�, ∀ t ∈ [0, τ ] .

It follows from (2.189) that the bounds in (2.185) hold for arbitrary t ∈ [0, τ ], i.e.,

|θ (t)| ≤ dθ (ρ,ρu) , |σ (t)| ≤ dσ (ρ,ρu) , ∀ t ∈ [0, τ ] . (2.191)

Let t1 ∈ (0, τ ] be the first time instant of the discontinuity of either of the derivatives of θ (t)and σ (t). Using the projection-based adaptation laws from (2.173), the following upperbound for V (t) can be obtained for t ∈ [0, t1):

V (t) ≤ −x�(t)Qx(t)+ 2

�

∣∣∣θ (t)θ (t)+ σ (t)σ (t)∣∣∣ . (2.192)

The projection algorithm ensures that for all t ∈ [0, t1),

ωl ≤ ω(t) ≤ ωu , |θ (t)| ≤ θb , |σ (t)| ≤� , (2.193)

and therefore

maxt∈[0, t1)

(1

�

(ω2(t)+ θ2(t)+ σ 2(t)

))≤ (ωu−ωl)2 +4θ2

b +4σ 2b

�. (2.194)

If at arbitrary t ′ ∈ [0, t1)

V (t ′)> θm(ρ, ρu)

�,

where θm(ρ,ρu) was defined in (2.190), then it follows from (2.194) that

x�(t ′)P x(t ′)> 4λmax(P )

�λmin(Q)(θbdθ (ρ,ρu)+�dσ (ρ,ρu)) .

Hence

x�(t ′)Qx(t ′) ≥ λmin(Q)

λmax(P )x�(t ′)P x(t ′)> 4

θbdθ (ρ,ρu)+�dσ (ρ,ρu)

�. (2.195)

Moreover, it follows from (2.187) and (2.193) that for all t ∈ [0, t1)

|θ (t)| ≤ 2θb , |σ (t)| ≤ 2� . (2.196)

Since θ (t) and σ (t) are continuous over [0, t1), the upper bounds in (2.191) and (2.196) leadto the following upper bound:

|θ (t)θ (t)+ σ (t)σ (t)|�

≤ 2θbdθ (ρ,ρu)+�dσ (ρ,ρu)

�. (2.197)


�

�

�

�

�

�

�

�


If V (t ′)> θm(ρ,ρu)�

, then from (2.192), (2.195), and (2.197) we have

V (t ′)< 0.

Thus, we have V (t) ≤ θm(ρ,ρu)�

for all t ∈ [0, t1).Since λmin(P )‖x(t)‖2 ≤ x�(t)P x(t) ≤ V (t), then from the continuity of V (·) we get

the following upper bound for all t ∈ [0, t1]:

‖x(t)‖∞ ≤ ‖x(t)‖ ≤√θm(ρ,ρu)

λmin(P )�.

Let t2 ∈ (t1, τ ] be the next time instant such that discontinuity of any of the derivatives θ (t)and σ (t) occurs. Using similar derivations as above, we can prove that

‖x(t)‖∞ ≤√θm(ρ,ρu)

λmin(P )�, t ∈ [t1, t2] .

Iterating the process until the time instant τ , we get

‖xτ‖L∞ ≤√θm(ρ,ρu)

λmin(P )�,

which concludes the proof. �

Theorem 2.4.1 Consider the closed-loop reference system in (2.175) and the closed-loopsystem consisting of the system in (2.163) and the L1 adaptive controller in (2.172)–(2.174)subject to the L1-norm condition (2.165). If the adaptive gain is chosen to verify the designconstraint

� ≥ θm(ρ,ρu)

λmin(P )γ 20

, (2.198)

then we have

‖x‖L∞ ≤ γ0 , (2.199)

‖xref −x‖L∞ ≤ γ1 , (2.200)

‖uref −u‖L∞ ≤ γ2 , (2.201)

where γ1 and γ2 are as defined in (2.167) and (2.170), respectively.

Proof. The proof is done by contradiction. Assume that the bounds (2.200) and (2.201) donot hold. Then, since ‖xref (0)−x(0)‖∞ = 0< γ1, uref (0)−u(0) = 0, and x(t), xref (t), u(t),uref (t) are continuous, there exists τ > 0 such that

‖xref (t)−x(t)‖∞ < γ1 , ‖uref (t)−u(t)‖∞ < γ2, ∀ t ∈ [0,τ )

and

‖xref (τ )−x(τ )‖∞ = γ1 , or ‖uref (τ )−u(τ )‖∞ = γ2 ,


�

�

�

�

�

�

�

�


which implies that at least one of the following equalities holds:


Taking into consideration the definitions of ρ and ρu in (2.166) and (2.168), it follows fromLemma 2.4.1 and the equalities in (2.202) that

‖xτ‖L∞ ≤ ρ , ‖uτ‖L∞ ≤ ρu . (2.203)

These bounds imply that the assumptions of Lemma 2.4.2 hold. Then, selecting the adaptivegain � according to the design constraint in (2.198), it follows that

‖xτ‖L∞ ≤ γ0 . (2.204)

Let η(t) � θ (t)‖x(t)‖∞ +σ (t). Then, it follows from (2.174) that

u(s) = −kD(s)(ωu(s)+η(s)+ η(s)−kgr(s)) ,

which can be rewritten as

u(s) = − kD(s)

1+kωD(s)(η(s)+ η(s)−kgr(s)) = −C(s)

ω(η(s)+ η(s)−kgr(s)) . (2.205)

The response of the closed-loop system in the frequency domain consequently takes theform

x(s) =G(s)η(s)−H (s)C(s)η(s)+H (s)C(s)kgr(s)+xin(s) .

This expression together with the response of the closed-loop reference system in (2.178)yields

xref (s)−x(s) =G(s)(ηref (s)−η(s))+H (s)C(s)η(s) .

Moreover, the relationship in (2.188) leads to

xref (s)−x(s) =G(s)(ηref (s)−η(s))+C(s)x(s) . (2.206)

Since for all t ∈ [0,τ ] the equalities

ηref (t)−η(t) = f (t ,xref (t))− (θ (t)‖x(t)‖∞ +σ (t)) = f (t ,xref (t))−f (t ,x(t))

hold, we have

‖(ηref −η)τ‖L∞ ≤ ‖(f (t ,xref )−f (t ,x))τ‖L∞ .

Taking into account that ‖x(t)‖∞ ≤ ρ = ρr (ρr ), and also ‖xref (t)‖∞ ≤ ρr < ρr (ρr ) for allt ∈ [0, τ ], Assumption 2.4.3 implies that for all t ∈ [0, τ ],

|f (t ,xref )−f (t ,x)| ≤ dfx (ρr (ρr ))‖xref (t)−x(t)‖∞ .

From the redefinition in (2.164) it follows that dfx (ρr (ρr ))< Lρr , and hence

‖(ηref −η)τ‖L∞ ≤ Lρr‖(xref −x)τ‖L∞ .


�

�

�

�

�

�

�

�


From this bound and the dynamics in (2.206), we get the following upper bound:

‖(xref −x)τ‖L∞ ≤ ‖G(s)‖L1Lρr‖(xref −x)τ‖L∞ +‖C(s)‖L1‖xτ‖L∞ .

Then, the upper bound in (2.204) and the L1-norm condition in (2.165) lead to the upperbound


1−‖G(s)‖L1Lρrγ0 = γ1 −β < γ1 , (2.207)

which contradicts the first equality in (2.202).To show that the second equation in (2.202) also cannot hold, we notice that from

(2.175) and (2.205) one can derive


ω(ηref (s)−η(s))+ C(s)

ωη(s) .

Further, since Lemma A.12.1 implies

C(s)

ωη(s) = 1

ωH1(s)x(s) ,

it follows from Lemma 2.4.2 that

‖(uref −u)τ‖L∞ ≤∥∥∥∥C(s)

ω

∥∥∥∥L1

Lρr‖(xref −x)τ‖L∞ +∥∥∥∥H1(s)

ω

∥∥∥∥L1

‖xτ‖L∞ .

The upper bounds in (2.204) and (2.207) and the definition of γ2 in (2.170) lead to

‖(uref −u)τ‖L∞ ≤∥∥∥∥C(s)

ω

∥∥∥∥L1

Lρr (γ1 −β)+∥∥∥∥H1(s)

ω

∥∥∥∥L1

γ0 < γ2 ,

which contradicts the second equality in (2.202). This proves the bounds in (2.200)–(2.201).Thus, the bounds in (2.203) hold uniformly, which implies that the bound in (2.204) alsoholds uniformly. This proves the bound in (2.199). �

Remark 2.4.3 It follows from (2.198) that one can prescribe the arbitrary desired per-formance bound γ0 by increasing the adaptive gain, which further implies from (2.167)and (2.170) that one can achieve arbitrarily small performance bounds γ1 and γ2 simulta-neously.

Remark 2.4.4 Similar to the previous section, notice that letting k→ ∞ leads toC(s) → 1,and thus the reference controller in the definition of the closed-loop reference systemin (2.140) leads, in the limit, to perfect cancelation of uncertainties and recovers the per-formance of the ideal desired system. As before, one can check that setting C(s) = 1 takesaway the uniform bound for the control signal, as the resultingH1(s) is an improper system.

Remark 2.4.5 Notice that the use of the parametrization f (t ,x(t)) = θ (t)||x(t)||∞ +σ (t)from Lemma A.8.1 led to the definition of a semiglobal positively invariant set, where thesolutions lie. The obtained uniform performance bounds, in particular, question the need forneural-network-based approximation schemes for adaptive control, where the uncertaintiesare limited to state-dependent nonlinearities and the approximation properties are of exis-tence nature, the convergence domains are local, and the obtained results are on ultimateboundedness.


�

�

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�


2.4.4 Simulation Example: Wing Rock

In this section, we explore application of the L1 adaptive controller to wing rock, whichcan be described by a second-order nonlinear system. Wing rock is the limit cycle behaviorin flight dynamics, caused by flow asymmetries in the result of nonlinear aerodynamicroll damping. It has been an active topic of research in the aerospace community overthe past three decades (see [26] and the references therein). We test the performance ofthe L1 adaptive controller for various flight conditions. We also verify robustness of theclosed-loop adaptive system to time delays. More details can be found in [26, 89].

Problem Formulation and Application of the L1 Adaptive Controller

An analytical model of wing rock for slender delta wings is given by [7, 65]

φ(t)+ a0

t2rφ(t)+ a1

trφ(t)+a2|φ(t)|φ(t)+ a3

t2rφ3(t)+ a4

trφ2(t)φ(t)

+ ω

t2ru(t)+d(t ,φ(t), φ(t)) = 0,

(2.208)

where φ(t) ∈ R is the roll angle, d(t ,φ(t), φ(t)) ∈ R models disturbances and unknownnonlinearities, u(t) ∈ R is the antisymmetric aileron deflection, ω ∈ R is the unknowncontrol effectiveness, and tr is the reference time conversion coefficient. Both the roll angleφ(t) and its derivative φ(t) are assumed to be available for feedback. In this model, ω = 1corresponds to the nominal control efficiency. The aerodynamic coefficients a0,a1,a2,a3,a4and the control efficiency ω depend upon the angle of attack and are given in Table 2.3 fortwo fixed values of angle of attack α [65]. Since the control efficiency is not precisely knownand depends upon the flight condition, it is treated as an unknown parameter in the design ofthe L1 adaptive controller. For the airspeed Vf = 30 m/s and the wingspan bw = 169 mm,we have tr = bw/(2Vf ) = 0.0028 s.

Table 2.3: Coefficients for wing rock motion.

α a0 a1 a2 a3 a4 ω

27.0 deg 0.0050 −0.0100 0.2000 −0.0025 0.0250 0.9000

35.0 deg 0.0060 −0.0120 0.2000 −0.0075 0.0400 1.2000

Letting x � [φ, φ]�, the system in (2.208) can be rewritten in state-space form as

x(t) = Ax(t)+b(ωu(t)+g0(t ,x(t)) , x(0) = x0

φ(t) = c�x(t) ,(2.209)

where

A�[

0 1−a0/t

2r −a1/tr

], b �

[0

1/t2r

], c �

[10

],

g0(t ,x) � −t2r a2|φ(t)|φ(t)−a3φ3(t)− tra4φ

2φ(t)− t2r d(t ,φ(t), φ(t)) .


�

�

�

�

�

�

�

�


For this problem, we consider the following control law:

u(t) = um(t)+uad(t) ,

um(t) = −k�mx(t) ,

where km ∈ R2 is the static feedback gain and uad(t) is the adaptive control signal. Substi-

tuting this control law in (2.209), we obtain the following partially closed-loop dynamics:

x(t) =(A−bk�m

)x(t)+b

(ωuad(t)+ (1−ω)k�mx(t)+g0(t ,x(t))

). (2.210)

Further, denoting

Am �(A−bk�m

), g(t ,x) � (1−ω)k�mx+g0(t ,x) , (2.211)

we can rewrite (2.210) as follows:

x(t) = Amx(t)+b (ωuad(t)+g(t ,x(t))) , x(0) = x0,

φ(t) = c�x(t) .(2.212)

We select the static feedback gain km so that the state matrixAm is Hurwitz and has its polesin desired locations. Then, letting

Am �[

0 1−am1 −am2

],


km =[

t2r am1−a0ω

t2r am2−tr a1ω

],

where am1, am2 are the design parameters specifying the desired closed-loop dynamics.Notice that the dynamics in (2.212) can be cast into the form given in Section 2.4.1. Thus,we can define the adaptive control uad(t) using (2.172), (2.173), and (2.174), subject to theL1-norm condition in (2.165).

Simulation Results

The phase portrait of the open-loop system for d(t ,φ, φ) ≡ 0 and for the angle of attack α =27.0 deg is given in Figure 2.36. One can see that the system’s state trajectories contain astable limit cycle. All trajectories starting inside the limit cycle converge to the limit cycle,while most of the trajectories outside the limit cycle are unstable.

Letting am1 = 50, am2 = 14.14 leads to km = [−0.0046, 0.0001]. The desired systemdynamics are given by

xid(t) = Amxid(t)+bkgr(t) ,

φid(t) = c�xid(t) .

Further, let k = 144, and let the filter D(s) be given by

D(s) = (s+500)(s+0.00400)2

s(s+368)(s+0.00439)2.


�

�

�

�

�

�

�

�


�150 �100 �50 0 50 100 150�2000

�1000

0

1000

2000

[deg]

[deg

/s]

Figure 2.36: Phase portrait for free wing rock motion.

Also, let the projection bounds be�= [0.3, 2],�= [−10, 10],�= 10, and let the adaptivegain be � = 107.

We consider two simulation scenarios for different values of angle of attack: α1 =27.0 deg and α2 = 35.0 deg. In both scenarios we inject the disturbance

d(t ,φ, φ) = 10sin(2πt)+20sin(πt)

into the plant dynamics and check the response of the closed-loop adaptive system to stepreference signals of different amplitude.

The simulation results for the first scenario are given in Figures 2.37 and 2.38(a) and forthe second scenario in Figures 2.39 and 2.38(b). In both cases we use the same controllerwithout any retuning. Notice that the response of the closed-loop system is close to thedesired system, which is scaled for different reference inputs. Moreover, the L1 adaptivecontroller is able to control the highly nonlinear system while working in different state-space regions with different stability properties (see Figure 2.38).

In Figures 2.37(b) and 2.39(b), we also plot the control history of the same system withd(t ,x, x) ≡ 0, along with the control signal of the closed-loop adaptive system. Comparisonof the control signals shows that the control system with the adaptive controller generatesadequate signals to compensate for disturbances, while in the absence of disturbances itproduces clean control signal without oscillatory components.

In Section 2.2.5, we demonstrated that for LTI systems the L1 adaptive controllerhas a bounded-away-from-zero time-delay margin in the presence of fast adaptation. In thefollowing simulations, we numerically verify a similar claim for the nonlinear wing rockexample, using the insights from Section 2.2.5. Assume that the control input in (2.208) isreplaced with the delayed signal ud (t), defined as

ud (t) ={

0, 0 ≤ t ≤ τ ,u(t− τ ) , t > τ ,


�

�

�

�

�

�

�

�

2.5. L1 Adaptive Controller in the Presence of Unmodeled Dynamics 93

0 1 2 3 4 50

20

40

60

80

time [s]

φ(t)r(t)φid(t)

(a) Control system response φ(t), desiredresponse φid(t), and the reference command

0 1 2 3 4 50

0.05

0.1

0.15

0.2

time [s]

r1(t)r2(t)r3(t)

(b) Control history u(t) in the presence ofuncertainties and without (dotted)

Figure 2.37: Simulation results for α = 27.5 deg.

−60 −40 −20 0 20 40 60−1000

−500

0

500

1000

φ [deg]

φ[d

eg/s]

r1(t)r2(t)r3(t)

(a) α = 27.0 deg

−60 −40 −20 0 20 40 60−1000

−500

0

500

1000

φ [deg]

φ[d

eg/s]

r1(t)r2(t)r3(t)

(b) α = 35.0 deg

Figure 2.38: Phase portraits for both simulation scenarios.

0 1 2 3 4 50

10

20

30

40

50

60

time [s]

φ(t)r(t)φid(t)

(a) Control system response φ(t), desiredresponse φid(t), and the reference command

0 1 2 3 4 5−0.05

0

0.05

0.1

0.15

time [s]

r1(t)r2(t)r3(t)

(b) Control history u(t) in the presence ofuncertainties and without (dotted)

Figure 2.39: Simulation results for α = 35.0 deg.

where τ is the time delay. Figure 2.40 demonstrates the tracking performance for the firstscenario without any retuning of the original L1 controller for τ = 5 ms. Figure 2.41demonstrates the simulation results for the second scenario with the same adaptive controller.We notice that the closed-loop system does not lose its stability in the presence of time delay.Also notice that the closed-loop performance in the presence of the time delay for bothscenarios is almost the same as in the systems without time delay (Figures 2.37 and 2.39).


�

�

�

�

�

�

�

�


0 1 2 3 4 50

20

40

60

80

time [s]

φ(t)r(t)φid(t)

(a) Control system response φ(t), ideal responseφid(t), and the reference command

0 1 2 3 4 50

0.05

0.1

0.15

0.2

time [s]

r1(t)r2(t)r3(t)

(b) Control history u(t)

Figure 2.40: Time histories for α = 27.5 deg in the presence of time delay of 5 ms.

0 1 2 3 4 50

10

20

30

40

50

60

time [s]

φ(t)r(t)φid(t)

(a) Control system response φ(t), ideal responseφid(t), and the reference command

0 1 2 3 4 5−0.05

0

0.05

0.1

0.15

time [s]

r1(t)r2(t)r3(t)

(b) Control history u(t)

Figure 2.41: Time histories for α = 35.0 deg in the presence of time delay of 5 ms.

2.5 L1 Adaptive Controller in the Presence of NonlinearUnmodeled Dynamics

In this section we integrate the tools from previous sections to expand the class of nonlinearsystems, for which the L1 controller can be designed and analyzed with similar claims. Weconsider unmodeled actuator dynamics and also unmodeled internal dynamics in a nonlin-ear system with time- and state-dependent nonlinearities. We prove that with appropriateredefinition of the reference system the performance bounds from previous sections holdsemiglobally [31].


In this section we present the L1 adaptive controller for nonlinear systems in the presenceof unmodeled dynamics. Thus, consider the following class of systems:

x(t) = Amx(t)+b(µ(t)+f (t ,x(t),z(t)))

, x(0) = x0 ,

xz(t) = g(t ,xz(t),x(t)) , xz(0) = xz0 ,

z(t) = g0(t ,xz(t)) ,

y(t) = c�x(t) ,

(2.213)


�

�

�

�

�

�

�

�


where x(t) ∈ Rn is the system state (measured); Am ∈ R

n×n is a known Hurwitz matrixspecifying the desired closed-loop dynamics; b, c ∈ R

n are known constant vectors; y(t) ∈ R

is the system output; f : R×Rn×R

l → R is an unknown nonlinear map, which representsunknown system nonlinearities; xz(t) ∈ R

m and z(t) ∈ Rl are the state and the output of

unmodeled nonlinear dynamics; g : R×Rm×R

n→ Rm and g0 : R×R

m→ Rl are unknown

nonlinear maps continuous in their arguments; and µ(t) ∈ R is the output of the followingsystem:

µ(s) = F (s)u(s) ,

where u(t) ∈ R is the control input and F (s) is an unknown BIBO-stable and proper transferfunction with known sign of its DC gain. The initial condition x0 is assumed to be inside anarbitrarily large known set, i.e., ‖x0‖∞ ≤ ρ0 <∞ with known ρ0 > 0. LetX � [x�, z�]�,and with a slight abuse of language let f (t ,X) � f (t ,x,z).

Assumption 2.5.1 (Uniform boundedness of f (t ,0)) There exists B > 0, such that|f (t ,0)| ≤ B holds for all t ≥ 0.

Assumption 2.5.2 (Semiglobal uniform boundedness of partial derivatives) For arbit-rary δ > 0 there exist positive constants dfx (δ) > 0 and dft (δ) > 0 independent of time,such that for all ‖X‖∞ ≤ δ the partial derivatives of f (t ,X) are piecewise-continuous andbounded, ∥∥∥∥∂f (t ,X)

∂X

∥∥∥∥1≤ dfx (δ),

∣∣∣∣∂f (t ,X)

∂t

∣∣∣∣≤ dft (δ) .

Assumption 2.5.3 (Stability of unmodeled dynamics) The xz-dynamics are BIBO stableboth with respect to initial conditions xz0 and input x(t), i.e., there exist L1, L2 > 0 suchthat for all t ≥ 0

‖zt‖L∞ ≤ L1‖xt‖L∞ +L2 .

Assumption 2.5.4 (Partial knowledge of actuator dynamics) There exists LF > 0, ver-ifying ‖F (s)‖L1 ≤ LF . Also, we assume that there exist known constants ωl , ωu ∈ R,satisfying

0< ωl ≤ F (0) ≤ ωu ,


The control objective is to design a full-state feedback adaptive controller to ensurethat the output of the system y(t) tracks a given bounded piecewise-continuous referencesignal r(t) with uniform and quantifiable performance bounds.



Similar to Section 2.2.2, the design of the L1 adaptive controller proceeds by consideringa feedback gain k > 0 and a strictly proper stable transfer function D(s), which imply that

C(s) � kF (s)D(s)

1+kF (s)D(s)(2.214)


�

�

�

�

�

�

�

�


is a strictly proper stable transfer function with DC gainC(0) = 1, for all F (s) ∈ F�. Similarto the previous sections, let xin(s) � (sI −Am)−1x0. Notice that from the fact that Am isHurwitz it follows that ‖xin‖L∞ ≤ ρin, where ρin � ‖s(sI−Am)−1‖L1ρ0. Further, let

Lδ � δ(δ)

δdfx (δ(δ)) , δ(δ) � max{δ+ γ1, L1(δ+ γ1)+L2} , (2.215)

wheredfx (·) was introduced inAssumption 2.5.2, and γ1> 0 is an arbitrary positive constant.For the proofs of stability and performance bounds, the choice of k andD(s) needs to

ensure that for a given ρ0, there exists ρr > ρin, such that the following L1-norm conditioncan be verified:

‖G(s)‖L1 <ρr −‖kgC(s)H (s)‖L1‖r‖L∞ −ρin

Lρr ρr +B, (2.216)

where G(s) �H (s)(1−C(s)), H (s) � (sI−Am)−1b, and kg � −1/(c�A−1m b) is the feed-

forward gain required for tracking of step-reference command r(t) with zero steady-stateerror.

To streamline the subsequent analysis of stability and performance bounds, we needto introduce some notation. Let Cu(s) � C(s)/F (s). Notice that from the definition of C(s)given in (2.214) and the fact thatD(s) is strictly proper and stable, while F (s) is proper andstable, it follows that Cu(s) is a strictly proper and stable transfer function. Further, similarto Section 2.3, we define

H1(s) � Cu(s)1

c�o H (s)c�o , (2.217)

where co ∈ Rn is a vector that renders H1(s) BIBO stable and proper. We refer to Lemma

A.12.1 for the existence of such co.We also define

ρ � ρr + γ1 ,

where

γ1 � ‖C(s)‖L1

1−‖G(s)‖L1Lρrγ0 +β , (2.218)

with β and γ0 being arbitrary small positive constants, such that γ1 ≤ γ1. Moreover, let

ρu � ρur +γ2 ,


ρur � ‖Cu(s)‖L1 (|kg|‖r‖L∞ +Lρr ρr +B) ,

γ2 � ‖Cu(s)‖L1Lρr γ1 +‖H1(s)‖L1γ0 . (2.219)

Finally, using the conservative knowledge of F (s), let

�1 � LρL2 +B+ ε , (2.220)

�2 � maxF (s)∈F�

‖F (s)− (ωl+ωu)/2‖L1ρu , (2.221)

��1 +�2 , (2.222)

ρu � ‖ksD(s)‖L1 (ρuωu+Lρρ+σb+|kg|‖r‖L∞ ) ,

where ε is an arbitrary positive constant.


�

�

�

�

�

�

�

�



The elements of the L1 controller are introduced next.

State Predictor


˙x(t) = Amx(t)+b(ω(t)u(t)+ θ (t)‖xt‖L∞ + σ (t)) , x(0) = x0 ,

y(t) = c�x(t) ,(2.223)

where x(t) ∈ Rn is the state of the predictor, while ω(t), θ (t), σ (t) ∈ R are the adaptive

estimates.

Adaptation Laws

The adaptive laws are defined via the projection operator as follows:


˙θ (t) = �Proj(θ (t),−x�(t)Pb‖xt‖L∞ ) , θ (0) = θ0 ,

˙σ (t) = �Proj(σ (t),−x�(t)Pb) , σ (0) = σ0 ,

(2.224)

where x(t) � x(t)−x(t), � ∈ R+ is the adaptation gain, and P = P� > 0 is the solution of

the algebraic Lyapunov equationA�mP +PAm = −Q for arbitrary symmetricQ=Q� > 0.

The projection operator ensures that ω(t) ∈�� [ωl , ωu], θ (t) ∈�= [−Lρ , Lρ], and also|σ (t)| ≤�.

Control Law


u(s) = −kD(s)(η(s)−kgr(s)) , (2.225)

where η(t) � ω(t)u(t)+ θ (t)‖xt‖L∞ + σ (t), while k andD(s) are as introduced in (2.214).The L1 adaptive controller is defined via (2.223), (2.224), and (2.225), subject to the

L1-norm condition in (2.216).




xref (t) = Amxref (t)+b(µref (t)+f (t ,xref (t),z(t)))

, xref (0) = x0 ,


uref (s) = Cu(s)(kgr(s)−ηref (s)) ,


(2.226)


�

�

�

�

�

�

�

�




Closed-Loop Adaptive System

Unmodeled Dynamics

f (t ,x,z)

f (t ,xref ,z)

x = Amx+b(·)y = c�x

xref = Amxref +b(·)yref = c�xref

Cu(s)

F (s)

F (s)kg

xz = g(t ,xz,x)z = g0(t ,xz)

r

ru

uref

µ

µref

x

xref

η

ηref

z

Figure 2.42: Closed-loop adaptive system and closed-loop reference system.

where xref (t) ∈ Rn is the state of the reference system, uref (t) ∈ R is the control input, and

ηref (s) denotes the Laplace transform of ηref (t) � f (t ,xref (t),z(t)). Note that z(t) is seenas an external disturbance to the closed-loop reference system. The block diagram of theclosed-loop system with the L1 adaptive controller and the closed-loop reference system isillustrated in Figure 2.42. The next lemma proves the stability of this closed-loop referencesystem subject to an additional assumption on z(t), which will later be verified in the proofof stability and performance bounds of the closed-loop adaptive system (Theorem 2.5.1).

Lemma 2.5.1 For the closed-loop reference system given in (2.226), subject to the L1-normcondition in (2.216), if for some τ ≥ 0

‖zτ‖L∞ ≤ L1(‖xref τ‖L∞ +γ1)+L2 , (2.227)

then the following bounds hold:

‖xref τ‖L∞ < ρr , ‖uref τ‖L∞ < ρur . (2.228)

Proof. The closed-loop reference system in (2.226) can be rewritten as

xref (s) =G(s)ηref (s)+H (s)C(s)kgr(s)+xin(s) . (2.229)

Lemmas A.6.2 and A.7.5 imply

‖xref τ‖L∞ ≤ ‖G(s)‖L1‖ηref τ‖L∞ +‖H (s)C(s)kg‖L1‖r‖L∞ +‖xin‖L∞ . (2.230)

Next, we use a contradictive argument to show stability of the closed-loop reference system.Assume that (2.228) does not hold. Then, since ‖xref (0)‖∞ = ‖x0‖∞ ≤ ρ0 < ρr and xref (t)is continuous, there exists a time instant τ1 ∈ (0, τ ], such that

‖xref (t)‖∞ < ρr , ∀ t ∈ [0,τ1) ,


�

�

�

�

�

�

�

�


and

‖xref (τ1)‖∞ = ρr ,

which implies that

‖xref τ1‖L∞ = ρr . (2.231)

The bound in (2.227) implies that ‖zτ1‖L∞ ≤ L1(ρr +γ1)+L2, which leads to

‖Xref τ1‖L∞ ≤ max{ρr +γ1, L1(ρr +γ1)+L2} ≤ ρr (ρr ) ,

where we have used the definition of δ(δ) in (2.215). Assumption 2.5.2 implies that, for all‖Xref ‖∞ ≤ ρr (ρr ), we have

|f (t ,Xref )−f (t ,0)| ≤ dfx (ρr (ρr ))‖Xref ‖∞ , ∀ t ∈ [0, τ ] .

Further, Assumption 2.5.1 and the definition of Lδ in (2.215) lead to

‖ηref τ1‖L∞ ≤ dfx (ρr (ρr ))ρr (ρr )+B = Lρr ρr +B . (2.232)

Thus, the bound in (2.230), together with the fact that ‖xin‖L∞ ≤ ρin, yields

‖xref τ1‖L∞ ≤ ‖G(s)‖L1 (Lρr ρr +B)+‖H (s)C(s)kg‖L1‖r‖L∞ +ρin .

The condition in (2.216) can be solved for ρr to obtain the upper bound

‖G(s)‖L1 (Lρr ρr +B)+‖kgC(s)H (s)‖L1‖r‖L∞ +ρin < ρr ,

which implies ‖xref τ1‖L∞ < ρr . This contradicts (2.231), which proves the first resultin (2.228). As this bound is strict and holds uniformly for all τ1 ∈ (0, τ ], one can rewrite thebound in (2.232) as a strict inequality,

‖ηref τ‖L∞ < Lρr ρr +B .

From the definition of the reference control signal in (2.226), it follows that

‖uref τ‖L∞ ≤ ‖Cu(s)‖L1 (|kg|‖r‖L∞ +‖ηref τ‖L∞ )

< ‖Cu(s)‖L1 (|kg|‖r‖L∞ +Lρr ρr +B) = ρur ,



In this section we transform the original nonlinear system with unmodeled dynamics in(2.213) into an equivalent (semi-)linear time-varying system with unknown time-varyingparameters and disturbances. This transformation requires us to impose the following as-sumptions on the signals of the system: the control signal u(t) is continuous, and moreoverthe following bounds hold

‖xτ‖L∞ ≤ ρ , ‖uτ‖L∞ ≤ ρu , ‖uτ‖L∞ ≤ ρu . (2.233)

These assumptions will be verified later in the proof of Theorem 2.5.1. Next we constructthe equivalent system in two steps.

L1book2010/7/22page 100

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First Equivalent System From the system dynamics in (2.213) and the first two boundsin (2.233), it follows that ‖xt‖L∞ is bounded for all t ∈ [0, τ ]. Thus, Lemma A.9.1 impliesthat there exist continuous θ (t) and σ1(t) with (piecewise)-continuous derivatives, definedfor t ∈ [0, τ ], such that

|θ (t)|< Lρ , |θ (t)| ≤ dθ , (2.234)

|σ1(t)|<�1 , |σ1(t)| ≤ dσ1 ,

and

f (t ,x(t),z(t)) = θ (t)‖xt‖L∞ +σ1(t) ,

where Lρ and �1 are defined in (2.215) and (2.220), while the algorithm for computationof dθ > 0, dσ1 > 0 is derived in the proof of Lemma A.9.1. Thus, the system in (2.213) canbe rewritten over t ∈ [0, τ ] as

x(t) = Amx(t)+b(µ(t)+ θ (t)‖xt‖L∞ +σ1(t))

, x(0) = x0 ,

y(t) = c�x(t) .(2.235)

Second Equivalent System Taking into account the assumption on u(t) and its derivativein (2.233), and using Lemma A.10.1, we can rewrite the signal µ(t) as

µ(t) = ωu(t)+σ2(t) ,

whereω ∈ (ωl , ωu) is an unknown constant and σ2(t) is a continuous signal with (piecewise)-continuous derivative, defined over t ∈ [0, τ ], such that

|σ2(t)| ≤�2 , |σ2(t)| ≤ dσ2 ,

with �2 as introduced in (2.221) and dσ2 � ‖F (s) − (ωl +ωu)/2‖L1ρu. This implies thatone can rewrite the system in (2.235) over t ∈ [0, τ ] as follows:

x(t) = Amx(t)+b(ωu(t)+ θ (t)‖xt‖L∞ +σ (t))

, x(0) = x0 ,

y(t) = c�x(t) ,(2.236)

where θ (t) was introduced in (2.234), σ (t) � σ1(t)+σ2(t) is an unknown continuous time-varying signal subject to |σ (t)|<�, with� as introduced in (2.222), and |σ (t)|< dσ , withdσ � dσ1 +dσ2 .


Using the equivalent (semi-)linear system in (2.236), one can write the prediction errordynamics over t ∈ [0, τ ] as

˙x(t) = Amx(t)+b(ω(t)u(t)+ θ (t)‖xt‖L∞ + σ (t)

), x(0) = 0, (2.237)


Lemma 2.5.2 For the prediction-error dynamics in (2.237), if u(t) is continuous, and more-over the following bounds hold:

‖xτ‖L∞ ≤ ρ , ‖uτ‖L∞ ≤ ρu , ‖uτ‖L∞ ≤ ρu ,

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then

‖xτ‖L∞ ≤√θm(ρ,ρu,ρu)

λmin(P )�, (2.238)

where

θm(ρ,ρu,ρu) � (ωu−ωl)2 +4L2ρ +4�2 +4

λmax(P )

λmin(Q)(Lρdθ +�dσ ) .


V (x(t), ω(t), θ (t), σ (t)) = x�(t)P x(t)+ 1

�(ω2(t)+ θ2(t)+ σ 2(t)) .

Similar to the proof of Lemma 5.1.2, we can use the adaptation laws in (2.224) and Prop-erty B.2 of the projection operator to derive the following upper bound on the derivative ofthe Lyapunov function:

V (t) ≤ −x�(t)Qx(t)+ 2

�|θ (t)θ (t)+ σ (t)σ (t)| .

Let t1 ∈ (0, τ ] be the time instant, when the first discontinuity of θ (t) or σ (t) occurs, ort1 = τ if there are no discontinuities. Notice that

maxt∈[0, t1]

1

�(ω2(t)+ θ2(t)+ σ 2(t)) ≤ (ωu−ωl)2 +4L2

ρ +4�2

�<θm(ρ,ρu,ρu)

�.

This, along with the fact that x(0) = 0, leads to

V (0)<θm(ρ,ρu,ρu)

�. (2.239)

If at arbitrary time t2 ∈ [0, t1]

V (t2)>θm(ρ,ρu,ρu)

�,

then from the fact that

V (t2) ≤ x�(t2)P x(t2)+ 1

�

((ωu−ωl)2 +4L2

ρ +4σ 2b

)= x�(t2)P x(t2)+�−1θm(ρ,ρu,ρu)−4

λmax(P )

�λmin(Q)(Lρdθ +�dσ ) ,

it follows that

x�(t2)P x(t2) ≥ 4λmax(P )

�λmin(Q)(Lρdθ +�dσ ) .

Further, one can write

x�(t2)Qx(t2) ≥ λmin(Q)

λmax(P )x�(t2)P (t2)x(t2) ≥ 4

�(Lρdθ +�dσ ) . (2.240)

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Notice also that

1

�|θ (t)θ (t)+ σ (t)σ (t)| ≤ 2

�(Lρdθ +�dσ ) ,

which, along with the bound in (2.240), leads to

V (t2)< 0.

Thus, the continuity of V (t2) along with the bound on the initial condition in (2.239) implies

V (t2) ≤ θm(ρ,ρu,ρu)

�.

Continuity of V (t) allows for repeating these derivations for all points of discontinuity ofθ (t) or σ (t), leading to the following bound for all t ∈ [0, τ ]:

V (t) ≤ θm(ρ,ρu,ρu)

�.

This further implies that

‖x(t)‖ ≤√θm(ρ,ρu,ρu)

λmin(P )�.

The result in (2.238) follows from the fact that this bound holds uniformly for all t ∈[0, τ ]. �

The next theorem specifies the uniform performance bounds that the L1 adaptivecontroller, defined via (2.223)–(2.225) and subject to the L1-norm condition in (2.216),guarantees in both transient and steady-state.

Theorem 2.5.1 If the adaptive gain verifies the design constraint

� ≥ θm(ρ,ρu,ρu)

λmin(P )γ 20

, (2.241)

where γ0 is as introduced in (2.219), then the following bounds hold:

‖u‖L∞ ≤ ρu , (2.242)

‖x‖L∞ ≤ ρ , (2.243)

‖x‖L∞ ≤ γ0 , (2.244)

‖xref −x‖L∞ ≤ γ1 , (2.245)

‖uref −u‖L∞ ≤ γ2 . (2.246)

Proof. We prove (2.245) and (2.246) following a contradicting argument. Assume that(2.245) and (2.246) do not hold. Then, since

‖xref (0)−x(0)‖∞ = 0, ‖uref (0)−u(0)‖∞ = 0,

continuity of xref (t), x(t), uref (t), u(t) implies that there exists time instant τ > 0, for which

‖xref (t)−x(t)‖∞ < γ1 , ‖uref (t)−u(t)‖∞ < γ2, ∀t ∈ [0,τ )

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and

‖xref (τ )−x(τ )‖∞ = γ1 , or ‖uref (τ )−u(τ )‖∞ = γ2 .

This consequently implies that at least one of the following equalities holds:


Assumption 2.5.3 and the first equality in (2.247) lead to

‖zτ‖L∞ ≤ L1(‖xref τ‖L∞ +γ1)+L2 . (2.248)

Then, since all the conditions of Lemma 2.5.1 hold, the following bounds are valid:

‖xref τ‖L∞ < ρr , ‖uref τ‖L∞ < ρur , (2.249)

which in turn lead to

‖xτ‖L∞ ≤ ρr +γ1 ≤ ρ , ‖uτ‖L∞ ≤ ρur +γ2 = ρu . (2.250)

Let η(t) be defined as

η(t) � ω(t)u(t)+ θ (t)‖xt‖L∞ + σ (t) .

Then for t ∈ [0, τ ], we have

η(t) = µ(t)+ η(t)+η(t) ,

where η(t) � f (t ,x(t),z(t)). Substituting the Laplace transform of η(t) into (2.225) yields

u(s) = −Cu(s)(η(s)+η(s)−kgr(s)) . (2.251)

Further, the closed-loop response of the system in (2.213) with the L1 adaptive controllercan be written in the frequency domain as

x(s) =G(s)η(s)−H (s)C(s)η(s)+H (s)C(s)kgr(s)+xin(s) .

Also, recall that in (2.229) the response of the closed-loop reference system is presented as


The two expressions above lead to

xref (s)−x(s) =G(s)(ηref (s)−η(s))+H (s)C(s)η(s) . (2.252)

Further, consider the control law in (2.225). From the properties of the projection operatorwe have

‖ητ‖L∞ ≤ ωuρu+Lρρ+� ,

and consequently

‖uτ‖ ≤ ‖ksD(s)‖L1 (ωuρu+Lρρ+�+|kg|‖r‖L∞ ) = ρu .

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These bounds imply that the conditions of Lemma 2.5.2 hold. Selecting the adaptive gain �according to the design constraint in (2.241), we get

‖xτ‖L∞ ≤ γ0 . (2.253)

It follows from (2.248) and the first bound in (2.249) that

‖zτ‖L∞ ≤ L1(ρr +γ1)+L2 ,

which, along with (2.250), leads to

‖Xτ‖L∞ ≤ max{ρr +γ1, L1(ρr +γ1)+L2} ≤ ρr (ρr ) .

Similarly, one can show that‖Xref τ‖L∞ ≤ ρr (ρr ) .

Thus, Assumption 2.5.2 yields the upper bound

‖(ηref −η)τ‖L∞ ≤ dfx (ρr (ρr ))‖(xref −x)τ‖L∞ ,

and since dfx (ρr (ρr ))< Lρr , it follows that


Moreover, the error dynamics in (2.237) can be rewritten in the frequency domain as x(s) =H (s)η(s), and therefore it follows from (2.252) that

‖(xref −x)τ‖L∞ ≤ ‖G(s)‖L1Lρr‖(xref −x)τ‖L∞ +‖C(s)‖L1‖xτ‖L∞ ,

which leads to


1−‖G(s)‖L1Lρr‖xτ‖L∞ .

Recalling the bound in (2.253) yields


1−‖G(s)‖L1Lρrγ0 = γ1 −β < γ1 . (2.254)

Hence, we obtain a contradiction to the first equality in (2.247).To show that the second equality in (2.247) also cannot hold, consider (2.226) and

(2.251), which lead to

uref (s)−u(s) = −Cu(s)(ηref (s)−η(s))+Cu(s)η(s) .

Using the bound on ‖(ηref −η)τ‖L∞ , Lemma A.12.1, and the definition ofH1(s) in (2.217),we can write

‖(uref −u)τ‖L∞ ≤ ‖Cu(s)‖L1Lρr‖(xref −x)τ‖L∞ +‖H1(s)‖L1‖xτ‖L∞ .

Further, we can use the upper bound on ‖(xref −x)τ‖L∞ in (2.254) to obtain

‖(uref −u)τ‖L∞ ≤ ‖Cu(s)‖L1Lρr (γ1 −β)+‖H1(s)‖L1γ0 < γ2 , (2.255)

which contradicts the second equality in (2.247).This proves the bounds in (2.254)–(2.255). Then, the bounds in (2.250) hold uniformly,

which implies that the bound in (2.253) also holds uniformly. This proves the boundsin (2.242)–(2.244). �

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Remark 2.5.2 It follows from (2.241) that one can prescribe the arbitrary desiredperformance bound γ0 by increasing the adaptive gain �, which further implies from(2.218) and (2.219) that one can achieve arbitrarily small performance bounds γ1 and γ2simultaneously.

Remark 2.5.3 Similar to previous sections, notice that letting k→ ∞ leads to C(s) → 1,and thus the reference controller in the definition of the closed-loop reference systemin (2.226) leads, in the limit, to perfect cancelation of uncertainties and recovers the perfor-mance of the ideal desired system. As before, setting C(s) = 1 leads to an improper H1(s),and hence the uniform bound for the control signal fails to hold.


To illustrate the results derived in this chapter, consider the following nonlinear system withunmodeled dynamics:

x(t) =[

0 1−1 −1.4

]x(t)+

[01

](µ(t)+f (t ,x(t),z(t))) , x(0) = x0 ,

y(t) = [1 0

]x(t) ,

(2.256)

where

f (t ,x(t),z(t)) = f1(t ,x(t),z(t)) = x1(t)+1.4x2(t)+x21 (t)+x2

2 (t)+ z2(t) ,

and

µ(s) = F (s)u(s) , F (s) = F1(s) = 75

s+75,

while the unmodeled dynamics are given by

z(s) = z1(s) = s−1

s2 +3s+2ν(s) , ν(t) = ν1(t) = sin(0.2t)x1(t)+x2(t) .

We implement the L1 adaptive controller according to (2.223), (2.224), and (2.225).In our design, we set the following parameters for the controller:

D(s) = 1

s, k = 50, � = 100000.

For the adaptation law, we set the following projection bounds:

�= [0.1, 3] , �= [−5, 5] , �= 100.

We consider bounded reference signals with ||r||L∞ ≤ 0.2. In order to track step-referencesignals with zero steady-state error for the ideal system

yid(s) � c�H (s)kgr(s) ,

we set kg = 1. SettingQ= I and solving the algebraic Lyapunov equation gives us

P =[

1.4143 0.50000.5000 0.7143

].

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0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

time [s]

r(t)y(t)yid(t)

(a) System output for r(t) ≡ 0.2

0 5 10 15 20 25 30−0.2

−0.1

0

0.1

0.2

time [s]

r(t)y(t)

(b) System output for r(t) = 0.2cos(0.5t)

0 5 10 15 20 25 30−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

time [s]

(c) Control history for r(t) ≡ 0.2

0 5 10 15 20 25 30−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

time [s]

(d) Control history for r(t) = 0.2cos(0.5t)

Figure 2.43: Performance of the L1 adaptive controller for µ(s) = F1(s)u(s), f (·) = f1(·),z(t) = z1(t), and ν(t) = ν1(t).

Next we verify the sufficient condition in (2.216). Letting ρr = 1.5, we compute the conser-vative estimate for Lρr according to (2.215) and obtain Lρr = 7.3. From (2.256) we have

H (s) =[

1s2+1.4s+1

s

s2+1.4s+1

].

The stability condition in (2.216) takes the form

‖G(s)‖L1Lρr < 1− ‖kgC(s)H (s)‖L1‖r‖L∞ρr

− ‖s(sI−Am)−1‖L1ρ0

ρr,

where we set ρ0 = 0.2. Using the definition of C(s) from (2.214) we compute numericallythe L1-norms of G(s) and kgC(s)H (s) and obtain the inequality

‖G(s)‖L1Lρr = 0.42< 0.56 ,

which implies that the L1-norm condition is satisfied. Notice that this condition is satisfiedfor arbitrary k > 48. The choice of k= 50 is explained from the fact that a smaller bandwidthof C(s) leads to better robustness (see Section 2.2.5).

Figure 2.43 shows the simulation results for the closed-loop system described abovefor the two reference signals r(t) ≡ 0.2 (step signal) and r(t) = 0.2cos(0.5t). One can see

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0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

time [s]

r(t)y(t)yid(t)


0 5 10 15 20 25 30−0.2

−0.1

0

0.1

0.2

time [s]

r(t)y(t)


0 5 10 15 20−0.1

0

0.1

0.2

0.3

time [s]


0 5 10 15 20 25 30−0.1

0

0.1

0.2

0.3

time [s]



that the closed-loop adaptive system has good tracking performance in both transient andsteady-state. The behavior of the system output y(t) is close to the ideal yid(t), which isgiven by a linear system. All performance specifications such as phase-lag for the sinusoidalsignal, overshoot, and settling time for the step-reference signal are very close to the idealsystem.

Next we change the unmodeled actuator dynamics to

F (s) = F2(s) = 50

s+50.

This change of the unmodeled dynamics does not violate the L1-norm condition in (2.216).The simulation results with these actuator dynamics and without any retuning of the adaptivecontroller are shown in Figure 2.44. One can see that the closed-loop system remains stableand there is no degradation in performance.

Next, we change the input to the unmodeled dynamics

ν(t) = ν2(t) = sin(5t)x1(t)+x2(t)+1.4sin(5t) .

Again, it can be verified that this change does not violate the L1-norm condition. Thesimulation results for this case without any retuning are given in Figure 2.45. While there

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0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

time [s]

r(t)y(t)yid(t)


0 5 10 15 20 25 30−0.2

−0.1

0

0.1

0.2

time [s]

r(t)y(t)


0 5 10 15 20−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

time [s]


0 5 10 15 20 25 30−0.2

−0.1

0

0.1

0.2

0.3

time [s]



are almost no changes in the system output, we see that the control signal changes tocompensate for the new type of the uncertainty.

Further, we change the unmodeled dynamics and the nonlinearities as follows:

z(s) = z2(s) = 1

s2 +30s+100ν1(s) ,

f (t ,x(t),z(t)) = f2(t ,x(t),z(t)) = x21 (t)+x2

2 (t)+x21 (t)x2

2 (t)+x1(t)x2(t)+ z2(t) .

The simulation results for these cases are shown in Figures 2.46 and 2.47. We see that theperformance of the system does not change significantly, and the conclusions made in theprevious scenario also hold here.

Next, we test the L1 adaptive system performance for nonzero initialization error.We set the initial conditions of the state predictor different from the plant: x0 = [0.1, 0.1],x0 = [1.5, − 1]. The simulation results are given in Figure 2.48. One can see that theestimation error is rapidly decaying.

In Section 2.2.5, we proved that the closed-loop system with the L1 adaptive controllerhas bounded-away-from-zero time-delay margin. Using the insights from that section, wetest the robustness of the closed-loop system in this simulation example to time delays.For this purpose, we introduce a time delay of 20 ms at the system input and repeat thesimulations for the system nonlinearities and unmodeled dynamics considered above. Wedo not retune the controller. The simulation results are shown in Figure 2.49. One can see

L1book2010/7/22page 109

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0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

time [s]

r(t)y(t)yid(t)


0 5 10 15 20 25 30−0.2

−0.1

0

0.1

0.2

time [s]

r(t)y(t)


0 5 10 15 20−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

time [s]


0 5 10 15 20 25 30−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

time [s]



0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

time [s]

r(t)y(t)yid(t)


0 5 10 15 20 25 30−0.2

−0.1

0

0.1

0.2

time [s]

r(t)y(t)


0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

time [s]


0 5 10 15 20 25 30−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

time [s]



L1book2010/7/22page 110

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y

0 5 10 15 20�0.5

0

0.5

1

1.5

time [s]

r(t)y(t)(t)

(a) System output

0 5 10 15 20−8

−6

−4

−2

0

2

time [s]

(b) Control history

Figure 2.48: Performance of the L1 adaptive controller for nonzero trajectory initializationerror.

System output Control history

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

time [s]

r(t)y(t)

(a) f1(·), F1(s), z1(s), and ν1(t)

0 5 10 15 20−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

time [s]

(b) f1(·), F1(s), z1(s), and ν1(t)

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

time [s]

r(t)y(t)

(c) f1(·), F2(s), z1(s), and ν1(t)

0 5 10 15 20−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

time [s]

(d) f1(·), F2(s), z1(s), and ν1(t)

Figure 2.49: Performance of the L1 adaptive controller in the presence of a time delayof 20 ms.

that the output of the closed-loop adaptive system for all considered cases of nonlinearitiesand unmodeled dynamics does not change significantly even in the presence of the timedelay. However, one can observe some expected oscillations in the control channel, whichindicate that the time delay of 20 ms is relatively close to the actual time-delay margin ofthe adaptive system.

L1book2010/7/22page 111

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2.6. Filter Design for Performance and Robustness Trade-Off 111


0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

time [s]

r(t)y(t)

(e) f1(·), F1(s), z1(s), and ν2(t)

0 5 10 15 20−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

time [s]

(f) f1(·), F1(s), z1(s), and ν2(t)

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

time [s]

r(t)y(t)

(g) f1(·), F1(s), z2(s), and ν1(t)

0 5 10 15 20−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

time [s]

(h) f1(·), F1(s), z2(s), and ν1(t)

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

time [s]

r(t)y(t)

(i) f2(·), F1(s), z1(s), and ν1(t)

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

time [s]

(j) f2(·), F1(s), z1(s), and ν1(t)

Figure 2.49: Performance of the L1 adaptive controller in the presence of a time delayof 20 ms.

2.6 Filter Design for Performance and RobustnessTrade-Off

In this section we address the problem of designing the filter C(s), which decides the trade-off between performance and robustness. We consider the system and the L1 adaptivecontroller given in Section 2.1. From the relationships in (2.7), (2.19) and (2.26), (2.27) it isstraightforward to notice that, in addition to increasing the rate of adaptation �, one needsto select C(s) to minimize ‖(1−C(s))H (s)‖L1 for performance improvement. One simplechoice was given by Lemma 2.1.5, which states that, by increasing the bandwidth of a

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first-order C(s), the L1-norm of (1−C(s))H (s) can be rendered arbitrarily small. It furtherfollows from (2.80) that increasing the bandwidth ofC(s) will reduce the time-delay marginto zero. For the purpose of identifying the optimal trade-off between the two objectives,we consider the following two constrained optimization problems separately [108].

Problem 1 (Optimization of performance retaining a desired time–delay margin)

minC(s)

‖(1−C(s))H (s)‖L1

s.t. C(0) = 1 and T (Lo(s)) ≥ τgr ,(2.257)

where τgr > 0 is a desired lower bound on the time-delay margin. Recall that the loop transferfunction Lo(s) was defined in (2.80).

Problem 2 (Maximization of time-delay margin retaining a desired performancebound)

maxC(s)

T (Lo(s))

s.t. C(0) = 1 and ‖(1−C(s))H (s)‖L1 ≤ λgp ,(2.258)

where λgp is a given upper bound on the L1-norm of (1−C(s))H (s), specifying the desiredperformance.

Let the state-space realization for H (s) be given by

x(t) = Amx(t)+bu(t) , x(0) = 0, (2.259)

where x(t) ∈ Rn is the system state vector; u(t) ∈ R is the control signal; b, c ∈ R

n areknown constant vectors; and Am ∈ R

n×n is a known Hurwitz matrix specifying the desiredclosed-loop dynamics. Also, let the state-space realization for C(s) be given by

xf (t) = Af xf (t)+bf uf (t) , xf (0) = 0,

yf (t) = c�f xf (t) ,(2.260)

where xf (t) ∈ Rnf is the state of the filter, yf (t) ∈ R is the output of the filter, uf (t) ∈ R is

the input, Af ∈ Rnf×nf is a Hurwitz matrix, and bf , cf ∈ R

nf are the input and the outputvectors, respectively. The dimension nf of the filter dynamics can be freely chosen by theuser a priori and is one of the parameters in the foregoing optimization problems.

The objective of the algorithms presented below is to determine the “optimal” state-space realization matrices of the filterC(s) in the context of the above-formulated problems.Since the optimization of the L1-norm of (1−C(s))H (s) over the parameters of C(s) is anonconvex problem by definition, we provide an overview of methods that can be employedfor investigation of this problem.

First we review some stochastic optimization methods, which have already proved tobe useful in control-related nonconvex optimization problems [87,123]. Then we formulatethe above-presented optimization problems in terms of LMIs, which provide an opportunityto use MATLAB tools to obtain a possible (conservative, but guaranteed) solution for thefilter realization.

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2.6.1 Overview of Stochastic Optimization Algorithms

Random search algorithms, which are also called Monte Carlo methods or stochastic algo-rithms [160, 184], refer to computational methods that use repetitive random samplings tosolve simulation problems in various applications. These algorithms are successfully usedto tackle a wide class of optimization problems [148,152]. We now illustrate the main ideaof stochastic optimization algorithms by starting with the following standard constrainedoptimization problem:

minJ (p)

s.t. p ∈ S .(2.261)

Herep denotes the decision variable that captures the entries of the filter realization given by(Af , bf , cf ); J (·) represents the objective function, which is given by ‖(1−C(s))H (s)‖L1

for (2.257) or T (Lo(s)) for (2.258); and S specifies the set of constraints, defined in (2.257)or (2.258) separately for each of the two cases.

The basic idea of the stochastic optimization algorithm for solving (2.261) is composedof the following two key steps:

1. Generate a sequence of random samples p(i) , i = 1, 2, . . . .

2. Obtain the optimal solution p∗ from the samples.

Next we introduce several widely used variances of stochastic optimization methods withcontrol applications.

Randomized Algorithms (RAs) In this approach, an optimal solution is selected from aset of samples {p(i), i = 1, . . . ,ns}, where ns is the number of samples. The elementsin the set are randomly independently identically distributed over the feasible solutionset S. The accuracy and the confidence levels of the solution depend upon the selectednumber of samples, which can be quantified as

ns ≥⌈

log( 1β

)

log( 11−ε )

⌉,

where ε ∈ (0,1) and β ∈ (0,1) represent the desired accuracy and the confidencelevels, respectively. Details of this algorithm are given in [87, 165, 166].

Adaptive Random Search Algorithms (ARSAs) Unlike RAs, in this method the opti-mization is done by iterations, where the sample point at the step i+1 is determined by

p(i+1) = p(i) + t (i)d(i) ,

where t (i) and d(i) represent the ith step size and a feasible direction. On each iterationof the algorithm, one performs a random search for t (i) and d(i), such that p(i+1) ∈ Sand J

(p(j+1)

)< J

(p(j )

). One can refer to [11, 21, 184] for more details.

Particle Swarm Optimization Algorithms (PSOAs). PSOA is a population-based ran-dom search algorithm, which considers a set of potential solutions {p(i)} ⊂ S, called

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particles. The set of potential solutions is initialized with a population of random sam-ples. During the optimization process the algorithm assigns a randomized velocityand/or acceleration to each particle based on comprehensive update law, which ensuresthat the population of particles remains in the feasible solution set. The optimal solu-tion is determined as the overall concurrent best value in the population [38,93,154].

Meta-Control Methodologies (MCMs). A meta-control methodology combines the ideaof well-known simulated annealing with population-based random search algorithms.This combination provides a way to escape possible local minima. The method sup-ports dynamical tuning of the algorithm parameters based on the observed values ateach iteration of the algorithm. Details can be found in [95].

Remark 2.6.1 Since it is impossible to compute the exact value of the time-delay marginT (Lo(s)) in the optimization problems of interest (2.257) and (2.258) due to the presenceof the unknown value θ in the transfer function Lo(s), the optimization method needsto estimate the value of T (Lo(s)). This can be done by employing a recently developedRA for sampling the uncertainty θ [166]. This approach considers the probabilistic worst-case or empirical mean performance, instead of relying on the deterministic worst-casevalues. Specifically, to check the feasibility of the constraint for guaranteed time-delaymargin in (2.257), for all θ ∈�, the method considers a set of randomly generated samples{θ (j ), j = 1, . . . ,Nθ } and evaluates T (Lo(s)) ≥ τgr for all θ (j ), j = 1, . . . ,Nθ , where Nθdenotes the number of random samples for the uncertain parameter θ . Similarly, to solvethe optimization problem in (2.258), the objective function T (Lo(s)) must be replaced bymin{θ (j )} T (Lo(s)). A brief summary of these methods is given in [92].

2.6.2 LMI-Based Filter Design

L1-Norm Optimization

In this section, we investigate the constraint-free L1-norm minimization problem of thecascaded system (1 −C(s))H (s). The L1-norm minimization problem is nonconvex bydefinition. We hence consider the ∗-norm, which serves as an upper bound of the L1-normof a stable LTI system [2] (see Definition C.2.1 in Appendix). From now on we fix bf torender our optimization problems convex and computationally tractable.

From (2.259) and (2.260), it follows that the state-space representation of the cascadedsystem (1−C(s))H (s) is given by

ξ (t) =[Am −bc�f0 Af

]ξ (t)+

[b

bf

]u(t) ,

y(t) = [I 0

]ξ (t) ,

where ξ (t) ∈ Rn+nf and y(t) ∈ R

n are the state and the output, respectively, and I and 0 arethe identity and the zero matrices of appropriate dimensions.

Theorem 2.6.1 If there exist a scalar α ∈ R+, a vector q ∈ R

nf , a matrixM ∈ Rnf×nf , and

positive definite matrices P1 ∈ Rn×n, P2 ∈ R

nf×nf , solving the LMIs

�(α,P1,P2,M ,q) ≤ 0, (2.262)

P1 ≤ λgpI , (2.263)

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where

�(α,P1,P2,M ,q) �

[

Qα −bq�−qb� M+M� +αP2

] [b

bf

][b� b�

f

]−α

,

for some λgp > 0 and bf ∈ Rnf , whereQα � AmP1 +P1A

�m+αP1, then

‖(1−C(s))H (s)‖L1 ≤ λgp ,

and the parameters of the corresponding filter are given by

Af =MP−12 and cf = P−1

2 q .

Proof. First, consider a structured Lyapunov matrix

Pα =[P1 00 P2

],

and its corresponding change of variables q � P2cf and M � AfP2. It is straightforwardto see that the inequality in (2.262) is equivalent to[

Am −bc�f0 Af

][P1 00 P2

]+[P1 00 P2

][Am −bc�f0 Af

]�

+ α[P1 00 P2

]+ 1

α

[b

bf

][b

bf

]�≤ 0.

Note that the state-space representation of (1−C(s))H (s) bears the same form of the systemin (C.2). Therefore, direct application of Theorem C.2.1 leads to the next result:

‖(1−C(s))H (s)‖L1 ≤∥∥∥∥[ I 0

][ P1 00 P2

][I

0

]∥∥∥∥2= ‖P1‖2 .

Thus, P1 ≤ λgpI implies ‖(1−C(s))H (s)‖L1 ≤ λgp. �We further refer to this theorem to find the minimal upper bound of the L1-norm.

Let λ ≥ ‖(1 −C(s))H (s)‖L1 be the upper bound to be minimized. We replace the boundλgp by a decision variable λ in (2.263) and formulate the following optimization problem(generalized eigenvalue problem (GEVP); see Appendix C.1):

minα,P1>0,P2>0,M ,q

λ ,

s.t. (2.262) holds and P1 ≤ λI .(2.264)

The optimal filter C(s) is then realized via (Af =MP−12 , bf , cf = P−1

2 q).

Remark 2.6.2 Since a necessary condition for the feasibility of (2.262) isQα ≤ 0, a feasiblesolution α for the GEVP (2.264) is bounded as 0< α ≤ −2�(λmax(Am)).

Time-Delay Margin Optimization

Similarly, we develop LMI tools for constraint-free time-delay margin optimization. Weconsider the time-delay margin maximization problem for the L1 adaptive controller.

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Let the state-space realization for Lo(s) in the presence of time delay τ be given by

xl(t) = Af +bf c�f bf c

�f 0

0 Af bf θ�

0 0 Am+bθ�

xl(t)

+ 0 0 0

−bf c�f −bf c�f 0

−bc�f −bc�f 0

xl(t− τ ) ,

(2.265)

where xl(t) ∈ Rn+2nf is the state of Lo(s).

Theorem 2.6.2 If for a fixed vector bf and prespecified lower bound τgr > 0 there exista vector q ∈ R

nf , a matrix M ∈ Rnf×nf , and positive definite matrices P1 ∈ R

nf×nf andP2 ∈ R

n×n satisfying

�(P1,P2,M ,q,τqr ) ≤ 0, (2.266)

where

�(P1,P2,M ,q,τqr ) �

Q1 +Q�

1 +Q2 +Q�2 Q�

1 Q�2 Q2

Q1 − 1τgrP 0 0

Q2 0 − 1τgrP 0

Q�2 0 0 − 1

2τgrP

,

Q1 �

M+bf q� bf q

� 00 M bf θ

�P2

0 0 AmP2 +bθ�P2

,

Q2 �

0 0 0

−bf q� −bf q� 0−bq� −bq� 0

,

P �

P1 0 0

0 P1 00 0 P2

,

then the system (2.265) is BIBS stable for arbitrary 0 ≤ τ ≤ τgr .Proof. The result of the theorem immediately follows from application of Lemma C.3.1 tothe LTI system in (2.265). �

Notice that from Theorem 2.6.2 it follows that the time-delay margin of the sys-tem (2.265) is greater than τgr . This result is summarized in the following corollary.

Corollary 2.6.1 If there exist matrices M ∈ Rnf×nf , q ∈ R

nf and positive definite matri-ces P1 ∈ R

nf×nf and P2 ∈ Rn×n satisfying the inequality in (2.266), then the time-delay

margin T of the closed-loop system (2.1) with the L1 adaptive controller, defined via (2.4)–(2.6), is lower bounded by τgr . The realization of the corresponding filter C(s) is then givenby (MP−1

1 , bf , P−11 q).

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Notice that τgr in (2.266) can be viewed as a generalized eigenvalue. We can formulatethe following GEVP for the time-delay margin maximization, where we use τs to denotethe optimization objective:

maxq, M , P1>0, P2>0

τs

s.t. �(P1,P2,M ,q,τs) ≤ 0.(2.267)

Remark 2.6.3 The augmented matrixQ1 in�(·) contains the unknown parameter θ , whichlies in the compact set�. This implies that the linear matrix inequalities (2.267) and (2.266)must be checked for all values of θ , thus rendering the numerical optimization intractable.However, the matrixQ1 is affine in θ ∈�, and because the uncertainty set� can be boundedby a polytope, one can use Corollary C.4.1 and check just a finite number of LMIs for thevertices of the polytope. If a solution for the filter realization is found for this finite numberof LMIs, then it will also solve the original optimization problem.

Constrained Optimization for L1-Norm and Time-Delay Margin

In this section, we address the constrained optimization problems in (2.257) and (2.258)via LMI formulations. To address the L1-norm optimization problem in (2.264) in thepresence of a constraint on the time-delay margin, we add the additional LMI conditionfrom Theorem 2.6.2 to the problem statement. Similarly, for the constrained time-delaymargin optimization, the GEVP optimization problem in (2.267) is used, together with theLMI condition for the L1-norm given in Theorem 2.6.1.

We start by proving the following theorem, which handles the constraint C(0) = 1 inthe LMI optimization problem.

Theorem 2.6.3 Let the state-space realization of C(s) be given as in (2.260). Let P ∈Rnf×nf be a nonsingular symmetric matrix, and also let M ∈ R

nf×nf be a nonsingularmatrix. Further, let m ∈ R

nf be the last row ofM�. If Af , bf , and cf are chosen as

Af � MP−1 , bf �[

0, · · · 0, −1]� , cf = P−1m ,

then we haveC(0) = 1.

Proof. Let p ∈ Rnf be the last column ofM−1. It is straightforward to verify that

C(0) = −c�f A−1f bf = −m�P−�PM−1bf = −m�M−1bf =m�q .

Then, fromMM−1 = I, it follows that m�q = 1, which leads to

C(0) = 1.

Next we address the constrained problem in (2.257) for performance optimizationretaining a desired time-delay margin.

Theorem 2.6.4 For a given desired lower bound for the time-delay margin τgr of the closed-loop L1 adaptive control system, if there exist scalars α, λ ∈ R, matrices P0 = P�

0 > 0,

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P1 = P�1 > 0, P2 = P�

2 > 0 of appropriate dimensions,M ∈ Rnf×nf , and a vector q ∈ R

nf

solving the GEVP

minα, P0, P1, P2, M , q

λ

such that

P0 ≤ λI,

�(α,P0,P1,M ,q) ≤ 0,(2.268)

and

�(P0,P2,M ,q,τgr ) ≤ 0, (2.269)

where M , q, and bf comply with the structure in Theorem 2.6.3 and �(·) and �(·) areas defined in (2.262) and (2.266), then the problem (2.257) is solved by via the followingrealization of the filter C(s):

(Af =MP−10 , bf , cf = P−1

0 q) .

Proof. Let λ be the upper bound of the L1-norm of (1−C(s))H (s) and let τgr be a givenlower bound on the time-delay margin to be satisfied. Consider the L1-norm optimiza-tion problem (2.264) and the LMI condition in (2.266) along with the definitions for � inTheorem 2.6.1 and � in Theorem 2.6.2, where M , q, and bf comply with the structurein Theorem 2.6.3. By substituting P0, M , and q into (2.264) and (2.266), we get (2.268)and (2.269), respectively. It follows from Theorem 2.6.3 that the choice of bf , M , andq ensures that C(0) = 1. Then the optimization problem (2.264), together with the con-straint (2.266), yields the filter C(s) via the realization of

(Af =MP−10 , bf , cf = P−1

0 q) ,


For the time-delay margin optimization problem in (2.258), we have the followingresult, where the proof is omitted since the idea is similar to Theorem 2.6.4.

Theorem 2.6.5 Let λgp be a desired upper bound of the L1-norm of (1−C(s))H (s). If thereexist α ∈ R, matrices P0 = P�

0 > 0, P1 = P�1 > 0, P2 = P�

2 > 0 of appropriate dimensions,M ∈ R

nf×nf and a vector q ∈ Rnf solving the GEVP

maxα, P0, P1, P2, M , q

τs ,

subject to

P0 ≤ λgpI ,

�(α,P0,P1,M ,q) ≤ 0,

and

�(P0,P2,M ,q,τs) ≤ 0,

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where M , q, and bf comply with the structure given in Theorem 2.6.3, and �(·) and �(·)are defined in (2.262) and (2.266), then the problem in (2.258) is solved by choosing thefollowing realization of the filter C(s):

(Af =MP−10 , bf , cf = P−1

0 q) .

Remark 2.6.4 We note that the existence of a solution to the above LMIs and GEVPs is notguaranteed in the general case. Theorems 2.6.4 and 2.6.5, however, suggest systematic waysto solve computationally tractable convex optimization problems for (2.257) and (2.258).

Obviously, by resorting the tuning of C(s) to the LMI algorithms presented above,we obtain overly conservative solution for the state-space realization of C(s). The structureimposed on the block-diagonal Lyapunov matrices Pα and P toward rendering the opti-mization problems convex is the first step, which increases the degree of conservatism. Thesecond step of introducing conservatism lies in the definition of the structure for the decisionvariablesM and q, considered in Theorem 2.6.3.

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Chapter 3

State Feedback in the Presenceof Unmatched Uncertainties

This chapter presents the extension of the ideas from previous sections to systems in thepresence of unmatched uncertainties. The class of uncertainties includes time- and state-dependent nonlinearities and unmodeled dynamics. We first present the L1 adaptive back-stepping controller for strict-feedback systems and proceed by considering systems of amore general type, which cannot be addressed by recursive design methods. We present anew piecewise-constant adaptive law, which uses the sampling rate of the available CPUto update the parametric estimates. The sufficient conditions in this case can be interpretedto determine the performance limitations due to the cross-coupling dynamics. In particular,the solution presented in Section 3.3 has been flight tested on NASA’s GTM (AirSTAR)and evaluated in mid- to high-fidelity simulations for the X-48B and X-29 aircraft.

3.1 L1 Adaptive Controller for NonlinearStrict-Feedback Systems

This section presents the L1 adaptive control architecture for nonlinear strict-feedbacksystems in the presence of uncertain system input gain and unknown time-varying nonlin-earities. Similar to earlier developments, we prove that the L1 adaptive control architectureensures guaranteed transient response for system’s input and output signals simultaneously.


Consider the following system:

x1(t) = f1(t ,x1(t))+x2(t) , x1(0) = x10 ,

x2(t) = f2(t ,x(t))+ωu(t) , x2(0) = x20 ,

y(t) = x1(t) ,

(3.1)

where x1(t), x2(t) ∈ R are the states of the system (measured), x(t) � [x1(t), x2(t)]�;u(t) ∈ R is the control signal; ω ∈ R is an unknown parameter; and f1(t ,x1) : R×R → R

and f2(t ,x) : R×R2 → R are unknown nonlinear maps continuous in their arguments. The

121

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122 Chapter 3. State Feedback in the Presence of Unmatched Uncertainties

initial condition x0 � [x10 , x20 ]� is assumed to be inside an arbitrarily large known set, sothat ‖x0‖∞ ≤ ρ0 <∞ for some known ρ0 > 0.


ω ∈�� [ωl , ωu] ,

where 0< ωl < ωu are known conservative bounds.

Assumption 3.1.2 (Uniform boundedness of fi(t ,0)) There exists B > 0, such that|fi(t ,0)| ≤ B, i = 1,2, holds for all t ≥ 0.

Assumption 3.1.3 (Semiglobal uniform boundedness of partial derivatives) For i =1,2, and arbitrary δ > 0, there exist positive constants dfxi (δ) > 0 and dfti (δ) > 0independent of time such that for all ‖x(t)‖∞ < δ, the partial derivatives of fi(·) arepiecewise-continuous and bounded:∣∣∣∣∂f1(t ,x1)

∂x1

∣∣∣∣≤ dfx1 (δ) ,

∣∣∣∣∂f1(t ,x1)

∂t

∣∣∣∣≤ dft1 (δ) ,∥∥∥∥∂f2(t ,x)

∂x

∥∥∥∥1≤ dfx2 (δ) ,

∣∣∣∣∂f2(t ,x)

∂t

∣∣∣∣≤ dft2 (δ) .

In this section we present theL1 adaptive (backstepping) controller, which ensures thatthe system output y(t) tracks a given bounded twice continuously differentiable referencesignal r(t) with uniform and quantifiable performance bounds.



The design of the L1 adaptive controller involves a stable low-pass filter C1(s) of relativedegree of at least 2 with unity DC gain, C1(0) = 1, and also a positive feedback gain k > 0and a strictly proper transfer function D(s), which lead, for all ω ∈ �, to a stable transferfunction

C2(s) � ωkD(s)

1+ωkD(s),

with unity DC gain C2(0) = 1. We assume zero initialization for the state-space realizationof these filters. Using the above notation, let

C(s) �[C1(s) 0

0 C2(s)

].

Also, let Am and Ag be defined as

Am �[ −a1 0

0 −a2

], Ag �

[ −a1 10 −a2

],

where a1 > 0 and a2 > 0 are positive constants specifying the desired closed-loop dynamics.

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3.1. L1 Adaptive Controller for Nonlinear Strict-Feedback Systems 123

Further, denote by rmax the (known) upper bound on the reference signals that wewant to track. Similarly, let rmax and rmax be the (known) upper bounds on their first andsecond derivatives. Let

ρin �∥∥∥s(sI−Ag)−1

∥∥∥L1

((1+a1)(ρ0 + rmax)+ rmax) . (3.2)

Notice that from the fact that Ag is Hurwitz and (sI −Ag)−1 is a strictly proper transfermatrix, it follows that ‖s(sI−Ag)−1‖L1 is bounded.

Moreover, for every δ > 0, let

Liδ � δ(δ)

δdfxi (δ(δ)) , δ(δ) � δ+ γ1 , (3.3)

where dfx (·) was introduced in Assumption 3.1.3 and γ1 is an arbitrarily small positiveconstant. Using the redefinitions in (3.3), let Lδ be defined as

Lδ � max{L1δ ,L2δ} . (3.4)

For the proofs of stability and performance bounds, the selection ofC1(s), k, andD(s) needsto ensure that, for given ρ0, there exists ρr > ρin such that the following L1-norm upperbound can be verified:

‖G(s)‖L1 <ρr −β0(ρr )

L�ρr ρr +β1(ρr ), (3.5)

where

G(s) �H (s) (I−C(s)) , H (s) �(sI−Ag

)−1 , (3.6)

while L�ρr , β0(ρr ) and β1(ρr ) are defined as

L�ρr �(1+a1 +‖C1(s)‖L1Lρr

)κ1(ρr ) , (3.7)

β0(ρr ) � ‖C1(s)‖L1B+ (1+‖C1(s)‖L1Lρr)rmax + rmax

+(1+a1 +‖C1(s)‖L1Lρr)ρin ,

β1(ρr ) �(1+a1 +‖C1(s)‖L1Lρr

)κ2 ,

with

κ1(ρr ) �(1+a1 +a2‖C1(s)‖L1 +‖sC1(s)‖L1

)Lρr +a2 (1+a1)+a1 , (3.8)

κ2 �(1+a1 +a2‖C1(s)‖L1 +‖sC1(s)‖L1

)B+a1a2rmax

+ (a1 +a2) rmax + rmax . (3.9)

To streamline the subsequent analysis of stability and performance bounds, we needto introduce some notations. Define

ρ � ρr + γ1 , (3.10)

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γ1 � κ3(ρr )

1−‖G(s)‖L1L�ρr

γ0 +β , (3.11)

where β and γ0 are arbitrarily small positive constants such that γ1 ≤ γ1, and κ3(ρr ) isdefined as

κ3(ρr ) �(1+a1 +‖C1(s)‖L1Lρr

)(‖G(s)‖L1κ4 +‖H (s)C(s)(sI−Am)‖L1

)+‖C1(s)(s+a1)‖L1

(3.12)

with

κ4 � a2 ‖C1(s)(s+a1)‖L1+‖sC1(s)(s+a1)‖L1

. (3.13)

Moreover, let

ρu � ρur +γ2 , (3.14)

where ρur and γ2 are given as

ρur �∥∥∥∥C2(s)

ω

∥∥∥∥L1

(κ1(ρr )ρr +κ2) , (3.15)

γ2 �∥∥∥∥C2(s)

ω

∥∥∥∥L1

κ1(ρr )γ1 +κ5γ0 , (3.16)

with κ5 being defined as

κ5 �∥∥∥∥C2(s)

ω

∥∥∥∥L1

κ4 +∥∥∥∥C2(s)

ω(s+a2)

∥∥∥∥L1

.

Finally, let

θb � max{dfx1(ρ),dfx2(ρ)} , �� B+ ε , (3.17)

where ε > 0 is an arbitrary constant.

The elements of L1 adaptive (backstepping) controller are introduced next.

State Predictor


˙x1(t) = −a1x1(t)+ θ1(t)|x1(t)|+ σ1(t)+x2(t) , x1(0) = x10 ,

˙x2(t) = −a2x2(t)+ θ2(t)‖x(t)‖∞ + σ2(t)+ ω(t)u(t) , x2(0) = x20 ,(3.18)

where x1(t), x2(t) ∈ R are the predictor states, x(t) � [x1(t), x2(t)]�; x(t) � [x1(t), x2(t)]� �[x1(t)−x1(t), x2(t)−x2(t)]� is the prediction error; and θ1(t), θ2(t), σ1(t), σ2(t), and ω(t)are the adaptive estimates.

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Adaptation Laws

The adaptive estimates are governed by

˙ω(t) = �Proj(ω(t),−x�(t)P [0, 1]�u(t)

), ω(0) = ω0 ,

˙θ1(t) = �Proj

(θ1(t),−x�(t)P [1, 0]�|x1(t)|

), θ1(0) = θ10 ,

˙θ2(t) = �Proj

(θ2(t),−x�(t)P [0, 1]�‖x(t)‖∞

), θ2(0) = θ20 ,

˙σ1(t) = �Proj(σ1(t),−x�(t)P [1, 0]�

), σ1(0) = σ10 ,

˙σ2(t) = �Proj(σ2(t),−x�(t)P [0, 1]�

), σ2(0) = σ20 ,

(3.19)

where� ∈ R+ is the adaptation gain, and the symmetric positive definite matrixP =P�> 0

solves the Lyapunov equationA�mP +PAm= −Q for arbitraryQ=Q�> 0. The projection

operator Proj(·, ·) ensures that |θi(t)| ≤ θb, σi(t) ≤�, i = 1,2, and ω(t) ∈�, where θb and�were defined in (3.17).

Control Law

Let

α(t) � −a1 (x1(t)− r(t))− η1C(t)+ r(t) , (3.20)

where η1C(t) is the signal with Laplace transform η1C(s) � C1(s)η1(s) with η1(t) �θ1(t)|x1(t)| + σ1(t). Then, the control signal is generated as the output of the (feedback)system

u(s) = −kD(s)ηu(s) , (3.21)

where ηu(s) is the Laplace transform of the signal

ηu(t) � η2(t)+a2 (x2(t)−α(t))− α(t) ,

with η2(t) � ω(t)u(t)+ θ2(t)‖x(t)‖∞ + σ2(t).



Next, we refer to Lemma A.8.1 to transform the nonlinear system in (3.1) into a(semi-)linear system with unknown time-varying parameters and disturbances. Since

‖x0‖∞ < ρr < ρ , u(0) = 0,

and x(t), u(t) are continuous, there always exists τ such that

‖xτ‖L∞ ≤ ρ , ‖uτ‖L∞ ≤ ρu .

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Then, Lemma A.8.1 implies that the nonlinear system in (3.1) can be rewritten over t ∈[0, τ ] as

x1(t) = θ1(t)|x1(t)|+σ1(t)+x2(t) , x1(0) = x10 ,

x2(t) = θ2(t)‖x(t)‖∞ +σ2(t)+ωu(t) , x2(0) = x20 ,(3.22)

where θi(t), σi(t), i = 1,2, are unknown time-varying signals subject to

|θi(t)|< θb , |σi(t)|<� , ∀ t ∈ [0, τ ] ,

|θi(t)|< dθi (ρ,ρu) , |σi(t)| ≤ dσi (ρ,ρu) , ∀ t ∈ [0, τ ] ,

with θb and� being defined in (3.17), and dθi (ρ,ρu)> 0 and dσi (ρ,ρu)> 0 being the boundsguaranteed by Lemma A.8.1.


In this section, we characterize the closed-loop reference system that the L1 adaptive con-troller tracks in both transient and steady-state and prove its stability. Toward that end,consider the ideal nonadaptive version of the adaptive controller and define the closed-loopreference system as

x1ref (t) = f1(t ,x1ref (t))+x2ref (t) , x1ref (0) = x10 ,

x2ref (t) = f2(t ,xref (t))+ωuref (t) , x2ref (0) = x20 ,

uref (s) = −C2(s)

ωηuref (s) ,

(3.23)

where xref (t) � [x1ref (t), x2ref (t)]�, xref (0) � x0, and

ηuref (t) � f2(t ,xref (t))+a2 (x2ref (t)−αref (t))− αref (t) , (3.24)

with αref (t) being defined as

αref (t) � −a1 (x1ref (t)− r(t))−η1Cref (t)+ r(t) , (3.25)

and with η1Cref (t) being the signal with the Laplace transform η1Cref (s) � C1(s)η1ref (s),where η1ref (t) � f1(t ,x1ref (t)). For convenience, we also define η2ref (t) � f2(t ,xref (t)).

Lemma 3.1.1 For the closed-loop reference system in (3.23), subject to the L1-norm con-dition in (3.5), if ‖x0‖∞ < ρ0, then


where ρr and ρur were defined in (3.5) and (3.15), respectively.

Proof. Suppose that the bound on ‖xref ‖L∞ is not true. Then, because ‖x0‖∞ < ρr , itfollows from the continuity of the solution that there exists a time instant τ > 0, such that

‖xref (t)‖∞ < ρr , ∀ t ∈ [0,τ ) ,

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and

‖xref (τ )‖∞ = ρr ,

which implies that

‖xref τ‖L∞ = ρr . (3.27)

Let

eref (t) �[e1ref (t)e2ref (t)

]�[

x1ref (t)− r(t)x2ref (t)−αref (t)

]. (3.28)


eref (t) =[

x1ref (t)− r(t)x2ref (t)− αref (t)

]=[

η1ref (t)+x2ref (t)− r(t)η2ref (t)+ωuref (t)− αref (t)

]

= Ageref (t)+[

η1ref (t)−η1Cref (t)a2e2ref (t)+η2ref (t)− αref (t)+ωuref (t)

].

In frequency domain, this expression can be rewritten as

eref (s) =H (s)

[η1ref (s)−η1Cref (s)

a2e2ref (s)+η2ref (s)−ηαref (s)+ωuref (s)

]+H (s)e0 , (3.29)

where ηαref (s) is the Laplace transform of αref (t) and e0 is the initial condition of eref (t),i.e., e0 � eref (0). Note that e0 can be upper bounded as follows:

‖e0‖∞ ≤ (1+a1)(ρ0 + rmax)+ rmax .

Also, it follows from (3.23) and (3.24) that

uref (s) = −C2(s)

ω(a2e2ref (s)+η2ref (s)−ηαref (s)) . (3.30)

Substituting (3.30) into (3.29), we have

eref (s) =G(s)ζref (s)+H (s)e0 , (3.31)

where ζref (s) is the Laplace transform of the signal

ζref (t) �[

η1ref (t)a2e2ref (t)+η2ref (t)− αref (t)

]. (3.32)

Next, we derive a bound for ‖ eref τ ‖L∞ . First, we note that∥∥ζref τ

∥∥L∞ ≤ max

{∥∥η1ref τ

∥∥L∞ ,

∥∥η2ref τ

∥∥L∞

}+a2 ‖e2ref τ‖L∞ +‖ αref τ‖L∞ ,

and, since e2ref (t) = x2ref (t)−αref (t), it follows that∥∥ζref τ

∥∥L∞ ≤ max

{∥∥η1ref τ

∥∥L∞ ,

∥∥η2ref τ

∥∥L∞

}+a2 ‖x2ref τ‖L∞

+ a2 ‖αref τ‖L∞ +‖ αref τ‖L∞ . (3.33)

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Next, taking into consideration Assumptions 3.1.2 and 3.1.3, the equality in (3.27), togetherwith the fact that ρr < ρr (ρr ), yields the following upper bound:∥∥ηiref τ

∥∥L∞ ≤ dfxi (ρr (ρr ))ρr (ρr )+B , i = 1,2 .

From the redefinition in (3.3), it follows that∥∥ηiref τ

∥∥L∞ ≤ Liρr ρr +B , i = 1,2 ,

and the definition of Lδ in (3.4) leads to∥∥ηiref τ

∥∥L∞ ≤ Lρr ρr +B , i = 1,2 . (3.34)

Moreover, from the definition of αref (t) in (3.25) and the bounds in (3.27) and (3.34), itfollows that

‖αref τ‖L∞ ≤ ‖C1(s)‖L1

∥∥η1ref τ

∥∥L∞ +a1

(‖x1ref τ‖L∞ +‖r‖L∞)+‖r‖L∞

≤ ‖C1(s)‖L1

(Lρr ρr +B

)+a1(ρr +‖r‖L∞

)+‖r‖L∞ . (3.35)

Further, from the definition ofαref (t) in (3.25) and the definition of the closed-loop referencesystem in (3.23), we have

α(t) = −a1 (x1ref (t)− r(t))− η1Cref (t)+ r(t)= −a1 (η1ref (t)+x2ref (t)− r(t))− η1Cref (t)+ r(t) ,

which, together with the bound in (3.34) and the equality in (3.27), leads to

‖ αref τ‖L∞ ≤ (a1 +‖sC1(s)‖L1

)∥∥η1ref τ

∥∥L∞ +a1 ‖x2ref τ‖L∞

+ a1 ‖r‖L∞ +‖r‖L∞≤ (a1 +‖sC1(s)‖L1

)(Lρr ρr +B

)+a1ρr +a1 ‖r‖L∞ +‖r‖L∞ . (3.36)

Then, the bounds in (3.33), (3.34), (3.35), and (3.36) lead to∥∥ζref τ

∥∥L∞ ≤

((1+a1 +a2‖C1(s)‖L1 +‖sC1(s)‖L1

)Lρr +a2 (1+a1)+a1

)ρr

+((

1+a1 +a2‖C1(s)‖L1 +‖sC1(s)‖L1

)B+a1a2‖r‖L∞

+ (a1 +a2)‖r‖L∞ +‖r‖L∞)

,

and the definitions of κ1(ρr ) and κ2 in (3.8) and (3.9) imply that∥∥ζref τ

∥∥L∞ ≤ κ1(ρr )ρr +κ2 . (3.37)

From this bound, the expression in (3.31), and the definition of ρin in (3.2), it follows that

‖eref τ‖L∞ ≤ ‖G(s)‖L1(κ1(ρr )ρr +κ2)+ρin . (3.38)

Using the bound above, we next prove that the L1-norm condition in (3.5) leads tothe first bound in (3.26). From the definition of eref (t) in (3.28) it follows that[

x1ref (t)x2ref (t)

]= eref (t)+

[r(t)αref (t)

],

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‖xref τ‖L∞ ≤ ‖eref τ‖L∞ +‖r‖L∞ +‖αref τ‖L∞ . (3.39)

From the definition of αref (t) in (3.25) we have

αref (t) = −a1e1ref (t)−η1Cref (t)+ r(t) ,

and consequently

‖αref τ‖L∞ ≤ a1 ‖e1ref τ‖L∞ +‖C(s)‖L1

(Lρr ‖x1ref τ‖L∞ +B)+‖r‖L∞

≤ a1 ‖e1ref τ‖L∞ +‖C(s)‖L1

(Lρr

(‖e1ref τ‖L∞ +‖r‖L∞)+B)+‖r‖L∞

≤ a1 ‖eref τ‖L∞ +‖C(s)‖L1

(Lρr

(‖eref τ‖L∞ +‖r‖L∞)+B)+‖r‖L∞ ,

which, together with the bound in (3.39), leads to

‖xref τ‖L∞ ≤ (1+a1 +‖C(s)‖L1Lρr

)‖eref τ‖L∞+(‖C1(s)‖L1

(Lρr ‖r‖L∞ +B)+‖r‖L∞ +‖r‖L∞

).

Then, the upper bound on ‖ eref τ ‖L∞ in (3.38) yields

‖xref τ‖L∞ ≤ (1+a1 +‖C(s)‖L1Lρr

)(‖G(s)‖L1(κ1(ρr )ρr +κ2)+ρin

)+(‖C1(s)‖L1

(Lρr ‖r‖L∞ +B)+‖r‖L∞ +‖r‖L∞

),

which implies that

‖xref τ‖L∞ ≤ ‖G(s)‖L1

(L�ρr ρr +β1(ρr )

)+β0(ρr ) .

The condition in (3.5) can be solved for ρr to obtain the bound

‖G(s)‖L1

(L�ρr ρr +β1(ρr )

)+β0(ρr )< ρr ,

which leads to

‖xref τ‖L∞ < ρr

and contradicts the equality in (3.27), thus proving the first bound in (3.26).To prove the second bound in (3.26), we first note that, from the expressions in (3.30)

and (3.32), it follows that

‖uref τ‖L∞ ≤∥∥∥∥C2(s)

ω

∥∥∥∥L1

∥∥ζref τ

∥∥L∞ .

Since the bound on ‖xref ‖L∞ in (3.26) implies that the upper bound in (3.37) holds for allt ∈ [0,τ ] with strict inequality, we have

‖uref ‖L∞ <

∥∥∥∥C2(s)

ω

∥∥∥∥L1

(κ1(ρr )ρr +κ2) = ρur ,

which proves the bound on ‖uref ‖L∞ in (3.26). �

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Using (3.18) and (3.22) one can write the error dynamics over t ∈ [t0, τ ]

˙x(t) = Amx(t)+[

θ1(t)|x1(t)|+ σ1(t)ω(t)u(t)+ θ2(t)‖x(t)‖∞ + σ2(t)

], x(0) = 0, (3.40)

where ω(t) � ω(t)−ω, θ1(t) � θ1(t)−θ1(t), σ1(t) � σ1(t)−σ1(t), θ2(t) � θ2(t)−θ2(t), andσ2(t) � σ2(t)−σ2(t).

Lemma 3.1.2 For the dynamics in (3.40), if

‖xτ‖L∞ ≤ ρ , ‖uτ‖L∞ ≤ ρu , (3.41)

and the adaptive gain is chosen to satisfy the design constraint

� >θm(ρr )

λmin(P )γ 20

, (3.42)

where γ0 was introduced in (3.11) and θm(ρr ) is defined as

θm(ρr ) = 8θ2b +8�2 + (ωu−ωl)2 +4

λmax(P )

λmin(Q)

(2∑i=1

(θbdθi +�dσi

)), (3.43)

the following bound holds:

‖xτ‖L∞ < γ0 .

Proof. First, we note that, if the bounds in (3.41) hold, then one can write the predictionerror dynamics in (3.40). Next, consider the Lyapunov function candidate

V (x(t), θi(t), σi(t), ω(t)) = x�(t)P x(t)+ 1

�

(θ2

1 (t)+ θ22 (t)+ σ 2

1 (t)+ σ 22 (t)+ ω2(t)

),

(3.44)

and let t1 ∈ [0,τ ) be the first instant of the discontinuity of any of the derivatives θi or σi .Next we prove that

V (t) ≤ θm(ρr )

�, ∀ t ∈ [0, t1] .

The projection-based adaptive laws in (3.19) ensure that for arbitrary t ∈ [0, t1),

V (t) ≤ −x�(t)Qx(t)+ 2

�

(|θ1(t)||θ1(t)|+ |θ2(t)||θ2(t)|+ |σ1(t)||σ1(t)|+ |σ2(t)||σ2(t)|

).

They also imply that

maxt∈[0, t1)

(θ2

1 (t)+ θ22 (t)+ σ 2

1 (t)+ σ 22 (t)+ ω2(t)

)≤(

4(

2θ2b +2�2

)+ (ωu−ωl)2

).

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Since θi(t) and σi(t) are continuous over t ∈ [0, t1), we conclude

maxt∈[0, t1)

(|θ1(t)||θ1(t)|+ |θ2(t)||θ2(t)|+ |σ1(t)||σ1(t)|

+ |σ2(t)||σ2(t)|)

≤ 22∑i=1

(θbdθi +�dσi

).

Next, notice that if at arbitrary time t ′ ∈ [0, t1), V (t ′) > θm(ρr )�

, it follows from (3.43)and (3.44) that

x�(t ′)P x(t ′)> 4λmax(P )

λmin(Q)

(2∑i=1

(θbdθi +�dσi

)),

and therefore

x�(t ′)Qx(t ′)> 42∑i=1

(θbdθi +�dσi

).

This implies that V (t ′)< 0. Since V (0)< θm(ρr )�

, it follows that V (t) ≤ θm(ρr )�

for arbitraryt ∈ [0, t1), which yields

‖x(t)‖∞ ≤√

θm(ρr )

(λmin(P )�), ∀ t ∈ [0, t1) .

The design constraint on � in (3.42) leads to

‖x(t)‖∞ < γ0 , ∀ t ∈ [0, t1) .

If t2 ∈ (t1, τ ] is the next instant of the discontinuity of the derivative, we can use similarderivations as above to prove that

‖x(τ )‖∞ < γ0 , ∀ t ∈ [t1, t2) .

Iterating the process until the time instant t , we get

‖xτ‖L∞ < γ0 ,


Theorem 3.1.1 For the system in (3.1), with the L1 adaptive controller defined via (3.18)–(3.21), subject to the L1-norm condition in (3.5) and the design constraint in (3.42), if‖x0‖∞ < ρ0, then

‖x‖L∞ ≤ ρ , (3.45)

‖u‖L∞ ≤ ρu , (3.46)

‖x‖L∞ ≤ γ0 , (3.47)

‖xref −x‖L∞ ≤ γ1 , (3.48)

‖uref −u‖L∞ ≤ γ2 , (3.49)

where γ1 and γ2 were defined in (3.11) and (3.16), respectively.

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Proof. The proof is done by contradiction. Assume that the bounds in (3.48) and (3.49) donot hold. Then, since ‖xref (0)−x(0)‖∞ = 0 < γ1, ‖uref (0)−u(0)‖∞ = 0 < γ2, and x(t),xref (t), u(t), and uref (t) are continuous, there exists a time τ > 0 such that

‖xref (τ )−x(τ )‖∞ = γ1 , or ‖uref (τ )−u(τ )‖∞ = γ2 ,

while

‖xref (t)−x(t)‖∞ < γ1 , ‖uref (t)−u(t)‖∞ < γ2 , ∀ t ∈ [0,τ ) ,

which implies that that at least one of the following equalities holds:∥∥ (xref −x) τ∥∥

L∞ = γ1 ,∥∥ (uref −u) τ

∥∥L∞ = γ2 . (3.50)

Taking into consideration the definitions of ρ and ρu in (3.10) and (3.14), it follows fromLemma 3.1.1 and the equalities in (3.50) that

‖x τ‖L∞ ≤ ρ , ‖uτ‖L∞ ≤ ρu . (3.51)

These bounds imply that the assumptions of Lemma 3.1.2 hold. Then, selecting the adaptivegain � according to the design constraint in (3.42), it follows that

‖ x τ‖L∞ ≤ γ0 . (3.52)

Next, let η1(t) and η2(t) be defined as

η1(t) � θ1(t)|x1(t)|+σ1(t) , η2(t) � θ2(t)‖x(t)‖∞ +σ2(t) .

From the bounds in (3.51) it follows that for all t ∈ [0,τ ] the following equalities hold:

η1(t) = f1(t ,x1(t)) , η2(t) = f2(t ,x(t)) .

Also, let e(t) be defined as

e(t) �[x1(t)− r(t)x2(t)−α(t)

],

and notice that, at time zero, since we assumed zero initialization of the filter C1(s) in thecontrol law, we have e(0) = eref (0) = e0 . Then, from the system dynamics in (3.1) and thedefinition of α(t) in (3.20), it follows that

e(t) =[x1(t)− r(t)x2(t)− α(t)

]=[η1(t)+x2(t)− r(t)η2(t)+ωu(t)− α(t)

]

= Age(t)+[

η1(t)− η1C(t)a2e2(t)+η2(t)− α(t)+ωu(t)

].

The dynamics above can be written in the frequency domain as

e(s) =H (s)

[η1(s)− η1C(s)

a2e2(s)+η2(s)−ηα(s)+ωu(s)

]+H (s)e0 , (3.53)

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where ηα(s) is the Laplace transform of α(t). Moreover, it follows from (3.21) that

u(s) = −C2(s)

ω

(a2e2(s)+η2(s)−ηα(s)+ ηω(s)+ η2(s)

), (3.54)

where η2(s) and ηω(s) are the Laplace transforms of η2(t) � θ2(t)‖x(t)‖∞ + σ2(t) andηω(t) � ω(t)u(t), respectively. Substituting (3.54) into (3.53) leads to

e(s) =G(s)ζ (s)−H (s)C(s)ζ (s)+H (s)e0 , (3.55)

where ζ (s) and ζ (s) are the Laplace transforms of the signals ζ (t) and ζ (t) defined as

ζ (t) �[

η1(t)a2e2(t)+η2(t)− α(t)

],

ζ (t) �[

η1(t)ω(t)u(t)+ η2(t)

],

with η1(t) � θ1(t)|x1(t)|+ σ1(t). The expression in (3.55), together with the response of theclosed-loop reference system in (3.31), yields

eref (s)− e(s) =G(s) (ζref (s)− ζ (s))+H (s)C(s)ζ (s) , (3.56)

where ζref (t) was introduced in (3.32). Also, the error dynamics in (3.40) lead to

x(s) = (sI−Am)−1 ζ (s) , (3.57)

which implies that the expression in (3.56) can be rewritten as

eref (s)− e(s) =G(s) (ζref (s)− ζ (s))+H (s)C(s) (sI−Am) x(s) . (3.58)

Next, we derive an upper bound on ‖ (eref − e) τ ‖L∞ in terms of ‖ (xref −x) τ ‖L∞ .From the definitions of ζ (t) and ζref (t), it follows that

ζref (t)− ζ (t) =[η1ref (t)−η1(t)

(η2ref (t)−η2(t))+a2 (x2ref (t)−x2(t))−a2 (αref (t)−α(t))+ (αref (t)− α(t))

],

(3.59)

which implies that∥∥ (ζref − ζ ) τ∥∥

L∞ ≤ max{∥∥ (η1ref −η1) τ

∥∥L∞ ,

∥∥ (η2ref −η2) τ∥∥

L∞

}+a2

∥∥ (x2ref −x2) τ∥∥

L∞ +a2∥∥ (αref −α) τ

∥∥L∞

+∥∥ (αref − α) τ∥∥

L∞ .

(3.60)

Taking into account that ‖x(t)‖∞ ≤ ρ = ρr (ρr ) and also ‖xref (t)‖∞ ≤ ρr < ρr (ρr ) for allt ∈ [0, τ ], Assumption 3.1.3 implies that∥∥ (ηiref −ηi) τ

∥∥L∞ ≤ dfxi (ρr (ρr ))

∥∥ (xref −x) τ∥∥

L∞ , i = 1,2 .

L1book2010/7/22page 134

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From the definitions in (3.3) and (3.4), it follows that dfxi (ρr (ρr ))<Lρr , i = 1,2, and hence∥∥ (ηiref −ηi) τ∥∥

L∞ ≤ Lρr∥∥ (xref −x) τ

∥∥L∞ , i = 1,2 . (3.61)

Moreover, it follows from the definitions of α(t) and αref (t) in (3.20) and (3.25) that

αref (s)−α(s) = −a1 (x1ref (s)−x1(s))−C1(s) (η1ref (s)−η1(s))+C1(s)η1(s) ,

and the error dynamics in (3.57) further imply that

αref (s)−α(s) = −a1 (x1ref (s)−x1(s))−C1(s) (η1ref (s)−η1(s))+C1(s)(s+a1)x1(s).

The above expression, together with the bounds in (3.61), yields∥∥ (αref −α) τ∥∥

L∞ ≤ a1∥∥ (x1ref −x1) τ

∥∥L∞ +‖C1(s)‖L1

Lρr∥∥ (xref −x) τ

∥∥L∞

+‖C1(s)(s+a1)‖L1‖ x τ‖L∞ .

Recalling that ηα(s) and ηαref (s) denote the Laplace transforms of the signals α(t) andαref (t), it follows from the definitions of α(t) and αref (t), together with the error dynamicsin (3.57), that

ηαref (s)−ηα(s) = −a1 (η1ref (s)−η1(s))−a1 (x2ref (s)−x2(s))

+ sC1(s) (η1ref (s)−η1(s))+ sC1(s)(s+a1)x1(s) ,

which, together with the bounds in (3.61), leads to the following upper bound:∥∥ (αref − α) τ∥∥

L∞ ≤ a1∥∥ (x1ref −x1) τ

∥∥L∞ +a1

∥∥ (x2ref −x2) τ∥∥

L∞+‖sC1(s)‖L1

∥∥ (x1ref −x1) τ∥∥

L∞+‖sC1(s)(s+a1)‖L1

‖ x τ‖L∞ .

(3.62)

Then, the expression in (3.58), along with the bounds in (3.60)–(3.62) and the definitionsof κ1(ρr ) and κ4 in (3.8) and (3.13), leads to

∥∥ (eref − e) τ∥∥

L∞ ≤ ‖G(s)‖L1

(κ1(ρr )

∥∥ (xref −x) τ∥∥

L∞ +κ4 ‖ x τ‖L∞

)+‖H (s)C(s) (sI−Am)‖L1

‖ x τ‖L∞ ,

or, equivalently,∥∥ (eref − e) τ∥∥

L∞ ≤ ‖G(s)‖L1κ1(ρr )

∥∥ (xref −x) τ∥∥

L∞+ (‖G(s)‖L1

κ4 +‖H (s)C(s) (sI−Am)‖L1

)‖ x τ‖L∞ .(3.63)

Next, noting that

xref (t)−x(t) = (eref (t)− e(t))+[

0αref (t)−α(t)

],

L1book2010/7/22page 135

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we can derive the following upper bound on ‖ (xref −x) τ ‖L∞ :∥∥ (xref −x) τ∥∥

L∞ ≤ ∥∥ (eref − e) τ∥∥

L∞ +∥∥ (αref −α) τ∥∥

L∞ . (3.64)

From the definitions of α(t) and αref (t) in (3.20) and (3.25) some straightforward manipu-lations lead to∥∥ (αref −α) τ

∥∥L∞ ≤ a1

∥∥ (eref − e) τ∥∥

L∞ +‖C1(s)‖L1Lρr

∥∥ (eref − e) τ∥∥

L∞+‖C1(s)(s+a1)‖L1

‖ x τ‖L∞ ,

which, together with the bound in (3.64), yields∥∥ (xref −x) τ∥∥

L∞ ≤ (1+a1 +‖C1(s)‖L1Lρr

)∥∥ (eref − e) τ∥∥

L∞+‖C1(s)(s+a1)‖L1

‖ x τ‖L∞ .

We can now apply the bound obtained in (3.63) to the expression above to find∥∥ (xref −x) τ∥∥

L∞ ≤ ‖G(s)‖L1L�ρr

∥∥ (xref −x) τ∥∥

L∞ +κ3(ρr )‖ x τ‖L∞ ,

where we have used the definitions ofL�ρr and κ3(ρr ) in (3.7) and (3.12), respectively. Then,noting that the L1-norm condition in (3.5) ensures that ‖G(s)‖L1

L�ρr < 1, we can derivethe bound

∥∥ (xref −x) τ∥∥

L∞ ≤ κ3(ρr )


‖ x τ‖L∞ ,


∥∥ (xref −x) τ∥∥

L∞ ≤ κ3(ρr )


γ0 .

The definition of γ1 in (3.11) implies that∥∥ (xref −x) τ∥∥

L∞ ≤ γ1 −β < γ1 , (3.65)

which contradicts the first equality in (3.50).On the other hand, it follows from (3.30) and (3.54) that

uref (s)−u(s) = −C2(s)

ω

((η2ref (s)−η2(s)

)+a2(x2ref (s)−x2(s)

)−a2 (αref (s)−α(s))− (ηαref (s)−ηα(s)

)− (ηω(s)+ η2(s)))

.

Using the expression in (3.59) and the error dynamics in (3.57), one can show that

∥∥ (uref −u) τ∥∥

L∞ ≤∥∥∥∥C2(s)

ω

∥∥∥∥L1

∥∥ (ζref − ζ ) τ∥∥

L∞ +∥∥∥∥C2(s)

ω(s+a2)

∥∥∥∥L1

‖ x τ‖L∞ ,

L1book2010/7/22page 136

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which leads to∥∥ (uref −u) τ∥∥

L∞ ≤∥∥∥∥C2(s)

ω

∥∥∥∥L1

(κ1(ρr )

∥∥ (xref −x) τ∥∥

L∞ +κ4 ‖ x τ‖L∞

)

+∥∥∥∥C2(s)

ω(s+a2)

∥∥∥∥L1

‖ x τ‖L∞ .

Finally, the bounds in (3.52) and (3.65), along with the definition of γ2 in (3.16), yield

∥∥ (uref −u) τ∥∥

L∞ ≤∥∥∥∥C2(s)

ω

∥∥∥∥L1

κ1(ρr ) (γ1 −β)+κ5γ0 < γ2 ,

which contradicts the second equality in (3.50). This proves the bounds in (3.48) and (3.49).Thus, the bounds in (3.51) hold uniformly, which implies that the bound in (3.52) also holdsuniformly. This proves the bounds in (3.45)–(3.47). �

Remark 3.1.1 It follows from the definitions of γ1 and γ2 in (3.11) and (3.16) that onecan achieve arbitrary desired performance bounds for the system’s signals, both input andoutput, simultaneously, by increasing the adaptive gain.

Remark 3.1.2 To understand how the performance bounds can be used for ensuring uniformtransient response with desired specifications, consider the ideal control law for the systemin (3.1):

uid(t) = ω−1 (−a2 (x2id(t)−αid(t))−η2id(t)+ αid(t)) ,

αid(t) = −a1 (x1id(t)− r(t))−η1id(t)+ r(t) ,

with η1id = f1(x1id(t), t) and η2id = f2(t ,xid(t)). This ideal nonadaptive controller leads tothe desired system response

eid(t) = Ageid(t) , (3.66)

where eid(t) � [(x1id(t) − r(t)), (x2id(t) − αid(t))]�. In the closed-loop reference system(3.23), uid(t) is further low-pass filtered to have a guaranteed low-frequency range. Similarto Section 2.1.4, the response of the closed-loop reference system can be made as close aspossible to (3.66) by reducing ‖G(s)‖L1 . From the definition ofG(s) in (3.6), and noting that(I−C(s)) has a diagonal structure with (1 −C1(s)) and (1 −C2(s)) as diagonal elements,we notice that ‖G(s)‖L1 can be made arbitrarily small by increasing the bandwidths of thelow-pass filters C1(s) and C2(s).


To verify numerically the results proved in this section, we consider the system given in (3.1).We perform simulations for the following scenarios:

– Scenario 1:

ω = ω1 � 0.8 ,

f1(t ,x1(t)) = f 11 (t ,x1(t)) � 0.1x1(t)+0.5x2

1 (t)−0.2sin(0.1t) ,

f2(t ,x(t)) = f 12 (t ,x(t)) � x1(t)+x2(t)+x2

1 (t)+x22 (t)+ sin(t) .

L1book2010/7/22page 137

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– Scenario 2:

ω = ω2 � 1.1 .

f1(t ,x1(t)) = f 11 (t ,x1(t)) ,

f2(t ,x(t)) = f 12 (t ,x(t)) .

– Scenario 3:

ω = ω1 ,

f1(t ,x(t)) = f 21 (t ,x(t)) � −x1(t)+0.3x2

1 (t)+ sin(x1(t))−0.1cos(t) ,

f2(t ,x(t)) = f 12 (t ,x(t)) .

– Scenario 4:

ω = ω1 ,

f1(t ,x(t)) = f 11 (t ,x(t)) ,

f2(t ,x(t)) = f 22 (t ,x(t)) � −x1(t)+x2(t)+ sin(x1(t))x2

2 (t)+x1(t)x2(t) ,

– Scenario 5:

ω = ω2 ,

f1(t ,x(t)) = f 21 (t ,x(t)) ,

f2(t ,x(t)) = f 22 (t ,x(t)) .

In Scenario 1 we consider an underactuated system with significant nonlinearities anda sinusoidal disturbance. Scenarios 2–4 are derived from Scenario 1 by changing only one ofthe unknown quantities at a time; the uncertainty in the input gain is changed in Scenario 2,and the unknown matched and unmatched nonlinearities are changed in Scenarios 3 and 4.In Scenario 5 we change all unknown quantities simultaneously.

We implement the L1 adaptive controller according to (3.18), (3.19), and (3.21). Forall five scenarios we use a single L1 controller design with the control parameters

k = 10, D(s) = 1

s, C1(s) = 0.1

(s+0.1)(s+1), a1 = 2, a2 = 2.

We set the projection bounds to be �= [0.1, 5], Lρ = 20, and �= 50 and the adaptationgain to � = 100000.

Figure 3.1 shows the simulation results for Scenarios 1 and 2; Figure 3.2 shows theresults for Scenarios 1 and 3; Figure 3.3 shows the results for Scenarios 1 and 4. In thesesimulations, we set the system reference input to r(t) = 0.5cos(0.5t). From the results onecan see that the fast adaptation ability of the L1 adaptive controller ensures uniform transientperformance for different uncertainties. We notice that the system’s output y(t) = x1(t) isnot significantly affected by changing the dynamics. However, the control signal changessignificantly to ensure adequate compensation for those.

Figure 3.4 shows the simulation results for Scenario 1 with reference signals of dif-ferent amplitudes:

r1(t) = 0.2cos(0.5t) , r2(t) = 0.5cos(0.5t) .

We observe that the system response scales with scaled reference inputs.

L1book2010/7/22page 138

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0 5 10 15 20 25−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

time [s]

r(t)x1(t) for ω1

x1(t) for ω2

(a) r(t) and x1(t)

0 5 10 15 20 25−3

−2

−1

0

1

2

time [s]

ω1ω2


0 5 10 15 20 25−0.4

−0.2

0

0.2

0.4

0.6

time [s]

ω1ω2

(c) x2(t)

0 5 10 15 20 25−10

−5

0

5

10

15

20

time [s]

ω1ω2

(d) Time history of u(t)

Figure 3.1: Performance of the L1 controller for Scenarios 1 and 2.

0 5 10 15 20 25−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

time [s]

r(t)x1(t) for f1

1 (t, x)x1(t) for f2

1 (t, x)

(a) r(t) and x1(t)

0 5 10 15 20 25−4

−3

−2

−1

0

1

2

time [s]

f11 (t, x)

f21 (t, x)


0 5 10 15 20 25−0.4

−0.2

0

0.2

0.4

0.6

time [s]

f11 (t, x)

f21 (t, x)

(c) x2(t)

0 5 10 15 20 25−20

−10

0

10

20

30

time [s]

f11 (t, x)

f21 (t, x)



L1book2010/7/22page 139

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0 5 10 15 20 25−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

time [s]

r(t)x1(t) for f1

2 (t, x)x1(t) for f2

2 (t, x)

(a) r(t) and x1(t)

0 5 10 15 20 25−3

−2

−1

0

1

2

time [s]

f12 (t, x)

f22 (t, x)


0 5 10 15 20 25−0.4

−0.2

0

0.2

0.4

0.6

time [s]

f12 (t, x)

f22 (t, x)

(c) x2(t)

0 5 10 15 20 25−10

−5

0

5

10

15

20

time [s]

f12 (t, x)

f22 (t, x)



0 2 4 6 8 10−0.5

−0.25

0

0.25

0.5

time [s]

(a) x1(t) for r1(t) and r2(t)

0 2 4 6 8 10−3

−2

−1

0

1

2

time [s]

r1(t)r2(t)


0 2 4 6 8 10−0.4

−0.2

0

0.2

0.4

0.6

time [s]

r1(t)r2(t)

(c) x2(t)

0 2 4 6 8 10−10

−5

0

5

10

15

20

time [s]

r1(t)r2(t)


Figure 3.4: Performance of the L1 controller for r1(t) and r2(t) reference signals (Sce-nario 1).

L1book2010/7/22page 140

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0 5 10 15 20 25−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

time [s]

r(t)x1(t)

(a) r(t) and x1(t) (Scenario 1)

0 5 10 15 20 25−6

−4

−2

0

2

4

time [s]

(b) Time history of u(t) (Scenario 1)

0 5 10 15 20 25−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

time [s]

r(t)x1(t)

(c) r(t) and x1(t) (Scenario 5)

0 5 10 15 20 25−2

−1

0

1

2

3

time [s]

(d) Time history of u(t) (Scenario 5)

Figure 3.5: Performance of the L1 controller with time delay of 80 ms.

Finally, we numerically test the robustness of the L1 adaptive controller to time delays.Figure 3.5 shows the simulation results for Scenarios 1 and 5 in the presence of a time delayof 0.08 s. One can see that the system has some expected degradation in the performancebut remains stable. Moreover, the system output in the presence of time delay remains closeto the one in the absence of time delay for both cases of uncertainties.

3.2 L1 Adaptive Controller for Multi-Input Multi-OutputSystems in the Presence of Unmatched NonlinearUncertainties

This section presents the L1 adaptive controller for a class of multi-input multi-output un-certain systems in the presence of uncertain system input gain and time- and state-dependentunknown nonlinearities, without enforcing matching conditions. The class of systems con-sidered includes general unmatched uncertainties that cannot be addressed by recursivedesign methods developed for strict-feedback systems in Section 3.1, semi-strict-feedbacksystems, pure-feedback systems, and block-strict-feedback systems [100, 181]. We showthat, subject to a set of mild assumptions, the system can be transformed into an equiva-lent (semi-)linear system with time-varying unknown parameters and disturbances. For thelatter, we apply the L1 adaptive controller, which yields semiglobal performance resultsfor the original nonlinear system. The adaptive algorithm ensures uniformly bounded tran-sient response for the system’s signals, both input and output, simultaneously, in additionto steady-state tracking.

L1book2010/7/22page 141

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3.2. L1 Adaptive Controller in the Presence of Unmatched Uncertainties 141


Consider the following system dynamics:

x(t) = Amx(t)+Bmωu(t)+f (t ,x(t),z(t)) , x(0) = x0 ,

xz(t) = g (t ,xz(t),x(t)) , xz(0) = xz0 ,

z(t) = go (t ,xz(t)) ,

y(t) = Cx(t) ,

(3.67)

where x(t) ∈ Rn is the system state vector (measured); u(t) ∈ R

m is the control signal(m≤ n); y(t) ∈ R

m is the regulated output;Am is a known Hurwitz n×nmatrix that definesthe desired dynamics for the closed-loop system; Bm ∈ R

n×m is a known full-rank constantmatrix, (Am,Bm) is controllable;C ∈ R

m×n is a known full-rank constant matrix, (Am,C) isobservable;ω ∈ R

m×m is the uncertain system input gain matrix; z(t) ∈ Rp and xz(t) ∈ R

l arethe output and the state vector of internal unmodeled dynamics; and f : R×R

n×Rp → R

n,go : R×R

l → Rp, and g : R×R

l×Rn → R

l are unknown nonlinear functions continuousin their arguments. The initial condition x0 is assumed to be inside an arbitrarily large knownset, i.e., ‖x0‖∞ ≤ ρ0 <∞ for some ρ0 > 0.

The system in (3.67) can also be written in the form

x(t) = Amx(t)+Bm (ωu(t)+f1(t ,x(t),z(t)))+Bumf2(t ,x(t),z(t)) , x(0) = x0 ,

xz(t) = g (t ,xz(t),x(t)) , xz(0) = xz0 ,

z(t) = go (t ,xz(t)) ,

y(t) = Cx(t) ,

(3.68)

whereBum ∈ Rn×(n−m) is a constant matrix such thatB�

mBum = 0 and also rank([Bm, Bum]) =n, while f1 : R×R

n×Rp → R

m and f2 : R×Rn×R

p → R(n−m) are unknown nonlinear

functions that verify[f1(t ,x(t),z(t))f2(t ,x(t),z(t))

]= [

Bm Bum]−1

f (t ,x(t),z(t)) .

In this problem formulation, f1(·) represents the matched component of the unknown non-linearities, whereas Bumf2(·) represents the unmatched uncertainties.

Let X � [x�, z�]�, and with a slight abuse of language let fi(t ,X) � fi(t ,x,z),i = 1,2. The system above verifies the following assumptions.

Assumption 3.2.1 (Boundedness of fi(t ,0)) There exists Bi > 0, such that ‖fi(t ,0)‖∞ ≤Bi holds for all t ≥ 0, and for i = 1,2.

Assumption 3.2.2 (Semiglobal uniform boundedness of partial derivatives) For i= 1,2and arbitrary δ > 0, there exist positive constants dfxi (δ) > 0 and dfti (δ) > 0 independentof time, such that for all ‖X(t)‖∞ < δ, the partial derivatives of fi(t ,X) are piecewise-continuous and bounded:∥∥∥∥∂fi(t ,X)

∂X

∥∥∥∥∞≤ dfxi (δ) ,

∥∥∥∥∂fi(t ,X)

∂t

∥∥∥∥∞≤ dfti (δ) ,

where the first is a matrix induced ∞-norm, and the second is a vector ∞-norm.

L1book2010/7/22page 142

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Assumption 3.2.3 (Stability of unmodeled dynamics) The xz-dynamics are BIBO stablewith respect to both initial conditions xz0 and input x(t), i.e., there exist Lz, Bz > 0 suchthat for all t ≥ 0

‖zt‖L∞ ≤ Lz‖xt‖L∞ +Bz .

Assumption 3.2.4 (Partial knowledge of the system input gain) The system input gainmatrix ω is assumed to be an unknown (nonsingular) strictly row-diagonally dominantmatrix with sgn(ωii) known. Also, we assume that there exists a known compact convexset � such that ω ∈�⊂ R

m×m.

Assumption 3.2.5 (Stability of matched transmission zeros) The transmission zeros ofthe transfer matrix Hm(s) = C(sI−Am)−1Bm lie in the open left half plane.

The control objective is to design an adaptive state-feedback controller to ensure thaty(t) tracks the output response of a desired systemM(s) defined as

M(s) � C (sI−Am)−1BmKg(s) ,

where Kg(s) is a feedforward prefilter, to a given bounded piecewise-continuous referencesignal r(t) in both transient and steady-state, while all other signals remain bounded.



To streamline the subsequent analysis, we need to introduce some notations. Let

Hxm(s) � (sIn−Am)−1Bm ,

Hxum(s) � (sIn−Am)−1Bum ,

Hm(s) � CHxm(s) = C (sIn−Am)−1Bm ,

Hum(s) � CHxum(s) = C (sIn−Am)−1Bum ,

and let xin(t) be the signal with Laplace transform xin(s) � (sIn −Am)−1x0 and ρin �‖s(sI−Am)−1‖L1ρ0. Since Am is Hurwitz and x0 is bounded, then ‖xin‖L∞ ≤ ρin.

Further, for every δ > 0, let

Liδ � δ(δ)

δdfxi (δ(δ)) , δ(δ) � max{δ+ γ1, Lz(δ+ γ1)+Bz} , (3.69)

where dfx (·) was introduced in Assumption 3.2.2 with γ1 being an arbitrary small positiveconstant.

The design of the L1 adaptive controller involves a feedback gain matrixK ∈ Rm×m

and an m×m strictly proper transfer matrix D(s), which lead, for all ω ∈ �, to a strictlyproper stable

C(s) � ωKD(s) (Im+ωKD(s))−1 , (3.70)

L1book2010/7/22page 143

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with DC gain C(0) = Im. The choice of D(s) needs to ensure also that C(s)H−1m (s) is a

proper stable transfer matrix. For a particular class of systems, a possible choice for D(s)might be D(s) = 1

sIm, which yields a strictly proper C(s) of the form

C(s) = ωK (sIm+ωK)−1 ,

with the condition that the choice ofK must ensure that −ωK is Hurwitz. It is easy to showthat, if one lets λmax(−ωK) be the maximum (negative) eigenvalue of the Hurwitz matrix−ωK , then one has

limλmax(−ωK)→−∞

‖Im−C(s)‖L1= 0.

For the proofs of stability and performance bounds, the choice of K and D(s) alsoneeds to ensure that, for a given ρ0, there exists ρr > ρin such that the following L1-normcondition holds:

‖Gm(s)‖L1+‖Gum(s)‖L1

�0 <ρr −

∥∥Hxm(s)C(s)Kg(s)∥∥

L1‖r‖L∞ −ρin

L1ρr ρr +B0, (3.71)

where

Gm(s) �Hxm(s) (Im−C(s)) , (3.72)

Gum(s) �(In−Hxm(s)C(s)H−1

m (s)C)Hxum(s) , (3.73)

while

�0 � L2ρr

L1ρr, B0 � max

{B10,

B20

�0

},

and Kg(s) is the (BIBO-stable) feedforward prefilter. Further, let ρ be defined as

ρ � ρr + γ1 , (3.74)


γ1 �∥∥Hxm(s)C(s)H−1

m (s)C∥∥

L1

1−‖Gm(s)‖L1L1ρr −‖Gum(s)‖L1

L2ρrγ0 +β , (3.75)

where γ0 and β are arbitrarily small positive constants such that γ1 ≤ γ1. Next, let

ρu � ρur +γ2 , (3.76)


ρur �∥∥∥ω−1C(s)

∥∥∥L1

(L1ρr ρr +B10

)+∥∥∥ω−1C(s)H−1m (s)Hum(s)

∥∥∥L1

(L2ρr ρr +B20

)+∥∥∥ω−1C(s)Kg(s)

∥∥∥L1

‖r‖L∞ ,

γ2 �(∥∥∥ω−1C(s)

∥∥∥L1L1ρr +

∥∥∥ω−1C(s)H−1m (s)Hum(s)

∥∥∥L1L2ρr

)γ1

+∥∥∥ω−1C(s)H−1

m (s)C∥∥∥

L1γ0 . (3.77)

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Finally, let

θbi � Liρ , σbi � LiρBz+Bi0 + εi , i = 1,2 , (3.78)

where εi > 0, i = 1, 2, are arbitrary numbers.

The L1 adaptive control architecture is introduced below.

State Predictor


˙x(t) = Amx(t)+Bm(ω(t)u(t)+ θ1(t)‖x t‖L∞ + σ1(t))

+Bum(θ2(t)‖x t‖L∞ + σ2(t)) , x(0) = x0 ,

y(t) = Cx(t) ,

(3.79)

where ω(t) ∈ Rm×m, θ1(t) ∈ R

m, σ1(t) ∈ Rm, θ2(t) ∈ R

n−m, and σ2(t) ∈ Rn−m are the

adaptive estimates.

Adaptation Laws

The adaptation laws for ω(t), θ1(t), σ1(t), θ2(t), and σ2(t) are defined as

˙ω(t) = �Proj(ω(t) ,−(x�(t)PBm)�u�(t)) , ω(0) = ω0 ,

˙θ1(t) = �Proj(θ1(t) ,−(x�(t)PBm)�‖ x t ‖L∞ ) , θ1(0) = θ10 ,

˙σ1(t) = �Proj(σ1(t) ,−(x�(t)PBm)�) , σ1(0) = σ10 ,

˙θ2(t) = �Proj(θ2(t) ,−(x�(t)PBum)�‖ x t ‖L∞ ) , θ2(0) = θ20 ,

˙σ2(t) = �Proj(σ2(t) ,−(x�(t)PBum)�) , σ2(0) = σ20 ,

(3.80)

where x(t) � x(t)−x(t), � ∈ R+ is the adaptation gain, P = P� > 0 is the solution to the

algebraic Lyapunov equation A�mP +PAm = −Q for arbitraryQ=Q� > 0, and Proj(· , ·)

denotes the projection operator defined in Definition B.3. The projection operator ensuresthat ω(t) ∈�, ‖θi(t)‖∞ ≤ θbi , ‖σi(t)‖∞ ≤ σbi , for i = 1, 2, where θbi and σbi are as definedin (3.78).

Control Law

The control law is generated as the output of the (feedback) system

u(s) = −KD(s)η(s) , (3.81)

where η(s) is the Laplace transform of the signal

η(t) � ω(t)u(t)+ η1(t)+ η2m(t)− rg(t) , (3.82)

with rg(s) � Kg(s)r(s), η2m(s) � H−1m (s)Hum(s)η2(s), and with η1(t) and η2(t) being de-

fined as ηi(t) � θi(t)‖x t‖L∞ + σi(t), i = 1,2.

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Conventional design methods from multivariable control theory can be used to designthe prefilterKg(s) to achieve desired decoupling properties (see, e.g., [54]).As an example, ifone choosesKg(s) as the constant matrixKg = −(CA−1

m Bm)−1, then the diagonal elementsof the desired transfer matrix M(s) = C(sIn−Am)−1BmKg have DC gain equal to one,while the off-diagonal elements have zero DC gain.

The L1 adaptive controller consists of (3.79)–(3.81), subject to the L1-norm conditionin (3.71).



In this section, we characterize the closed-loop reference system that the L1 adaptive con-troller tracks both in transient and steady-state and prove its stability. Toward this end, weconsider the ideal nonadaptive version of the adaptive controller and define the closed-loopreference system as

xref (t) = Amxref (t)+Bm (ωuref (t)+f1(t ,xref (t),z(t)))

+Bumf2(t ,xref (t),z(t)) , xref (0) = x0 ,

uref (s) = −ω−1C(s)(η1ref (s)+H−1

m (s)Hum(s)η2ref (s)−Kg(s)r(s))

,

yref (t) = Cxref (t) ,

(3.83)

where η1ref (s) and η2ref (s) are the Laplace transforms of the signals ηiref (t) � fi(t ,xref (t),z(t)), i = 1,2.

Lemma 3.2.1 For the closed-loop reference system in (3.83), subject to the L1-norm con-dition (3.71), if ‖x0‖∞ ≤ ρ0 and

‖ z τ ‖L∞ ≤ Lz(‖ xref τ ‖L∞ +γ1)+Bz , (3.84)

then

‖ xref τ ‖L∞ < ρr , (3.85)

‖ uref τ ‖L∞ < ρur . (3.86)

Proof. It follows from (3.83) and the definitions ofGm(s) andGum(s) in (3.72) and (3.73)that

xref (s) =Gm(s)η1ref (s)+Gum(s)η2ref (s)+Hxm(s)C(s)Kg(s)r(s)+xin(s) . (3.87)

Then, for all t ∈ [0,τ ] we have

‖xref t‖L∞ ≤ ‖Gm(s)‖L1

∥∥η1ref t

∥∥L∞ +‖Gum(s)‖L1

∥∥η2ref t

∥∥L∞

+∥∥Hxm(s)C(s)Kg(s)∥∥

L1‖r‖L∞ +ρin .

(3.88)

If the bound (3.85) is not true, since ‖xref (0)‖∞ = ‖x0‖∞ < ρr and xref (t) is continuous,there exists a time τ1 ∈ (0,τ ] such that

‖xref (t)‖∞ < ρr , ∀t ∈ [0,τ1),

‖xref (τ1)‖∞ = ρr ,

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which implies that ∥∥xref τ1

∥∥L∞ = ρr . (3.89)

It follows from the assumption in (3.84) and the bound in (3.89) that∥∥z τ1∥∥L∞ ≤ Lz (ρr +γ1)+Bz ,

and hence, from the definition of δ(δ) in (3.69), we have∥∥Xref τ1

∥∥L∞ =

∥∥∥[ x�ref z�

]�τ1

∥∥∥L∞

≤ ρr (ρr ) = max {ρr + γ1,Lz (ρr + γ1)+Bz} .

Then, it follows from Assumptions 3.2.1 and 3.2.2 that∥∥ηiref τ1

∥∥L∞ ≤ dfxi (ρr (ρr ))

∥∥Xref τ1

∥∥L∞ +Bi0 ≤ dfxi (ρr (ρr ))ρr (ρr )+Bi0 , i = 1,2 ,

and the redefinition in (3.69) leads to the following bounds:∥∥ηiref τ1

∥∥L∞ ≤ Liρr ρr +Bi0 , i = 1,2 . (3.90)

These bounds, together with the upper bound in (3.88), lead to∥∥xref τ1

∥∥L∞ ≤ ‖Gm(s)‖L1

(L1ρr ρr +B10

)+‖Gum(s)‖L1

(L2ρr ρr +B20

)+∥∥Hxm(s)C(s)Kg(s)

∥∥L1

‖r‖L∞ +ρin .

The condition in (3.71) can be solved for ρr to obtain the bound(‖Gm(s)‖L1+‖Gum(s)‖L1

�0)(L1ρr ρr +B0

)+∥∥Hxm(s)C(s)Kg(s)

∥∥L1

‖r‖L∞ +ρin < ρr ,

which leads to ∥∥xref τ1

∥∥L∞ < ρr .

This contradicts the equality in (3.89), thus proving the bound in (3.85). This further impliesthat the upper bounds in (3.90) hold for all t ∈ [0,τ ] with strict inequality, which in turnimplies that ∥∥η1ref τ

∥∥L∞ < L1ρr ρr +B10 ,

∥∥η2ref τ

∥∥L∞ < L2ρr ρr +B20 .

The bound on uref (t) follows from (3.83) and the two bounds above,

‖uref τ‖L∞ <∥∥∥ω−1C(s)

∥∥∥L1

(L1ρr ρr +B10

)+∥∥∥ω−1C(s)H−1

m (s)Hum(s)∥∥∥

L1

(L2ρr ρr +B20

)+∥∥∥ω−1C(s)Kg(s)

∥∥∥L1

‖r‖L∞ ,

which proves (3.86). �

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In this section, we refer to Lemma A.9.1 to demonstrate that the nonlinear system withunmodeled dynamics in (3.68) can be transformed into a (semi-)linear system with unknowntime-varying parameters and time-varying disturbances. According to Lemma A.9.1 for thesystem in (3.68), if u(t) is continuous, and moreover the following bounds hold:

‖x τ‖L∞ ≤ ρ , ‖uτ‖L∞ ≤ ρu ,

then, for all t ∈ [0,τ ], there exist continuous θ1(t) ∈ Rm, σ1(t) ∈ R

m, θ2(t) ∈ Rn−m, and

σ2(t) ∈ Rn−m with (piecewise)-continuous derivative such that

‖θi(t)‖∞ < θbi = θbi (ρr ) , ‖θi(t)‖∞ < dθi = dθi (ρr ) ,

‖σi(t)‖∞ < σbi = σbi (ρr ) , ‖σi(t)‖∞ < dσi = dσi (ρr ) ,

fi(t ,x(t),z(t)) = θi(t)‖xt‖L∞ +σi(t) ,

for i = 1,2, where θbi and σbi were defined in (3.78). Thus the system in (3.68) can berewritten over t ∈ [0,τ ] as

x(t) = Amx(t)+Bm(ωu(t)+ θ1(t)‖x t‖L∞ +σ1(t)

)+Bum

(θ2(t)‖x t‖L∞ +σ2(t)

), x(0) = x0 ,

y(t) = Cx(t) .

(3.91)


Let θm(ρr ) be defined as

θm(ρr ) � 4

(maxω∈� tr(ω�ω)+

(θ2b1

+σ 2b1

)m+

(θ2b2

+σ 2b2

)(n−m)

)

+ 4λmax(P )

λmin(Q)

((θb1dθ1 +σb1dσ1

)m+ (θb2dθ2 +σb2dσ2

)(n−m)

).

(3.92)

Also, let

ω(t) = ω(t)−ω , θi(t) = θi(t)− θi(t) , σi(t) = σi(t)−σi(t) , i = 1,2 .

Using the above notations, the following error dynamics can be derived from (3.79) and (3.91):

˙x(t) = Amx(t)+Bm (ω(t)u(t)+ η1(t))+Bumη2(t) , x(0) = 0, (3.93)

where ηi(t) � ηi(t)−ηi(t), with ηi(t) � θi(t)‖x t‖L∞ +σi(t), i = 1,2.Next, we show that if the adaptive gain � is chosen to verify the lower bound

� >θm(ρr )

λmin(P )γ 20

, (3.94)

and the projection is confined to the bounds

ω(t) ∈� , ‖θi(t)‖∞ ≤ θbi , ‖σi(t)‖∞ ≤ σbi , i = 1,2 , (3.95)

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then the prediction error x(t) between the state of the system and the state predictor can besystematically reduced in both transient and steady-state by increasing the adaptation gain.The following lemma summarizes this result.

Lemma 3.2.2 Let the adaptive gain be lower bounded as in (3.94) and the projection beconfined to the bounds in (3.95). Given the system in (3.68) and the L1 adaptive controllerdefined via (3.79)–(3.81) subject to the L1-norm condition in (3.71), if

‖x τ‖L∞ ≤ ρ , ‖uτ‖L∞ ≤ ρu , (3.96)

then we have

‖ x τ‖L∞ < γ0 ,

where γ0 was introduced in (3.75).

Proof. It follows from the assumption in (3.96) and Lemma A.9.1 that the system in (3.68)can be rewritten as in (3.91) for all t ∈ [0,τ ], with

‖θi(t)‖∞ < θbi (ρr ) , ‖θi(t)‖∞ < dθi (ρr ) , (3.97)

‖σi(t)‖∞ < σbi (ρr ) , ‖σi(t)‖∞ < dσi (ρr ) , ∀ t ∈ [0,τ ] . (3.98)

Consider the following Lyapunov function candidate:

V (x(t), ω(t), θi(t), σi(t)) = x�(t)P x(t)

+ 1

�

(tr(ω�(t)ω(t))+

2∑i=1

(θ�i (t)θi(t)+ σ�

i (t)σi(t)))

.(3.99)

Next we prove that

V (t) ≤ θm(ρr )

�, ∀ t ∈ [0,τ ] .

Toward that end, first notice that

V (0) ≤ 4

�

(maxω∈� tr(ω�ω)+ θ2

b1m+σ 2

b1m+ θ2

b2(n−m)+σ 2

b2(n−m)

)≤ θm(ρr )

�.

Let τ1 ∈ (0,τ ] be the first time instant of discontinuity of either of the derivatives of θi(t)and σi(t). Using the projection-based adaptive laws in (3.80), one has for arbitrary t ∈ [0,τ1)the following upper bound:

V (t) ≤ −x�(t)Qx(t)+ 2

�

∣∣∣∣∣2∑i=1


i (t)σi(t))∣∣∣∣∣ .

Since θi(t) and σi(t) are continuous for all t ∈ [0,τ1), the upper bounds in (3.97) and (3.98)lead to

V (t) ≤ −x�(t)Qx(t)+ 4

�

(θb1dθ1m+σb1dσ1m+ θb2dθ2 (n−m)+σb2dσ2 (n−m)

).

(3.100)

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The projection operator ensures that for all t ∈ [0,τ1)

ω(t) ∈� , ‖θi(t)‖∞ ≤ θbi , ‖σi(t)‖∞ ≤ σbi , i = 1,2 ,

and therefore

maxt∈[0,τ1)

(tr(ω�(t)ω(t))+

2∑i=1


i (t)σi(t)))

≤ 4(

maxω∈� tr(ω�ω)+

(θ2b1

+σ 2b1

)m+

(θ2b2

+σ 2b2

)(n−m)

).

If at arbitrary τ ′ ∈ (0,τ1) we haveV (τ ′)> θm(ρr )�

, then it follows from the Lyapunov functionin (3.99), the definition of θm(ρr ) in (3.92), and the bound in (3.101) that

x�(τ ′)P x(τ ′)> 4

�

λmax(P )

λmin(Q)

(θb1dθ1m+σb1dσ1m+ θb2dθ2 (n−m)+σb2dσ2 (n−m)

),

and hence

x�(τ ′)Qx(τ ′) ≥ λmin(Q)

λmax(P )x�(τ ′)P x(τ ′)

>4

�

((θb1dθ1 +σb1dσ1

)m+ (θb2dθ2 +σb2dσ2

)(n−m)

).

(3.101)

Thus, if V (τ ′)> θm(ρr )�

, then from (3.100) and (3.101) we have

V (τ ′)< 0. (3.102)

It follows from (3.102) that V (t) ≤ θm(ρr )�

for arbitrary t ∈ [0,τ1). Since

λmin(P )‖x(t)‖22 ≤ x�(t)P x(t) ≤ V (t) ,

then for arbitrary t ∈ [0,τ1)

‖x(t)‖2∞ ≤ ‖x(t)‖22 ≤ θm(ρr )

λmin(P )�.

Since V (t) is continuous, we further have

‖x(t)‖∞ ≤√

θm(ρr )

λmin(P )�.

Continuity of θi(t), σi(t), ω(t), θi(t), and σi(t) implies that

V (τ1) ≤ θm(ρr )

�.

Next, let τ2 ∈ (τ1,τ ] be the next time instant such that the discontinuity of any of thederivatives of θi(t) and σi(t) occurs. Using similar derivations as above, we can prove that

‖x(t)‖∞ ≤√

θm(ρr )

λmin(P )�, ∀ t ∈ (τ1,τ2] .

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Iterating this process until the time instant τ , we get

‖ x τ‖L∞ ≤√

θm(ρr )

λmin(P )�,

and the choice for the adaptive gain � in (3.94) leads to

‖ x τ‖L∞ < γ0 ,

which concludes the proof. �

We note that the closed-loop reference system is not implementable since it uses theunknown system input gain matrix ω, the unknown signal z(t), and the unknown functionsf1 and f2. This auxiliary closed-loop system is used only for analysis purposes and is notinvolved in the implementation of the L1 adaptive controller. In the following theorem, weprove the stability and derive the performance bounds of the actual closed-loop adaptivesystem with the L1 adaptive controller with respect to this reference system.

Theorem 3.2.1 Let the adaptive gain be lower bounded as in (3.94) and the projectionbe confined to the bounds in (3.95). Given the closed-loop system with the L1 adaptivecontroller defined via (3.79)–(3.81), subject to the L1-norm condition in (3.71), and theclosed-loop reference system in (3.83), if

‖x0‖∞ ≤ ρ0 ,

then we have

‖x‖L∞ ≤ ρ , (3.103)

‖u‖L∞ ≤ ρu , (3.104)

‖x‖L∞ ≤ γ0 , (3.105)

‖xref −x‖L∞ ≤ γ1 , (3.106)

‖uref −u‖L∞ ≤ γ2 , (3.107)

‖yref −y‖L∞ ≤ ‖C‖∞ γ1 , (3.108)

where γ1 and γ2 were defined in (3.75) and (3.77), respectively.

Proof. Assume that the bounds in (3.106) and (3.107) do not hold. Then, since ‖xref (0)−x(0)‖∞ = 0< γ1, ‖uref (0)−u(0)‖∞ = 0< γ2, and x(t), xref (t), u(t), and uref (t) are con-tinuous, there exists τ such that

‖xref (τ )−x(τ )‖∞ = γ1 or

‖uref (τ )−u(τ )‖∞ = γ2 ,

while

‖xref (t)−x(t)‖∞ < γ1 , ‖uref (t)−u(t)‖∞ < γ2 , ∀ t ∈ [0,τ ) .

This implies that that at least one of the following equalities holds:∥∥ (xref −x) τ∥∥

L∞ = γ1 ,∥∥ (uref −u) τ

∥∥L∞ = γ2 . (3.109)

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‖z τ‖L∞ ≤ Lz(‖xref τ‖L∞ +γ1

)+Bz . (3.110)

Then, Lemma 3.2.1 implies that

‖xref τ‖L∞ ≤ ρr , ‖uref τ‖L∞ ≤ ρur . (3.111)

Using the definitions of ρ and ρu in (3.74) and (3.76), it follows from the bounds in (3.109)and (3.111) that

‖x τ‖L∞ ≤ ρr +γ1 ≤ ρ ,

‖uτ‖L∞ ≤ ρur +γ2 ≤ ρu .

Hence, if one chooses the adaptive gain according to (3.94) and the projection is confinedto the bounds in (3.95), Lemma 3.2.2 implies that

‖ x τ‖L∞ < γ0 . (3.112)

Next, let η(t) � ω(t)u(t) + η1(t) + η2m(t), where η2m(t) is the signal with the Laplacetransform η2m(s) �H−1

m (s)Hum(s)η2(s). It follows from (3.81) that

u(s) = −KD(s)(ωu(s)+η1(s)+H−1

m (s)Hum(s)η2(s)−Kg(s)r(s)+ η(s))

,

where η1(s), η2(s), and η(s) are the Laplace transforms of the signals η1(t), η2(t), and η(t),respectively. Consequently

u(s) = −KD(s) (Im+ωKD(s))−1(η1(s)+ H−1


,

which leads to

ωu(s) = −ωKD(s) (Im+ωKD(s))−1(η1(s)+H−1


.

(3.113)Using the definition of C(s) in (3.70), one can write

ωu(s) = −C(s)(η1(s)+H−1


,


x(s) =Gm(s)η1(s)+Gum(s)η2(s)−Hxm(s)C(s)η(s)

+Hxm(s)C(s)Kg(s)r(s)+xin(s) .(3.114)

Next, from (3.87) and (3.114) we have

xref (s)−x(s) =Gm(s) (η1ref (s)−η1(s))+Gum(s) (η2ref (s)−η2(s))

+Hxm(s)C(s)η(s) .

Moreover, it follows from the error dynamics in (3.93) that

H−1m (s)Cx(s) = ηω(s)+ η1(s)+ η2m(s) = η(s) ,

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with ηω(s) being the Laplace transform of the signal ηω(t) � ω(t)u(t), which leads to


+Hxm(s)C(s)H−1m (s)Cx(s) .

Therefore, we have∥∥ (xref −x) τ∥∥

L∞ ≤ ‖Gm(s)‖L1

∥∥ (η1ref −η1) τ∥∥

L∞ +‖Gum(s)‖L1

∥∥ (η2ref −η2) τ∥∥

L∞

+∥∥∥Hxm(s)C(s)H−1

m (s)C∥∥∥

L1‖ x τ‖L∞ .

(3.115)

Substituting (3.111) into (3.110) one obtains

‖z τ‖L∞ ≤ Lz (ρr +γ1)+Bz ,

and hence, from the definition of δ(δ) in (3.69), we have

‖Xτ‖L∞ ≤ max {ρr +γ1, Lz (ρr +γ1)+Bz} ≤ ρr (ρr ) ,

‖Xref τ‖L∞ ≤ max {ρr , Lz (ρr +γ1)+Bz} ≤ ρr (ρr ) .

Since for all t ∈ [0,τ ], the following equalities hold:

ηiref (t)−ηi(t) = fi(t ,Xref (t))− (θi(t)‖xt‖L∞ +σi(t))= fi(t ,Xref (t))−fi(t ,X(t)) , i = 1,2 ,

Assumption 3.2.2 implies that, for i = 1,2, we have∥∥ (ηiref −ηi) τ∥∥

L∞ ≤ dfxi (ρr (ρr ))∥∥ (Xref −X) τ

∥∥L∞ = dfxi (ρr (ρr ))

∥∥ (xref −x) τ∥∥

L∞ .

(3.116)

Then, from (3.115) we have∥∥ (xref −x) τ∥∥

L∞ ≤ ‖Gm(s)‖L1dfx1 (ρr (ρr ))

∥∥ (xref −x) τ∥∥

L∞+‖Gum(s)‖L1

dfx2 (ρr (ρr ))∥∥ (xref −x) τ

∥∥L∞


m (s)C∥∥∥

L1‖ x τ‖L∞ .

From the redefinition in (3.69) it follows that dfx1 (ρr (ρr ))<L1ρr and dfx2 (ρr (ρr ))<L2ρr ,and, therefore, we obtain∥∥ (xref −x) τ

∥∥L∞ ≤ ‖Gm(s)‖L1

L1ρr

∥∥ (xref −x) τ∥∥

L∞+‖Gum(s)‖L1

L2ρr

∥∥ (xref −x) τ∥∥

L∞


m (s)C∥∥∥

L1‖ x τ‖L∞ .

The upper bound in (3.112) and the L1-norm condition in (3.71) lead to the upper bound

∥∥ (xref −x) τ∥∥

L∞ ≤∥∥Hxm(s)C(s)H−1

m (s)C∥∥

L1


L2ρrγ0 ,

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which along with the definition of γ1 in (3.75) leads to∥∥ (xref −x) τ∥∥

L∞ ≤ γ1 −β < γ1 . (3.117)

On the other hand, it follows from (3.83) and (3.113) that

uref (s)−u(s) = −ω−1C(s) (η1ref (s)−η1(s))

−ω−1C(s)H−1m (s)Hum(s) (η2ref (s)−η2)+ω−1C(s)H−1

m (s)Cx(s) .

One can write∥∥ (uref −u) τ∥∥

L∞ ≤∥∥∥ω−1C(s)

∥∥∥L1

∥∥ (η1ref −η1) τ∥∥

L∞

+∥∥∥ω−1C(s)H−1


L1

∥∥ (η2ref −η2) τ∥∥

L∞

+∥∥∥ω−1C(s)H−1

m (s)C∥∥∥

L1‖ x τ‖L∞ ,

and the bound in (3.116) leads to

∥∥ (uref −u) τ∥∥

L∞ ≤∥∥∥ω−1C(s)

∥∥∥L1L1ρr

∥∥ (xref −x) τ∥∥

L∞

+∥∥∥ω−1C(s)H−1


L1L2ρr

∥∥ (xref −x) τ∥∥

L∞

+∥∥∥ω−1C(s)H−1

m (s)C∥∥∥

L1‖ x τ‖L∞ .

The bounds (3.112) and (3.117) and the definition of γ2 in (3.77) lead to

∥∥ (uref −u) τ∥∥

L∞ ≤(∥∥∥ω−1C(s)

∥∥∥L1L1ρr

+∥∥∥ω−1C(s)H−1


L1L2ρr

)(γ1 −β)

+∥∥∥ω−1C(s)H−1

m (s)C∥∥∥

L1γ0 < γ2 .

(3.118)

Finally, we note that the upper bounds in (3.117) and (3.118) contradict the equalitiesin (3.109), which proves the bounds in (3.106) and (3.107). The results in (3.103)–(3.105)and (3.108) follow directly from the bounds in (3.111) and (3.112) and from the fact thatyref (t)−y(t) = C(xref (t)−x(t)) . �

Remark 3.2.1 Thus, the tracking error between y(t) and yref (t), as well asu(t) anduref (t), isuniformly bounded by a constant inverse proportional to the square root of the adaptive gain.This implies that in both transient and steady-state one can achieve arbitrary close trackingperformance for both signals simultaneously by increasing �. To understand how thesebounds can be used to ensure transient response with desired specifications, we considerthe ideal control signal for the system in (3.68),

uid(s) = −ω−1(η1(s)+H−1

m (s)Hum(s)η2(s)−Kg(s)r(s))

, (3.119)

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which leads to the desired system output response

yid(s) =Hm(s)Kg(s)r(s) (3.120)

by canceling the uncertainties exactly. In the closed-loop reference system in (3.83), uid(t) isfurther low-pass filtered by C(s) to have guaranteed low-frequency range. Thus the closed-loop reference system has a different response as compared to (3.120) achieved with (3.119).Similar to Section 2.1.4, the response of yref (t) can be made as close as possible to (3.120) byreducing (‖Gm(s)‖L1 +‖Gum(s)‖L1�0) arbitrarily. In the absence of unmatched uncertain-ties, we can make ‖Gm(s)‖L1 arbitrarily small by increasing the bandwidth of the low-passfilterC(s). However, for the general case with unmatched uncertainties, the design ofK andD(s) which satisfy (3.71) is an open problem. We note also that the presence of unmatcheduncertainties may limit the choice of the desired state matrix Am.


Consider the system

x(t) = (Am+A�)x(t)+Bmωu(t)+f�(t ,x(t),z(t)) ,

y(t) = Cx(t) , x(0) = x0 ,

where

Am = −1 0 0

0 0 10 −1 −1.8

, Bm =

1 0

0 01 1

,

C =[

1 0 00 1 0

],

while A� ∈ R3×3 and ω ∈ R

2×2 are unknown constant matrices satisfying

‖A�‖∞ ≤ 1, ω ∈[

[0.6,1.2] [−0.2,0.2][−0.2,0.2] [0.6,1.2]

]=� ,

and f� is the (unknown) nonlinear function

f�(t ,x,z) = k1

3 x�x+ tanh( k2

2 x1)x1 +k3zk42 sech(x2)x2 + k5

5 x23 + k6

2 (1− e−λt )+ k72 z

k8x3 cos(ωut)+k9z2

,

with ki ∈ [−1,1], i = 1, . . . ,9, and λ,ωu ∈ R+. The internal unmodeled dynamics are

given by

xz1(t) = xz2(t) ,

xz2(t) = −xz1(t)+0.8(

1−x2z1(t)

)xz2(t) ,

z(t) = 0.1(xz1(t)−xz2(t))+ zu(t) ,

zu(s) = −s+1s2

0.12 + 0.8s0.1 +1

[1 −2 1

]x(s) ,

L1book2010/7/22page 155

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with [xz1(0), xz2(0)] = [xz10 , xz20 ]. The control objective is to design a control u(t) so thatthe output y(t) of the system tracks the output of the desired model M(s) in response tobounded reference inputs r(t) (‖r‖L∞ ≤ 1).

In the implementation of the L1 controller, we set

Q= I3 , � = 80000, K =[

8 00 8

],

D(s) = 1

s( s25 +1)( s70 +1)( s2

402 + 1.8s40 +1)

I2 ,

Kg(s) ≡Kg = −(CA−1m Bm)−1 =

[1 0

−1 1

].

We choose conservatively L1ρ = L2ρ = 40 and B10 = B20 = 5, and thus the projectionbounds can be chosen as

θ1(t) ∈ [−40, 40]1m , σ1(t) ∈ [−5, 5]1m ,

θ2(t) ∈ [−40, 40]1(n−m) , σ2(t) ∈ [−5, 5]1(n−m) ,

ω11(t), ω22(t) ∈ [0.25, 3] , ω12(t), ω21(t) ∈ [−0.2, 0.2] ,

where 1r ∈ Rr represents the vector with all elements 1.

To illustrate the performance of the L1 adaptive controller we consider five differentscenarios:

– Scenario 1:

A� = 0.2 −0.2 −0.3

−0.2 −0.2 0.6−0.1 0 −0.9

, ω =

[0.6 −0.20.2 1.2

],

k1 = −1, k2 = 1, k3 = 0, k4 = 1, k5 = 0,

k6 = 0.2 , λ= 0.3 , k7 = 1, k8 = 0.6 , ωu = 5, k9 = −0.7 .

– Scenario 2:

A� = 0.2 −0.3 0.5

0 0 0−0.1 0.4 0.5

, ω =

[1 00 1

],

k1 = 0, k2 = 0, k3 = 0, k4 = 0, k5 = 0,

k6 = 0, k7 = 0, k8 = 0, k9 = 0.

– Scenario 3:

A� = 0.1 −0.4 0.5

−0.5 0.5 0−0.2 0.3 0.5

, ω =

[0.8 −0.10.1 0.8

],

k1 = 0, k2 = 0, k3 = 0, k4 = 0, k5 = 0,

k6 = 0, k7 = 0, k8 = 0, k9 = 0.

L1book2010/7/22page 156

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– Scenario 4:

A� = 0.1 0.4 0.2

−0.4 −0.2 0.3−0.2 0.6 −0.1

, ω =

[0.6 00.1 1.2

],

k1 = 1, k2 = 1, k3 = 1, k4 = 1, k5 = 1,

k6 = 1, λ= 0.1 , k7 = 1, k8 = 1, ωu = 1, k9 = 1.

– Scenario 5:

A� = 0.2 −0.2 −0.3

0.1 −0.4 0.3−0.1 0 −0.9

, ω =

[0.9 −0.20.1 1.1

],

k1 = 1, k2 = 1, k3 = 1, k4 = 1, k5 = −1,

k6 = −1, λ= 0.3 , k7 = 1, k8 = −1, ωu = 1, k9 = 1.

All the scenarios above consider a significant amount of unmatched uncertainties inthe dynamics of the system, except for Scenario 2, in which the uncertainty (affecting onlythe state matrix of the system) remains matched. Also, in Scenarios 2 and 3 the uncertaintiesaffect only the state matrix and appear therefore in a linear fashion; instead, Scenarios 1, 4,and 5 consider nonlinear uncertain dynamics, both matched and unmatched. Moreover,Scenarios 1, 3, 4, and 5 include uncertainty in the system input gain matrix; in particu-lar, Scenarios 1 and 5 consider significant coupling between the control channels, whileScenario 3 introduces a 20% reduction in the control efficiency in both control channels.

Figures 3.6 and 3.7 show, respectively, the response of the closed-loop system forScenario 1 with the L1 adaptive controller (i) to a series of doublets of different amplitudesin the different channels, and (ii) to the sinusoidal reference signals r(t) = [sin(π3 t), 0.2+0.8cos(π6 t)] and r(t) = [0.5sin(π3 t), 0.1+0.4cos(π6 t)]. One can observe that the L1 adap-tive controller leads to scaled system output for scaled reference signals, similar to linearsystems.

Figure 3.8 presents the closed-loop response to the same doublets as in Figure 3.6 butnow for Scenarios 2, 3, 4, and 5. The L1 controller guarantees smooth and uniform transientperformance in the presence of different nonlinearities affecting both the matched and theunmatched channel. Note that the control signals required to track the reference signals andcompensate for the uncertainties are significantly different for each scenario. Also, notethat, despite the large adaptation rate, the control signals are well in the low-frequencyrange. In order to show the benefits of the compensation for unmatched uncertainties, werepeat the same four scenarios (Scenarios 2–5) without the unmatched component in thecontrol law (term η2m(t) in equation (3.82)). We notice that, in this case, the L1 adaptivecontroller reduces to the MIMO version of the L1 controller introduced in Section 2.5. Theresults are shown in Figure 3.9. Since Scenario 2 only considered matched uncertainties,the performance in this case remains the same. For the other three scenarios, however, theclosed-loop performance degrades significantly, especially for the second output y2(t).

Finally, it is important to emphasize that, for all the simulations provided above(Figures 3.6–3.9), the L1 adaptive controller has not been redesigned or retuned and that asingle set of control parameters has been used for all the scenarios.

L1book2010/7/22page 157

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0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

time [s]

r1(t)y1id(t)y1(t)

(a) y1(t)

0 5 10 15 20 25 30−2

−1

0

1

2

time [s]

u1(t)u2(t)

(b) u(t)

0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

time [s]

r2(t)y2id(t)y2(t)

(c) y2(t)

0 5 10 15 20 25 30−20

−10

0

10

20

time [s]

du1/dt(t)du2/dt(t)

(d) ddt u(t)

Figure 3.6: Scenario 1. Performance of the L1 adaptive controller for doublet commands.

0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

time [s]

r1(t)y1id(t)y1(t)

(a) y1(t)

0 5 10 15 20 25 30−2

−1

0

1

2

time [s]

u1(t)u2(t)

(b) u(t)

0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

time [s]

r2(t)y2id(t)y2(t)

(c) y2(t)

0 5 10 15 20 25 30−10

−5

0

5

10

time [s]

du1/dt(t)du2/dt(t)

(d) ddt u(t)

Figure 3.7: Scenario 1. Performance of the L1 adaptive controller for sinusoidal referencesignals.

L1book2010/7/22page 158

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0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

time [s]

r1(t)y1id(t)y1(t)

(a) y1(t)

0 5 10 15 20 25 30−5

0

5

time [s]

u1(t)u2(t)

(b) u(t)

0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

time [s]

r2(t)y2id(t)y2(t)

(c) y2(t)

0 5 10 15 20 25 30−20

−10

0

10

20

time [s]

du1/dt(t)du2/dt(t)

(d) ddt u(t)

Figure 3.8: Scenarios 2 (solid), 3 (dashed), 4 (dash-dot), and 5 (dotted). Performance of theL1 adaptive controller for doublet commands.

0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

time [s]

r1(t)y1id(t)y1(t)

(a) y1(t)

0 5 10 15 20 25 30

−5

0

5

time [s]

u1(t)u2(t)

(b) u(t)

0 5 10 15 20 25 30−4

−2

0

2

4

time [s]

r2(t)y2id(t)y2(t)

(c) y2(t)

0 5 10 15 20 25 30−20

−10

0

10

20

time [s]

du1/dt(t)du2/dt(t)

(d) ddt u(t)

Figure 3.9: Scenarios 2 (solid), 3 (dashed), 4 (dash-dot), and 5 (dotted). Performance ofthe L1 adaptive controller for doublet commands without compensation for unmatcheduncertainties.

L1book2010/7/22page 159

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3.3. Adaptive Laws for Systems with Unmatched Uncertainties 159

3.3 Piecewise-Constant Adaptive Laws forL1 Adaptive Control in the Presence of UnmatchedNonlinear Uncertainties

This section presents a different estimation scheme for the L1 adaptive control architecturedeveloped in the previous section. We consider again multi-input multi-output uncertainsystems in the presence of uncertain system input gain and time- and state-dependent un-known nonlinearities, without enforcing matching conditions. In particular, the L1 adaptivecontroller developed in this section uses a fast estimation scheme based on a piecewise-constant adaptive law, first introduced in [33], and whose adaptation rate can be directlyassociated with the sampling rate of the available CPU. Similar to the previous section,the class of considered systems includes general unmatched uncertainties. The adaptivealgorithm guarantees semiglobal uniform performance bounds for the system’s signals,both input and output, simultaneously, and thus ensures uniform transient response in ad-dition to steady-state tracking. This extension of the L1 adaptive controller, summarizedin [180], has been successfully applied to the design of Flight Control Systems for NASA’sGTM (AirSTAR) [59] and for the Boeing X-48B [106]. Results from piloted simulationevaluations on NASA’s GTM are included in Section 6.1.2.



x(t) = Amx(t)+Bmωu(t)+f (t ,x(t),z(t)) , x(0) = x0 ,

xz(t) = g (t ,xz(t),x(t)) , xz(0) = xz0 ,

z(t) = go (t ,xz(t)) ,

y(t) = Cx(t) ,

(3.121)

where x(t) ∈ Rn is the system state vector (measured); u(t) ∈ R

m is the control signal(m≤ n); y(t) ∈ R

m is the regulated output;Am is a known Hurwitz n×nmatrix that definesthe desired dynamics for the closed-loop system; Bm ∈ R

n×m is a known full-rank constantmatrix, (Am,Bm) is controllable;C ∈ R

m×n is a known full-rank constant matrix, (Am,C) isobservable;ω ∈ R

m×m is the uncertain system input gain matrix; z(t) ∈ Rp and xz(t) ∈ R

l arethe output and the state vector of internal unmodeled dynamics; and f : R×R

n×Rp → R

n,go : R×R

l → Rp, and g : R×R

l ×Rn → R

l are unknown nonlinear functions satisfyingthe standard assumptions on existence and uniqueness of solutions. The initial conditionx0 is assumed to be inside an arbitrarily large known set, i.e., ‖x0‖∞ ≤ ρ0 <∞ for someρ0 > 0.

Again, we note that the system in (3.121) can also be written in the form

x(t) = Amx(t)+Bm (ωu(t)+f1(t ,x(t),z(t)))+Bumf2(t ,x(t),z(t)), x(0) = x0 ,

xz(t) = g (t ,xz(t),x(t)) , xz(0) = xz0 ,

z(t) = go (t ,xz(t)) ,

y(t) = Cx(t) , (3.122)

where Bum ∈ Rn×(n−m) is a constant matrix such that B�

mBum = 0 and alsorank([Bm, Bum]) = n, while f1 : R × R

n × Rp → R

m and f2 : R × Rn × R

p → R(n−m)

L1book2010/7/22page 160

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are unknown nonlinear functions that verify[f1(t ,x(t),z(t))f2(t ,x(t),z(t))

]= B−1f (t ,x(t),z(t)) , B �

[Bm, Bum

]. (3.123)

In this problem formulation, f1(·) represents the matched component of the uncertainties,whereas Bumf2(·) represents the unmatched component.

Let X � [x�, z�]�, and with a slight abuse of language let fi(t ,X) � fi(t ,x,z),i = 1,2. The system above verifies the following assumptions.

Assumption 3.3.1 (Boundedness of fi(t ,0)) There exists Bi > 0, such that ‖fi(t ,0)‖∞ ≤Bi holds for all t ≥ 0 and for i = 1,2.

Assumption 3.3.2 (Semiglobal Lipschitz condition) For arbitrary δ > 0, there exist pos-itive K1δ , K2δ , such that

‖fi(t ,X1)−fi(t ,X2)‖∞ ≤Kiδ‖X1 −X2‖∞ , i = 1,2 ,

for all ‖Xj‖∞ ≤ δ, j = 1,2, uniformly in t .

Assumption 3.3.3 (Stability of unmodeled dynamics) The xz-dynamics are BIBO stablewith respect to both initial conditions xz0 and input x(t), i.e., there exist Lz, Bz > 0 suchthat for all t ≥ 0

‖zt‖L∞ ≤ Lz‖xt‖L∞ +Bz .

Assumption 3.3.4 (Partial knowledge of the system input gain) The system input gainmatrix ω is assumed to be an unknown (nonsingular) strictly row-diagonally dominantmatrix with sgn(ωii) known. Also, we assume that there exists a known compact convexset �, such that ω ∈�⊂ R

m×m, and that a nominal system input gain ω0 ∈� is known.

Assumption 3.3.5 (Stability of matched transmission zeros) The transmission zeros ofthe transfer matrix Hm(s) = C(sI−Am)−1Bm lie in the open left half plane.

As in the previous section, the control objective is to design an adaptive state feedbackcontroller to ensure that y(t) tracks the output response of a desired systemM(s) defined as

M(s) � C (sI−Am)−1BmKg(s) ,

where Kg(s) is a feedforward prefilter, to a given bounded piecewise-continuous referencesignal r(t) in both transient and steady-state, while all other signals remain bounded.



Let

Hxm(s) � (sIn−Am)−1Bm ,

Hxum(s) � (sIn−Am)−1Bum ,

Hm(s) � CHxm(s) = C (sIn−Am)−1Bm ,

Hum(s) � CHxum(s) = C (sIn−Am)−1Bum ,

L1book2010/7/22page 161

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and also let xin(t) be the signal with Laplace transform xin(s) � (sIn−Am)−1x0, and ρin �‖s(sI−Am)−1‖L1ρ0. SinceAm is Hurwitz and x0 is finite, then ‖xin‖L∞ ≤ ρin. Further, forevery δ > 0, let

Liδ � δ(δ)

δKiδ(δ) , δ(δ) � max{δ+ γ1, Lz(δ+ γ1)+Bz} , (3.124)

where Kiδ was introduced in Assumption 3.3.2, and γ1 is an arbitrarily small positiveconstant.

The design of the L1 adaptive controller involves a feedback gain matrixK ∈ Rm×m

and an m×m strictly proper transfer matrix D(s), which lead, for all ω ∈ �, to a strictlyproper stable

C(s) � ωKD(s) (Im+ωKD(s))−1 (3.125)

with DC gain C(0) = Im. The choice of D(s) needs to ensure also that C(s)H−1m (s) is a

proper stable transfer matrix.For the proofs of stability and performance bounds, the choice of K and D(s) also

needs to ensure that, for a given ρ0, there exists ρr > ρin such that the following L1-normcondition holds:

‖Gm(s)‖L1+‖Gum(s)‖L1

�0 <ρr −

∥∥Hxm(s)C(s)Kg(s)∥∥

L1‖r‖L∞ −ρin

L1ρr ρr +B0, (3.126)

where

Gm(s) �Hxm(s) (Im−C(s)) ,

Gum(s) �(In−Hxm(s)C(s)H−1

m (s)C)Hxum(s) ,

while

�0 � L2ρr

L1ρr, B0 � max

{B10,

B20

�0

},

and Kg(s) is the (BIBO-stable) feedforward prefilter. Further, let ρ be defined as

ρ � ρr + γ1 , (3.127)


γ1 �∥∥Hxm(s)C(s)H−1

m (s)C∥∥

L1


L2ρrγ0 +β , (3.128)

where γ0 and β are arbitrarily small positive constants such that γ1 ≤ γ1. Let

ρu � ρur +γ2 , (3.129)


ρur �∥∥∥ω−1C(s)

∥∥∥L1

(L1ρr ρr +B10

)+∥∥∥ω−1C(s)H−1m (s)Hum(s)

∥∥∥L1

(L2ρr ρr +B20

)+∥∥∥ω−1C(s)Kg(s)

∥∥∥L1

‖r‖L∞ ,

γ2 �(∥∥∥ω−1C(s)

∥∥∥L1L1ρr +

∥∥∥ω−1C(s)H−1m (s)Hum(s)

∥∥∥L1L2ρr

)γ1

+∥∥∥ω−1C(s)H−1

m (s)C∥∥∥

L1γ0 . (3.130)

L1book2010/7/22page 162

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Further, let Ts > 0 be the adaptation sampling time, which can be associated with thesampling rate of the available CPU, and let ς (Ts) be

ς (Ts) � κ1(Ts)�1 +κ2(Ts)�2 ,

where κ1(Ts) and κ2(Ts) are defined as

κ1(Ts) �∫ Ts

0

∥∥∥eAm(Ts−τ )Bm

∥∥∥2dτ , (3.131)

κ2(Ts) �∫ Ts

0

∥∥∥eAm(Ts−τ )Bum

∥∥∥2dτ , (3.132)

while �1 and �2 are given by

�1 � (maxω∈� {‖ω−ω0‖2}ρu+L1ρρ+B10)

√m , (3.133)

�2 �(L2ρρ+B20

)√n−m ,

with ω0 being the best available guess of ω introduced in Assumption 3.3.4. Also, let α1(t),α2(t), α3(t), and α4(t) be defined as

α1(t) �∥∥∥eAmt

∥∥∥2

,

α2(t) �∫ t

0

∥∥∥eAm(t−τ )�−1(Ts)eAmTs

∥∥∥2dτ ,

α3(t) �∫ t

0

∥∥∥eAm(t−τ )Bm

∥∥∥2dτ ,

α4(t) �∫ t

0

∥∥∥eAm(t−τ )Bum

∥∥∥2dτ , (3.134)

where �(Ts) is an n×n matrix defined as

�(Ts) � A−1m

(eAmTs − In

). (3.135)

Let

α1(Ts) � maxt∈[0, Ts ]

α1(t) , α2(Ts) � maxt∈[0,Ts ]

α2(t) , (3.136)

α3(Ts) � maxt∈[0, Ts ]

α3(t) , α4(Ts) � maxt∈[0, Ts ]

α4(t) . (3.137)

Finally, let

γ0(Ts) � (α1(Ts)+ α2(Ts))ς (Ts)+ α3(Ts)�1 + α4(Ts)�2 . (3.138)

Lemma 3.3.1 The following limiting relationship is true:

limTs→0

γ0(Ts) = 0.

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Proof. We note that since α1(Ts), α2(Ts), �1, and �2 are bounded, it is enough to provethat

limTs→0

ς (Ts) = 0, (3.139)

limTs→0

α3(Ts) = 0, limTs→0

α4(Ts) = 0. (3.140)

From the definition of κ1(Ts) and κ2(Ts) in (3.131)–(3.132) we have

limTs→0

κ1(Ts) = 0, limTs→0

κ2(Ts) = 0,

and, then, since �1 and �2 are bounded, we have

limTs→0

ς (Ts) = 0,

which proves (3.139). Since α3(t) and α4(t) are continuous, it follows from (3.137) that

limTs→0

α3(Ts) = limt→0

α3(t) = 0, limTs→0

α4(Ts) = limt→0

α4(t) = 0,

which prove (3.140). Boundedness of α1(Ts), α2(Ts), �1, and �2 implies that

limTs→0

((α1(Ts)+ α2(Ts))ς (Ts)+ α3(Ts)�1 + α4(Ts)�2) = 0,


The L1 adaptive control architecture is introduced next.

State Predictor


˙x(t) = Amx(t)+Bm (ω0u(t)+ σ1(t))+Bumσ2(t) , x(0) = x0 ,

y(t) = Cx(t) ,(3.141)

where σ1(t) ∈ Rm and σ2(t) ∈ R

n−m are the adaptive estimates.

Adaptation Laws

The adaptation laws for σ1(t) and σ2(t) are defined as[σ1(t)σ2(t)

]=[σ1(iTs)σ2(iTs)

], t ∈ [iTs , (i+1)Ts) ,[

σ1(iTs)σ2(iTs)

]= −

[Im 00 In−m

]B−1�−1(Ts)µ(iTs) ,

(3.142)

for i = 0,1,2, . . . , where B and�(Ts) were introduced in (3.123) and (3.135), respectively,while

µ(iTs) = eAmTs x(iTs) , x(t) = x(t)−x(t) .

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Control Law

The control signal is generated as the output of the (feedback) system

u(s) = −KD(s)η(s) , (3.143)

where η(s) is the Laplace transform of the signal

η(t) � ω0u(t)+ η1(t)+ η2m(t)− rg(t) , (3.144)

with rg(s) �Kg(s)r(s), η2m(s) �H−1m (s)Hum(s)η2(s), and with η1(t) � σ1(t) and η2(t) �

σ2(t).As before, we repeat that conventional design methods from multivariable control

theory can be used to design the prefilter Kg(s) to achieve desired decoupling properties.As an example, if one choosesKg(s) as the constant matrixKg = −(CA−1

m Bm)−1, then thediagonal elements of the desired transfer matrix M(s) = C(sIn−Am)−1BmKg have DCgain equal to one, while the off-diagonal elements have zero DC gain.

The L1 adaptive controller consists of (3.141)–(3.143), subject to the L1-norm con-dition in (3.126).



We consider the same closed-loop reference system as in Section 3.2:

xref (t) = Amxref (t)+Bm (ωuref (t)+f1(t ,xref (t),z(t)))

+Bumf2(t ,xref (t),z(t)) , xref (0) = x0 ,

uref (s) = −ω−1C(s)(η1ref (s)+H−1

m (s)Hum(s)η2ref (s)−Kg(s)r(s))

,

yref (t) = Cxref (t) ,

(3.145)

where ηiref (s) is the Laplace transform of ηiref (t) � fi(t ,xref (t),z(t)), i = 1, 2.

Lemma 3.3.2 For the closed-loop reference system in (3.145), subject to the L1-normcondition (3.126), if ‖x0‖∞ ≤ ρ0 and

‖ z t ‖L∞ ≤ Lz(‖ xref t ‖L∞ +γ1)+Bz ,

then

‖ xref t ‖L∞ < ρr ,

‖ uref t ‖L∞ < ρur .

Proof. The proof of this lemma is similar to the proof of Lemma 3.2.1 in Section 3.2 andis therefore omitted. �

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The error dynamics can be derived from (3.122) and (3.141),

˙x(t) = Amx(t)+Bmη1(t)+Bumη2(t) , x(0) = 0, (3.146)

where

η1(t) � σ1(t)− ((ω−ω0)u(t)+η1(t)) , (3.147)

η2(t) � σ2(t)−η2(t) , (3.148)

with ηi(t) = fi(t ,x(t),z(t)), i = 1,2.Next we show that if Ts is chosen to ensure that

γ0(Ts)< γ0 , (3.149)

then the tracking error between the state of the system and the state predictor can be sys-tematically reduced in both transient and steady-state by reducing Ts .

Lemma 3.3.3 Let the adaptation rate be chosen to satisfy the design constraint in (3.149).Given the system in (3.122) and the L1 adaptive controller defined via (3.141)–(3.143),subject to the L1-norm condition in (3.126), if

‖x τ‖L∞ ≤ ρ , ‖uτ‖L∞ ≤ ρu , (3.150)

we have

‖ x τ‖L∞ < γ0 ,

where γ0 is as introduced in (3.128).

Proof. If the bounds in (3.150) hold, then it follows fromAssumption 3.3.3 that ‖ z τ ‖L∞ ≤Lzρ+Bz, which leads to

‖Xτ‖L∞ ≤ ρ(ρ) .

From Assumptions 3.3.2 and 3.3.1, and the redefinition in (3.124), one finds∥∥η1 τ

∥∥L∞ ≤ L1ρρ+B10 .

This implies that

‖η1(t)‖2 ≤ (L1ρρ+B10)√m , ∀t ∈ [0,τ ] . (3.151)

Similarly, one can show that

‖η2(t)‖2 ≤ (L2ρρ+B20)√n−m , ∀t ∈ [0,τ ] . (3.152)

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It follows from the error dynamics in (3.146) that

x(iTs + t) = eAmt x(iTs)+∫ iTs+t

iTs

eAm(iTs+t−ξ )Bmσ1(iTs)dξ

+∫ iTs+t

iTs

eAm(iTs+t−ξ )Bumσ2(iTs)dξ

−∫ iTs+t

iTs

eAm(iTs+t−ξ )Bm ((ω−ω0)u(ξ )+η1(ξ ))dξ

−∫ iTs+t

iTs

eAm(iTs+t−ξ )Bumη2(τ )dξ ,


x(iTs + t) = eAmt x(iTs)+∫ t

0eAm(t−ξ )B

[σ1(iTs)σ2(iTs)

]dξ

−∫ t

0eAm(t−ξ )Bm ((ω−ω0)u(iTs + ξ )+η1(iTs + ξ ))dξ

−∫ t

0eAm(t−ξ )Bumη2(iTs + ξ )dξ .

Define the signals ζ1(iTs + t) and ζ2(iTs + t) as

ζ1(iTs + t) � eAmt x(iTs)+∫ t

0eAm(t−ξ )B

[σ1(iTs)σ2(iTs)

]dξ ,

ζ2(iTs + t) � −∫ t

0eAm(t−ξ )Bumη2(iTs + ξ )dξ

−∫ t

0eAm(t−ξ )Bm ((ω−ω0)u(iTs + ξ )+η1(iTs + ξ ))dξ .

Next, we prove that

‖x(iTs)‖2 ≤ ς (Ts) , ∀ iTs ≤ τ . (3.153)

Because x(0) = 0, it follows that ‖x(0)‖2 <ς (Ts). Consider now the time interval [jTs , (j+1)Ts), with (j +1)Ts < τ . The prediction error at the sampling instant (j +1)Ts is given by

x((j +1)Ts) = ζ1((j +1)Ts)+ ζ2((j +1)Ts) ,

with

ζ1((j +1)Ts) = eAmTs x(jTs)+∫ Ts

0eAm(Ts−ξ )B

[σ1(jTs)σ2(iTs)

]dξ , (3.154)

ζ2((j +1)Ts) = −∫ Ts

0eAm(Ts−ξ )Bumη2(jTs + ξ )dξ

−∫ Ts

0eAm(Ts−ξ )Bm ((ω−ω0)u(jTs + ξ )+η1(jTs + ξ ))dξ .

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Substituting the adaptive law (3.142) into (3.154) leads to

ζ1((j +1)Ts) = 0.

Then, it follows that

x((j +1)Ts) = ζ2((j +1)Ts) ,

and the bounds in (3.151) and (3.152) together with the definitions of κ1(Ts), κ2(Ts), �1,and �2, imply that

‖x((j +1)Ts)‖2 ≤ κ1(Ts)�1 +κ2(Ts)�2 = ς (Ts) .

This confirms the upper bound for arbitrary (j + 1)Ts ≤ τ , and hence, the upper boundin (3.153) holds for all iT ≤ τ .

For all iT + t ≤ τ , with t ∈ (0, Ts], we can write

x(iTs + t) = eAmt x(iTs)+∫ t

0eAm(t−ξ )B

[σ1(iTs)σ2(iTs)

]dξ

−∫ t

0eAm(t−ξ )Bm ((ω−ω0)u(iTs + ξ )+η1(iTs + ξ ))dξ

−∫ t

0eAm(t−ξ )Bumη2(iTs + ξ )dξ .

The bounds in (3.151) and (3.152) and the definitions of �1, �2, α1(·), α2(·), α3(·), andα4(·) in (3.133)–(3.134) imply that

‖x(iTs + t)‖2 ≤ α1(t)‖x(iTs)‖2 +α2(t)‖x(iTs)‖2 +α3(t)�1 +α4(t)�2 .

Next, for all iTs + t ≤ τ , the upper bound in (3.153) and the definitions of α1(Ts), α4(Ts),α3(Ts), α4(Ts) in (3.136)–(3.137) lead to

‖x(iTs + t)‖2 ≤ (α1(Ts)+ α2(Ts))ς (Ts)+ α3(Ts)�1 +α4(Ts)�2 .

Further, because the right-hand side coincides with the definition of γ0(Ts) in (3.138), wehave ‖x(t)‖2 ≤ γ0(Ts), which holds for all t ∈ [0,τ ], which, along with the design constraintin (3.149), yields

‖x(t)‖2 < γ0 , ∀ t ∈ [0,τ ] ,

and consequently implies that

‖ x τ‖L∞ < γ0 .

The proof is complete. �

Similar to the previous section, in the following theorem we prove the stability andderive the performance bounds of the adaptive closed-loop system with the L1 adaptivecontroller. Although the proof of this result is similar to the proof of Theorem 3.2.1 inSection 3.2, we present it for the sake of completeness.

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Theorem 3.3.1 Let the adaptation rate be chosen to satisfy (3.149). Given the closed-loopsystem with the L1 controller defined via (3.141)–(3.143), subject to the L1-norm conditionin (3.126), and the closed-loop reference system in (3.145), if

‖x0‖∞ ≤ ρ0 ,

then we have

‖x‖L∞ ≤ ρ , (3.155)

‖u‖L∞ ≤ ρu , (3.156)

‖x‖L∞ ≤ γ0 , (3.157)

‖xref −x‖L∞ ≤ γ1 , (3.158)

‖uref −u‖L∞ ≤ γ2 , (3.159)

‖yref −y‖L∞ ≤ ‖C‖∞ γ1 . (3.160)

Proof. Assume that the bounds in (3.158) and (3.159) do not hold. Then, since‖xref (0)−x(0)‖∞ = 0< γ1, ‖uref (0)−u(0)‖∞ = 0< γ2, and x(t), xref (t), u(t), and uref (t)are continuous, there exists τ such that

‖xref (τ )−x(τ )‖∞ = γ1 or

‖uref (τ )−u(τ )‖∞ = γ2 ,

while

‖xref (t)−x(t)‖∞ < γ1 , ‖uref (t)−u(t)‖∞ < γ2 , ∀ t ∈ [0,τ ) .

This implies that at least one of the following equalities holds:∥∥ (xref −x) τ∥∥

L∞ = γ1 ,∥∥ (uref −u) τ

∥∥L∞ = γ2 . (3.161)


‖z τ‖L∞ ≤ Lz(‖xref τ‖L∞ +γ1

)+Bz . (3.162)

Then, Lemma 3.3.2 implies that

‖xref τ‖L∞ ≤ ρr , ‖uref τ‖L∞ ≤ ρur . (3.163)

Using the definitions ofρ andρu in (3.127) and (3.129), it follows from the bounds in (3.161)and (3.163) that

‖x τ‖L∞ ≤ ρr +γ1 ≤ ρ , (3.164)

‖uτ‖L∞ ≤ ρur +γ2 ≤ ρu . (3.165)

Hence, if one chooses the adaptation sampling time Ts according to (3.149), Lemma 3.3.3implies that

‖ x τ‖L∞ < γ0 . (3.166)

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Next, let η(t) � η1(t) + η2m(t), with η2m(t) being the signal with its Laplace transformη2m(s) �H−1

m (s)Hum(s)η2(s), where η1(t) and η2(t) are as introduced in (3.147) and (3.148).It follows from (3.143) that

u(s) = −KD(s)(ωu(s)+η1(s)+H−1


,

where η1(s), η2(s), and η(s) are the Laplace transforms of the signals η1(t), η2(t), and η(t),respectively. Consequently

u(s) = −KD(s) (Im+ωKD(s))−1(η1(s)+H−1


,

which leads to

ωu(s) = −ωKD(s) (Im+ωKD(s))−1(η1(s)+H−1


.

(3.167)Using the definition of C(s) in (3.125), one can write

ωu(s) = −C(s)(η1(s)+H−1


,


x(s) =Gm(s)η1(s)+Gum(s)η2(s)−Hxm(s)C(s)η(s)

+Hxm(s)C(s)Kg(s)r(s)+xin(s) .(3.168)

Next, from the definition of the closed-loop reference system in (3.145) and (3.168)we have


+Hxm(s)C(s)η(s) .

Moreover, it follows from the error dynamics in (3.146) that

H−1m (s)Cx(s) = η1(s)+ η2m(s) = η(s) ,

which leads to

xref (s)−x(s) =Gm(s) (η1ref (s)−η1(s))

+Gum(s) (η2ref (s)−η2(s))+Hxm(s)C(s)H−1m (s)Cx(s) .

Therefore, we have∥∥ (xref −x) τ∥∥

L∞ ≤ ‖Gm(s)‖L1

∥∥ (η1ref −η1) τ∥∥

L∞ +‖Gum(s)‖L1

∥∥ (η2ref −η2) τ∥∥

L∞


m (s)C∥∥∥

L1‖ x τ‖L∞ .

(3.169)

Substituting (3.163) in (3.162) one obtains

‖z τ‖L∞ ≤ Lz (ρr +γ1)+Bz ,

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and hence, from the definition of δ(δ) in (3.124), we have

‖Xτ‖L∞ ≤ max {ρr +γ1, Lz (ρr +γ1)+Bz} ≤ ρr (ρr ) ,

‖Xref τ‖L∞ ≤ max {ρr , Lz (ρr +γ1)+Bz} ≤ ρr (ρr ) .

Assumption 3.3.2 implies that, for i = 1,2, we have∥∥ (ηiref −ηi) τ∥∥

L∞ ≤Kiρr (ρr )∥∥ (Xref −X) τ

∥∥L∞ =Kiρr (ρr )

∥∥ (xref −x) τ∥∥

L∞ . (3.170)

Then, from (3.169) we have∥∥ (xref −x) τ∥∥

L∞ ≤ ‖Gm(s)‖L1K1ρr (ρr )

∥∥ (xref −x) τ∥∥

L∞+‖Gum(s)‖L1

K2ρr (ρr )∥∥ (xref −x) τ

∥∥L∞


m (s)C∥∥∥

L1‖ x τ‖L∞ .

From the redefinition in (3.124), it follows that K1ρr (ρr ) < L1ρr and K2ρr (ρr ) < L2ρr , andtherefore, we obtain∥∥ (xref −x) τ

∥∥L∞ ≤ ‖Gm(s)‖L1

L1ρr

∥∥ (xref −x) τ∥∥

L∞+‖Gum(s)‖L1

L2ρr

∥∥ (xref −x) τ∥∥

L∞


m (s)C∥∥∥

L1‖ x τ‖L∞ .

The upper bound in (3.166) and the L1-norm condition in (3.126) lead to the upper bound

∥∥ (xref −x) τ∥∥

L∞ ≤∥∥Hxm(s)C(s)H−1

m (s)C∥∥

L1


L2ρrγ0 ,

which along with the definition of γ1 in (3.128) leads to∥∥ (xref −x) τ∥∥

L∞ ≤ γ1 −β < γ1 . (3.171)

On the other hand, it follows from (3.145) and (3.167) that

uref (s)−u(s) = −ω−1C(s) (η1ref (s)−η1(s))

−ω−1C(s)H−1m (s)Hum(s) (η2ref (s)−η2)+ω−1C(s)H−1

m (s)Cx(s) .

One can write∥∥ (uref −u) τ∥∥

L∞ ≤∥∥∥ω−1C(s)

∥∥∥L1

∥∥ (η1ref −η1) τ∥∥

L∞

+∥∥∥ω−1C(s)H−1


L1

∥∥ (η2ref −η2) τ∥∥

L∞

+∥∥∥ω−1C(s)H−1

m (s)C∥∥∥

L1‖ x τ‖L∞ ,

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and the bound in (3.170) leads to∥∥ (uref −u) τ∥∥

L∞ ≤∥∥∥ω−1C(s)

∥∥∥L1L1ρr

∥∥ (xref −x) τ∥∥

L∞

+∥∥∥ω−1C(s)H−1


L1L2ρr

∥∥ (xref −x) τ∥∥

L∞

+∥∥∥ω−1C(s)H−1

m (s)C∥∥∥

L1‖ x τ‖L∞ .

The bounds (3.166) and (3.171) and the definition of γ2 in (3.130) lead to∥∥ (uref −u) τ∥∥

L∞ ≤(∥∥∥ω−1C(s)

∥∥∥L1L1ρr

+∥∥∥ω−1C(s)H−1


L1L2ρr

)(γ1 −β)

+∥∥∥ω−1C(s)H−1

m (s)C∥∥∥

L1γ0 < γ2 .

(3.172)

Finally, we note that the upper bounds in (3.171) and (3.172) contradict the equalitiesin (3.161), which proves the bounds in (3.158) and (3.159). The results in (3.155)–(3.157)and (3.160) follow directly from the bounds in (3.164)–(3.166) and from the fact thaty(t)−yref (t) = C(x(t)−xref (t)) . �

Remark 3.3.1 Thus, the tracking error between y(t) and yref (t), as well as u(t) and uref (t),is uniformly bounded by an arbitrary small constant, which implies that in both transientand steady-state one can achieve arbitrarily close tracking performance for both signalssimultaneously by reducing Ts . To understand how these bounds can be used for ensuringtransient response with desired specifications, we consider the ideal control signal for thesystem in (3.122),

uid(s) = −ω−1(η1(s)+H−1

m (s)Hum(s)η2(s)−Kg(s)r(s))

, (3.173)

which leads to the desired system output response

yid(s) =Hm(s)Kg(s)r(s) , (3.174)

by canceling the uncertainties exactly. In the closed-loop reference system in (3.145), uid(t)is further low-pass filtered byC(s) to have guaranteed low-frequency range. Thus the closed-loop reference system has a different response as compared to (3.174) achieved with (3.173).Similar to Section 3.2, the response of yref (t) can be made as close as possible to (3.174) byreducing (‖Gm(s)‖L1 +‖Gum(s)‖L1�0) arbitrarily. In the absence of unmatched uncertain-ties, we can make ‖Gm(s)‖L1 arbitrarily small by increasing the bandwidth of the low-passfilterC(s). However, for the general case with unmatched uncertainties, the design ofK andD(s) that satisfy (3.126) is an open problem. We note also that the presence of unmatcheduncertainties may limit the choice of the desired state matrix Am.

Remark 3.3.2 It is important to notice that the performance bounds given by (3.158)–(3.160) are exactly the same as the ones in (3.106)–(3.108), analyzed in Section 3.2 usingprojection-based adaptive laws. These results make clear that the key elements for the deriva-tion of the performance bounds of the L1 adaptive controller are (i) an appropriate filtering

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structure in the control channel and (ii) the use of a predictor-based fast estimation scheme.Therefore, one would like to conjecture that other estimation algorithms from the literaturecan also be successfully employed in the design of L1 adaptive control architectures withsimilar performance bounds.

Remark 3.3.3 Similar to Section 2.1.6, the state predictor in (3.141) can be modified asfollows:

˙x(t) = Amx(t)+Bm (ω0u(t)+ σ1(t))+Bumσ2(t)−Kspx(t) , x(0) = x0 ,

y(t) = Cx(t) ,

where the constant matrix Ksp ∈ Rn×n is used to assign faster poles for the prediction

error dynamics, defined via As � Am−Ksp. This modification of the state predictor addsdamping to the adaptation loop inside the L1 adaptive controller and can be used to tune thefrequency response and robustness margins of the closed-loop adaptive system. In general,this modification may require an increase of the adaptation sampling rate. The proofs ofstability and performance bounds for the L1 adaptive controller presented in this sectionwith the modified state predictor can be found in [180].


In this section, we use the example in Section 3.2.4 in order to demonstrate that a similarlevel of performance can be achieved with this new adaptive law (a different fast estimationscheme). For the sake of completeness we repeat the system dynamics and the simulationparameters. Thus, consider again the system

x(t) = (Am+A�)x(t)+Bmωu(t)+f�(t ,x(t),z(t)) , x(0) = x0 ,

y(t) = Cx(t) ,

where

Am = −1 0 0

0 0 10 −1 −1.8

,Bm =

1 0

0 01 1

,

C =[

1 0 00 1 0

],

while A� ∈ R3×3 and ω ∈ R

2×2 are unknown constant matrices satisfying

‖A�‖∞ ≤ 1, ω ∈[

[0.6, 1.2] [−0.2, 0.2][−0.2, 0.2] [0.6, 1.2]

]=� ,

and f� is the (unknown) nonlinear function

f�(t ,x,z) =

k13 x

�x+ tanh( k22 x1)x1 +k3z

k42 sech(x2)x2 + k5

5 x23 + k6

2 (1− e−λt )+ k72 z

k8x3 cos(ωut)+k9z2

,

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for ki ∈ [−1, 1], i = 1, . . . ,9, and λ,ωu ∈ R+. The internal unmodeled dynamics are given by

xz1(t) = xz2(t) ,

xz2(t) = −xz1(t)+0.8(

1−x2z1(t)

)xz2(t) ,

z(t) = 0.1(xz1(t)−xz2(t))+ zu(t)

zu(s) = −s+1s2

0.12 + 0.8s0.1 +1

[1 −2 1

]x(s) ,

with [xz1(0), xz2(0)] = [xz10 , xz20 ]. The control objective is to design a control u(t) sothat the output y(t) of the system tracks the output of the desired model M(s) to boundedreference inputs r(t) (‖r‖L∞ ≤ 1).

In the implementation of the L1 controller, we set

Ts = 1

100s , ω0 = I2 , K =

[8 00 8

],

D(s) = 1

s( s25 +1)( s70 +1)( s2

402 + 1.8s40 +1)

I2 ,

Kg(s) ≡Kg = −(CA−1m Bm)−1 =

[1 0

−1 1

].

To illustrate the performance of the L1 adaptive controller we consider the same fivescenarios that we used in Section 3.2.4:

– Scenario 1:

A� = 0.2 −0.2 −0.3

−0.2 −0.2 0.6−0.1 0 −0.9

, ω =

[0.6 −0.20.2 1.2

],

k1 = −1, k2 = 1, k3 = 0, k4 = 1, k5 = 0,

k6 = 0.2 , λ= 0.3 , k7 = 1, k8 = 0.6 , ωu = 5, k9 = −0.7 .

– Scenario 2:

A� = 0.2 −0.3 0.5

0 0 0−0.1 0.4 0.5

, ω =

[1 00 1

],

k1 = 0, k2 = 0, k3 = 0, k4 = 0, k5 = 0,

k6 = 0, k7 = 0, k8 = 0, k9 = 0.

– Scenario 3:

A� = 0.1 −0.4 0.5

−0.5 0.5 0−0.2 0.3 0.5

, ω =

[0.8 −0.10.1 0.8

],

k1 = 0, k2 = 0, k3 = 0, k4 = 0, k5 = 0,

k6 = 0, k7 = 0, k8 = 0, k9 = 0.

L1book2010/7/22page 174

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– Scenario 4:

A� = 0.1 0.4 0.2

−0.4 −0.2 0.3−0.2 0.6 −0.1

, ω =

[0.6 00.1 1.2

],

k1 = 1, k2 = 1, k3 = 1, k4 = 1, k5 = 1,

k6 = 1, λ= 0.1 , k7 = 1, k8 = 1, ωu = 1, k9 = 1.

– Scenario 5:

A� = 0.2 −0.2 −0.3

0.1 −0.4 0.3−0.1 0 −0.9

, ω =

[0.9 −0.20.1 1.1

],

k1 = 1, k2 = 1, k3 = 1, k4 = 1, k5 = −1,

k6 = −1, λ= 0.3 , k7 = 1, k8 = −1, ωu = 1, k9 = 1.

A detailed explanation of these scenarios was presented in Section 3.2.4. For the sakeof completeness, we repeat the same explanation here. All the scenarios above consider asignificant amount of unmatched uncertainties in the dynamics of the system, except forScenario 2, in which the uncertainty (affecting only the state matrix of the system) remainsmatched. Also, in Scenarios 2 and 3 the uncertainties affect only the state matrix and appeartherefore in a linear fashion; instead, Scenarios 1, 4, and 5 consider nonlinear uncertaindynamics, both matched and unmatched. Moreover, Scenarios 1, 3, 4, and 5 include uncer-tainty in the system input gain matrix; in particular, Scenarios 1 and 5 consider significantcoupling between the control channels, while Scenario 3 introduces a 20% reduction in thecontrol efficiency in both control channels.

Figures 3.10 and 3.11 show, respectively, the response of the closed-loop systemfor Scenario 1 with the L1 adaptive controller (i) to a series of doublets of differentamplitudes in the different channels and (ii) to the sinusoidal reference signals r(t) =[sin(π6 t), 0.2 + 0.8cos(π3 t)] and r(t) = [0.5sin(π6 t), 0.1 + 0.4cos(π3 t)]. One can observethat the L1 adaptive controller leads to scaled system output for scaled reference signals,similar to linear systems. Moreover, although the reference signals are different from theones in Section 3.2.4, we can see that the L1 architecture with the new fast estimationscheme achieves a similar level of performance as the L1 controller with projection-basedadaptation laws.

Next, we demonstrate that the L1 adaptive controller is able to guarantee a similartransient response of the closed-loop system for different (admissible) uncertainties in theplant. Figure 3.12 presents the closed-loop response to the same doublets as in Figure 3.10,but now for Scenarios 2, 3, 4, and 5. The L1 controller guarantees smooth and uniformtransient performance in the presence of different nonlinearities affecting both the matchedand the unmatched channel. Note that the control signals required to track the referencesignals and compensate for the uncertainties are significantly different for each scenario.Also note that despite the high adaptation sampling rate, the control signals are well withinthe low-frequency range. In order to show the benefits of the compensation for unmatcheduncertainties, we repeat the same four scenarios (Scenarios 2–5) without the unmatchedcomponent in the control law (term η2m(t) in equation (3.144)). The results are shown inFigure 3.13. Since Scenario 2 considered only matched uncertainties, the performance in this

L1book2010/7/22page 175

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0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

time [s]

r1(t)y1id(t)y1(t)

(a) y1(t)

0 5 10 15 20 25 30−2

−1

0

1

2

time [s]

u1(t)u2(t)

(b) u(t)

0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

time [s]

r2(t)y2id(t)y2(t)

(c) y2(t)

0 5 10 15 20 25 30−20

−10

0

10

20

time [s]

du1/dt(t)du2/dt(t)

(d) ddt u(t)

Figure 3.10: Scenario 1. Performance of the L1 adaptive controller for doublet commands.

0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

time [s]

r1(t)y1id(t)y1(t)

(a) y1(t)

0 5 10 15 20 25 30−2

−1

0

1

2

time [s]

u1(t)u2(t)

(b) u(t)

0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

time [s]

r2(t)y2id(t)y2(t)

(c) y2(t)

0 5 10 15 20 25 30−10

−5

0

5

10

time [s]

du1/dt(t)du2/dt(t)

(d) ddt u(t)

Figure 3.11: Scenario 1. Performance of the L1 adaptive controller for sinusoidal referencesignals.

L1book2010/7/22page 176

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0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

time [s]

r1(t)y1id(t)y1(t)

(a) y1(t)

0 5 10 15 20 25 30−5

0

5

time [s]

u1(t)u2(t)

(b) u(t)

0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

time [s]

r2(t)y2id(t)y2(t)

(c) y2(t)

0 5 10 15 20 25 30−20

−10

0

10

20

time [s]

du1/dt(t)du2/dt(t)

(d) ddt u(t)

Figure 3.12: Scenarios 2 (solid), 3 (dashed), 4 (dash-dot), and 5 (dotted). Performance ofthe L1 adaptive controller for doublet commands.

0 5 10 15 20 25 30−3

−2

−1

0

1

2

3

time [s]

r1(t)y1id(t)y1(t)

(a) y1(t)

0 5 10 15 20 25 30−10

−5

0

5

10

time [s]

u1(t)u2(t)

(b) u(t)

0 5 10 15 20 25 30−3

−2

−1

0

1

2

3

time [s]

r2(t)y2id(t)y2(t)

(c) y2(t)

0 5 10 15 20 25 30−20

−10

0

10

20

time [s]

du1/dt(t)du2/dt(t)

(d) ddt u(t)

Figure 3.13: Scenarios 2 (solid), 3 (dashed), 4 (dash-dot), and 5 (dotted). Performance ofthe L1 adaptive controller for doublet commands without compensation for unmatcheduncertainties.

L1book2010/7/22page 177

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case remains the same. For the other three scenarios, however, the closed-loop performancedegrades significantly, especially for the second output y2(t). In particular, one can observethat for Scenario 3, the system becomes unstable.

Finally, it is important to emphasize that, for all the simulations provided above(Figures 3.10–3.13), the L1 adaptive controller has not been redesigned or retuned, andthat a single set of control parameters has been used for all the scenarios. Moreover, theresults in this section and in Section 3.2.4 illustrate that the design of the control law inL1 adaptive architectures (that is, the design of k, D(s), and Kg(s)) is independent of theestimation scheme used, as long as the latter is estimating fast.

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Chapter 4

Output Feedback

This chapter extends the results to output feedback. We limit the discussion to single-inputsingle-output (SISO) systems in the presence of unknown time-varying nonlinearities. Westate the problem formulation in the frequency domain and consider two cases of perfor-mance specifications. The first case corresponds to strictly positive real (SPR) referencesystems, and for simplicity we reduce the design and analysis here to a first-order referencesystem. In the second section, we present the solution for an arbitrary strictly proper, mini-mum phase and stable reference system. In this case, we use the piecewise-constant adaptivelaw to establish the desired performance bounds. The major difference of the analysis inoutput feedback as compared to that in state feedback is that the uncertainty is not decou-pled in the L1-norm condition and enters directly into the underlying transfer function, forwhich the L1-norm must be computed. This adds an additional constraint on the choice ofthe filter and the desired reference system to ensure that this transfer function is stable andhas bounded L1-norm. The two-cart benchmark example is used for numerical illustration.Chapter 6 summarizes flight test results obtained using the architectures presented in thischapter.

4.1 L1 Adaptive Output Feedback Controller forFirst-Order Reference Systems

This section presents the L1 adaptive output feedback controller for a SISO system of un-known dimension in the presence of unmodeled dynamics and time-varying disturbances.The methodology ensures uniformly bounded transient response for system’s both signals,input and output, simultaneously, as compared to the same signals of a first-order stable ref-erence system. The L∞-norm bounds for the error signals between the closed-loop adaptivesystem and the closed-loop reference LTI system can be systematically reduced by increas-ing the adaptation gain [32]. The flight test results using this solution are summarized inChapter 6.

179

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180 Chapter 4. Output Feedback


Consider the SISO system

y(s) = A(s)(u(s)+d(s)) , (4.1)

where u(s) is the Laplace transform of the system’s input signal u(t); y(s) is the Laplacetransform of the system’s output signal y(t); A(s) is a strictly proper unknown transferfunction; d(s) is the Laplace transform of the time-varying nonlinear uncertainties anddisturbances, denoted by d(t) � f (t ,y(t)); and f : R×R → R is an unknown map, subjectto the following assumptions.

Assumption 4.1.1 (Lipschitz continuity) There exist constantsL> 0 andL0> 0, possiblyarbitrarily large, such that the following inequalities hold uniformly in t :

|f (t ,y1)−f (t ,y2)| ≤ L|y1 −y2| , |f (t ,y)| ≤ L|y|+L0 .

Assumption 4.1.2 (Uniform boundedness of the rate of variation of uncertainties)There exist constants L1 > 0, L2 > 0, and L3 > 0, possibly arbitrarily large, such thatfor all t ≥ 0,

|d(t)| ≤ L1|y(t)|+L2|y(t)|+L3 .

The control objective is to design an adaptive output feedback controller u(t) suchthat the system output y(t) tracks the given bounded piecewise-continuous reference inputr(t) following a desired reference model M(s). In this section, we consider a first-orderreference system, i.e.,

M(s) = m

s+m , m> 0. (4.2)



We note that the system in (4.1) can be rewritten in terms of the reference system, definedbyM(s), as

y(s) =M(s) (u(s)+σ (s)) , (4.3)

where the uncertainties due to A(s) and d(s) are lumped into the signal σ (s), which isgiven by

σ (s) = (A(s)−M(s))u(s)+A(s)d(s)

M(s). (4.4)

The design of L1 adaptive controller proceeds by considering a strictly proper filterC(s) with C(0) = 1, such that

H (s) � A(s)M(s)

C(s)A(s)+ (1−C(s))M(s)is stable (4.5)

and the following L1-norm condition holds:

‖G(s)‖L1L < 1, (4.6)

where G(s) �H (s)(1−C(s)).

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4.1. Feedback Controller for First-Order Reference Systems 181

Letting

A(s) = An(s)

Ad (s), C(s) = Cn(s)

Cd (s), M(s) = Mn(s)

Md (s), (4.7)


H (s) = Cd (s)Mn(s)An(s)

Md (s)Cn(s)An(s)+ (Cd (s)−Cn(s))Mn(s)Ad (s). (4.8)

We note that a strictly proper C(s) implies that the orders of Cd (s) −Cn(s) and Cd (s) areequal. Since the order of Ad (s) is higher than the order of An(s), we note that the transferfunction H (s) is strictly proper.

Next, we introduce some notations that will be useful for the proofs of stability andperformance bounds. First, define

H0(s) � A(s)

C(s)A(s)+ (1−C(s))M(s), (4.9)

H1(s) � (A(s)−M(s))C(s)

C(s)A(s)+ (1−C(s))M(s), (4.10)

H2(s) � C(s)H (s)

M(s), (4.11)

H3(s) � − M(s)C(s)

C(s)A(s)+ (1−C(s))M(s). (4.12)

Using (4.7) in (4.9)–(4.12), we have

H0(s) = Cd (s)An(s)Md (s)

Hd (s),

H1(s) = Cn(s)An(s)Md (s)−Cn(s)Ad (s)Mn(s)

Hd (s), (4.13)

H3(s) = −Cn(s)Ad (s)Mn(s)

Hd (s), (4.14)

where

Hd (s) � Cn(s)An(s)Md (s)+Mn(s)Ad (s)(Cd (s)−Cn(s)) .

Since deg(Cd (s) −Cn(s)) is larger than deg(Cn(s)), then deg(Mn(s)Ad (s)(Cd (s) −

Cn(s)))

is larger than deg(Cn(s)Ad (s)Mn(s)). Since deg(Ad (s)) is larger than deg(An(s)),while deg(Md (s))−deg(Mn(s)) = 1, we note that deg(Mn(s)Ad (s)(Cd (s)−Cn(s))) is higherthan deg(Cn(s)An(s)Md (s)). Therefore, H1(s) is strictly proper. We note from (4.8) and(4.13) thatH1(s) has the same denominator as H (s), and it follows from (4.5) that H1(s) isstable. Using similar arguments, it can be verified that H0(s) and H3 are proper and stabletransfer functions.

Let ρr be defined as

ρr � ‖H (s)C(s)‖L1‖r‖L∞ +‖G(s)‖L1L0

1−‖G(s)‖L1L. (4.15)

L1book2010/7/22page 182

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Also, let

�� ‖H1(s)‖L1‖r‖L∞ +‖H0(s)‖L1 (Lρr +L0)

+(

‖H1(s)/M(s)‖L1 +‖H0(s)‖L1

‖H2(s)‖L1

1−‖G(s)‖L1LL

)γ0 ,

where γ0 > 0 is an arbitrary constant. It can be verified easily that H2(s) is strictly properand stable and hence ‖H2(s)‖L1

is bounded. Since H1(s) is stable and strictly proper, wenote that ‖H1(s)/M(s)‖L1 exists and, hence, � is bounded.

Further, let

β1 � 4�‖H0(s)‖L1

(β01L1 + ‖H2(s)‖L1

1−‖G(s)‖L1LL2

),

β2 � 4�(‖sH1(s)‖L1

(‖r‖L∞ +2�)+‖H0(s)‖L1 (β02L1 +L2ρr +L3)

),

(4.16)

where β01 and β02 are defined as

β01 � ‖sH (s)(1−C(s))‖L1

‖H2(s)‖L1

1−‖G(s)‖L1LL ,

β02 � ‖sH (s)C(s)‖L1

(‖r‖L∞ +2�)+‖sH (s)(1−C(s))‖L1 (Lρr +L0).

(4.17)

SinceH (s) andH1(s) are strictly proper and stable, we note that‖sH1(s)‖L1 ,‖sH (s)C(s)‖L1 ,and ‖sH (s)(1−C(s))‖L1 are bounded.

Finally, choose arbitrary P ∈ R+, and letQ� 2mP , while

β3 � P

Qβ1 = β1

2m,

β4 � 4�2 + P

Qβ2 = 4�2 + β2

2m. (4.18)

The elements of L1 adaptive controller are introduced below.

Output Predictor

We consider the following output predictor:

˙y(t) = −my(t)+m (u(t)+ σ (t)) , y(0) = 0, (4.19)

where σ (t) is the adaptive estimate.

Adaptation Law

The adaptation of σ (t) is defined as

˙σ (t) = �Proj(σ (t), − y(t)) , σ (0) = 0, (4.20)

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y(s) = A(s)(u(s)+d(s))

˙y(t) = −my(t)+m (u(t)+ σ (t))

˙σ (t) = �Proj(σ (t), − y(t))

C(s)

System

Output Predictor

Adaptive Law

r u y

y

y

σ

Figure 4.1: Closed-loop system with L1 adaptive controller.

where y(t) � y(t) − y(t) is the error signal between the output of the system in (4.3) andthe predictor in (4.19), � ∈ R

+ is the adaptation rate subject to the lower bound

� >max

{αβ2

3

(α−1)2β4P,αβ4

P γ 20

}, (4.21)

with α > 1 being an arbitrary constant, and the projection is performed with the followingbound:

|σ (t)| ≤� , ∀ t ≥ 0. (4.22)

Control Law

The control signal is generated according to the following law, assuming zero initializationfor C(s):

u(s) = C(s)(r(s)− σ (s)) . (4.23)

The complete L1 adaptive controller consists of (4.19), (4.20), and (4.23) subject tothe L1-norm condition in (4.6). The closed-loop system is illustrated in Figure 4.1.

Remark 4.1.1 The sufficient condition for stability in (4.6) restricts the class of systemsA(s) in (4.1) that can be stabilized by the controller architecture in this section. However,as discussed in Section 4.1.4, the class of such systems is not empty.

4.1.3 Analysis of the L1 Adaptive Output Feedback Controller



yref (s) =M(s)(uref (s)+σref (s)) , (4.24)

uref (s) = C(s)(r(s)−σref (s)) , (4.25)

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where

σref (s) = (A(s)−M(s))uref (s)+A(s)dref (s)

M(s), (4.26)

and dref (s) is the Laplace transform of dref (t) � f (t ,yref (t)). We note that there is noalgebraic loop involved in the definition of σ (s), u(s) and σref (s), uref (s).

The next lemma establishes the stability of the closed-loop reference system in (4.24)–(4.25).

Lemma 4.1.1 Let C(s) and M(s) verify the L1-norm condition in (4.6). Then the closed-loop reference system in (4.24)–(4.25) is BIBO stable.

Proof. It follows from (4.25) and (4.26) that

uref (s) = C(s)r(s)− C(s)((A(s)−M(s))uref (s)+A(s)dref (s))

M(s),

and hence

uref (s) = C(s)M(s)r(s)−C(s)A(s)dref (s)

C(s)A(s)+ (1−C(s))M(s). (4.27)

From (4.24)–(4.26) we have

yref (s) = A(s)(uref (s)+dref (s)) . (4.28)

Substituting (4.27) into (4.28), it follows from (4.5) that

yref (s) = A(s)

(C(s)M(s)r(s)−C(s)A(s)dref (s)

C(s)A(s)+ (1−C(s))M(s)+dref (s)

)

= A(s)M(s)C(s)r(s)+ (1−C(s))dref (s)

C(s)A(s)+ (1−C(s))M(s)

=H (s) (C(s)r(s)+ (1−C(s))dref (s)) .

(4.29)

Since H (s) and G(s) are strictly proper and stable, the following upper bound holds:∥∥yref t

∥∥L∞ ≤ ‖H (s)C(s)‖L1‖r‖L∞ +‖G(s)‖L1

(L∥∥yref t

∥∥L∞ +L0

).

Then, the L1-norm condition in (4.6), together with the definition of ρr in (4.15), impliesthat ∥∥yref t

∥∥L∞ ≤ ρr , (4.30)

which holds uniformly, and hence‖yref ‖L∞ is bounded. Therefore, the closed-loop referencesystem in (4.24)–(4.25) is BIBO stable. �

Remark 4.1.2 We notice that the ideal control signal

uid(t) = r(t)−σref (t)

is the one that leads to the desired system response

yid(s) =M(s)r(s)

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by canceling the uncertainties exactly. Thus, the reference system in (4.24)–(4.25) has adifferent response as compared to the ideal one. It cancels only the uncertainties withinthe bandwidth of C(s), which can be selected to be compatible with the control channelspecifications. This is exactly what one can hope to achieve with any feedback in the presenceof uncertainties.


In this section, we analyze the stability of the closed-loop adaptive system with the L1 adap-tive controller and derive the uniform performance bounds. Toward this end, let γ0 begiven by

γ0 �√αβ4

�P. (4.31)

Then, it follows from (4.21) that γ0 < γ0, and hence

�≥ ‖H1(s)‖L1‖r‖L∞ +‖H0(s)‖L1 (Lρr +L0)

+(∥∥∥∥H1(s)

M(s)

∥∥∥∥L1

+L‖H0(s)‖L1

‖H2(s)‖L1

1−‖G(s)‖L1L

)γ0 .

Theorem 4.1.1 Consider the system in (4.1) and the L1 adaptive controller in (4.19), (4.20),and (4.23), subject to the L1-norm condition in (4.6). If the adaptive gain is chosen to satisfythe design constraint in (4.21), then the following bounds hold:

‖y‖L∞ < γ0 , (4.32)

‖yref −y‖L∞ ≤ γ1 , (4.33)

‖uref −u‖L∞ ≤ γ2 , (4.34)

where y(t) � y(t)−y(t), γ0 is defined in (4.31), and

γ1 �‖H2(s)‖L1

1−‖G(s)‖L1Lγ0 ,

γ2 � ‖H2(s)‖L1Lγ1 +‖H3(s)/M(s)‖L1γ0 .

Proof. Let σ (t) � σ (t)−σ (t), where σ (t) is defined in (4.4). It follows from (4.23) that

u(s) = C(s)r(s)−C(s)(σ (s)+ σ (s)) , (4.35)


y(s) =M(s)(C(s)r(s)+ (1−C(s))σ (s)−C(s)σ (s)

). (4.36)

Substituting (4.35) into (4.4), it follows from the definitions of H (s), H0(s), and H1(s)in (4.5), (4.9), and (4.10) that

σ (s) =H1(s)(r(s)− σ (s))+H0(s)d(s) . (4.37)

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Substituting (4.37) into (4.36), we have

y(s) =M(s)(C(s)+H1(s)(1−C(s)))(r(s)− σ (s))

+H0(s)M(s)(1−C(s))d(s) .(4.38)

It can be verified from (4.5) and (4.10) that

M(s)(C(s)+H1(s)(1−C(s))) =H (s)C(s) and H (s) =H0(s)M(s) ,

and, hence, the expression in (4.38) can be rewritten as

y(s) =H (s)(C(s)r(s)−C(s)σ (s))+H (s)(1−C(s))d(s) . (4.39)

Let e(t) � yref (t)−y(t). From (4.29) and (4.39), one has

e(s) =H (s)(1−C(s))de(s)+H (s)C(s)σ (s) ,

where de(s) is introduced to denote the Laplace transform of de(t) � f (t ,yref (t))−f (t ,y(t)).Moreover, it follows from (4.3) and (4.19) that

y(s) =M(s)σ (s) , (4.40)

which implies that

C(s)H (s)σ (s) = C(s)H (s)

M(s)M(s)σ (s) = C(s)H (s)

M(s)y(s) .

Lemma A.7.1 and Assumption 4.1.1 give the following upper bound:

‖et‖L∞ ≤ ‖H (s)(1−C(s))‖L1L‖et‖L∞ +‖H2(s)‖L1‖yt‖L∞ ,

which leads to

‖et‖L∞ ≤ ‖H2(s)‖L1

1−‖G(s)‖L1L‖yt‖L∞ . (4.41)

First, we prove the bound in (4.32) by contradiction. Since y(0) = 0 and y(t) iscontinuous, then assuming the opposite implies that there exists t ′ such that

‖y(t)‖< γ0, ∀ 0 ≤ t < t ′ ,‖y(t ′)‖ = γ0 ,

which leads to

‖yt ′ ‖L∞ = γ0 . (4.42)

Since y(t) = yref (t)− e(t), it follows from (4.30) and (4.42) that

‖yt ′ ‖L∞ ≤ ‖yref t ′ ‖L∞ +‖et ′ ‖L∞ ≤ ρr +‖H2(s)‖L1

1−‖G(s)‖L1Lγ0 . (4.43)

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σ (s) =H1(s)r(s)− H1(s)

M(s)y(s)+H0(s)d(s) ,

and hence, the equality in (4.42) implies that

‖σt ′ ‖L∞ ≤ ‖H1(s)‖L1‖r‖L∞ +∥∥∥∥H1(s)

M(s)

∥∥∥∥L1

γ0 +‖H0(s)‖L1 (L‖yt ′ ‖L∞ +L0) .

Along with (4.43) this leads to

‖σt ′ ‖L∞ ≤� . (4.44)

Consider the following candidate Lyapunov function:

V (y(t), σ (t)) = P y2(t)+�−1σ 2(t) . (4.45)

The adaptive law in (4.20) implies that for all 0 ≤ t ≤ t ′,V (t) ≤ −Qy2(t)+2�−1|σ (t)σ (t)| . (4.46)


σd (s) = sH1(s)(r(s)− σ (s))+H0(s)dd (s) , (4.47)

where σd (s) and dd (s) are the Laplace transformations of σ (t) and d(t), respectively.From (4.22) and (4.44), we have

‖σt ′ ‖L∞ ≤ 2� . (4.48)


‖dt ′ ‖L∞ ≤ Lρr +‖H2(s)‖L1

1−‖G(s)‖L1LLγ0 +L0 . (4.49)

From the definitions of β01 and β02 in (4.17), (4.39), and (4.49), we have ‖yt ′ ‖L∞ ≤β01γ0 +β02 . It follows from Assumption 4.1.2 that

‖dt ′ ‖L∞ ≤ L2‖yt ′ ‖L∞ +L1(β01γ0 +β02)+L3 . (4.50)

From (4.43), (4.47), (4.50), and the definitions of β1 and β2 in (4.16), it follows that

‖σt ′ ‖L∞ ≤ β1γ0 +β2

4�. (4.51)

Therefore, from (4.46), (4.48), and (4.51) we have

V (t) ≤ −Qy2(t)+�−1(β1γ0 +β2

), ∀ 0 ≤ t ≤ t ′ . (4.52)

The projection algorithm ensures that |σ (t)| ≤� for all t ≥ 0, and therefore

maxt ′≥t≥0

�−1σ 2(t) ≤ 4�2/� . (4.53)

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Let θmax � β3γ0 + β4, where β3 and β4 are defined in (4.18). If at arbitrary t ∈ [0, t ′],V (t)> θmax/�, then it follows from (4.45) and (4.53) that

P y2(t)>P (β1γ0 +β2)

�Q,

and hence

Qy2(t) = (Q/P )P y2(t)> (β1γ0 +β2)/� . (4.54)

From (4.52) and (4.54) it follows that if V (t)> θmax/� for some t ∈ [0, t ′], then

V (t)< 0. (4.55)

Since y(0) = 0, we can verify that V (0) ≤(β3γ0 +β4

)/�. It follows from (4.55) that

V (t) ≤ θmax/� , ∀ 0 ≤ t ≤ t ′ . (4.56)

Since P ‖y(t)‖2 ≤ V (t), then it follows from (4.56) that

|y(t)|2 ≤ β3γ0 +β4

�P, ∀ 0 ≤ t ≤ t ′ ,

and from the definition of γ0 in (4.31), we get

|y(t)|2 ≤(β3

αβ4

√αβ4

�P+ 1

α

)γ 2

0 , ∀ 0 ≤ t ≤ t ′ .

Then, the design constraint in (4.21) leads to

|y(t)|2 < γ 20 , ∀ 0 ≤ t ≤ t ′ ,

which contradicts the assumption in (4.42), and thus (4.32) holds. It follows from (4.6),(4.32), and (4.41) that

‖et‖L∞ ≤ ‖H2(s)‖L1

1−‖G(s)‖L1Lγ0 ,

which holds uniformly for all t ≥ 0 and therefore leads to the bound in (4.33).On the other hand, it follows from (4.4) and (4.35) that

u(s) = M(s)(C(s)r(s)−C(s)σ (s))−C(s)A(s)d(s)

C(s)A(s)+ (1−C(s))M(s).

To prove the bound in (4.34), we notice that from (4.27) one can derive

uref (s)−u(s) = −H2(s)de(s)−H3(s)σ (s)

= −H2(s)de(s)− (H3(s)/M(s))M(s)σ (s) .(4.57)


‖uref −u‖L∞ ≤ ‖H2(s)‖L1L‖yref −y‖L∞ +‖H3(s)/M(s)‖L1‖y‖L∞ ,

which leads to (4.34). �

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Thus, the tracking error between y(t) and yref (t), as well as between u(t) and uref (t), isuniformly bounded by a constant inverse proportional to

√�. This implies that, during

the transient phase, one can achieve arbitrarily close tracking performance for both signalssimultaneously by increasing �.

Remark 4.1.3 We notice that if we setC(s) = 1, then the bound in (4.34) is not well defined,since the second term degenerates into the L1-norm of an improper transfer function.

4.1.4 Design for the L1-Norm Condition

In this section, we discuss the classes of systems that can satisfy (4.5) via the choice ofM(s)and C(s). For simplicity, we consider first-order C(s), given by

C(s) = ω

s+ω , (4.58)

and first-orderM(s), such as that in (4.2). It follows from (4.2) and (4.58) that

H (s) = m(s+ω)An(s)

ω(s+m)An(s)+msAd (s).

Stability of H (s) is equivalent to stabilization of A(s) by a PI controller KPI (s) of thestructure

KPI (s) = ω(s+m)

ms, (4.59)

wherem andω are the same as in (4.2) and (4.58). The loop-transfer function of the cascadedsystem A(s) with the PI controller will be

LPI (s) = ω(s+m)

msA(s) ,

leading to the following closed-loop system:

HPI (s) = ω(s+m)An(s)

ω(s+m)An(s)+msAd (s). (4.60)

Hence, the stability of H (s) is equivalent to that of (4.60), and the problem can be reducedto identifying the class of systems A(s) that can be stabilized by a PI controller. It alsopermits the use of root locus methods for checking the stability ofH (s) via the loop-transferfunction LPI (s). We note that the PI controller KPI (s) in (4.59) adds a pole at the originand a zero at −m to the loop-transfer function, while ω/m is the proportional gain of thecontroller. In the absence of nonlinearity, one has L = 0 in (4.6), and hence the stabilityof the closed-loop system follows from the stability of H (s), which can be verified usingmethods from H∞ robust control.

Minimum Phase Systems with Relative Degree 1 or 2

Consider a minimum phase system H (s) with relative degree 1. Notice that the zeros ofLPI (s) are located in the open left half plane. As the gainω/m increases, it follows from theclassical control theory that all the closed-loop poles approach the open-loop zeros except 1,

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which tends to ∞ along the negative real axis. This implies that all the closed-loop polesare located in the open left half plane. Hence, the transfer function in (4.60) is stable, andso is H (s).

For a minimum phase system H (s) with relative degree 2, with the increase of thegain ω/m there are two closed-loop poles approaching ∞ along the direction of −π/2and π/2 in the complex plane. Let the abscissa of the intersection of the asymptotes and thereal axis be δ. We note that the two infinite poles approach δ± j∞. If the choice of M(s)ensures negative δ, then the closed-loop system can be stabilized by increasing the loopgain. Therefore, by choosing appropriate M(s), we can ensure stability of minimum phasesystems with relative degree 1 or 2.

Other Systems

We note that nonminimum phase systems can also be stabilized by a PI controller. However,the choice of m and ω is not straightforward. Reference [67] has addressed the problem ofthe filter design using standard root locus analysis from classical control. It has been shownthat, if the system is nonminimum phase, then the choice of the low-pass filter and thereference system that would verify stability of H (s) might be very limited.

Remark 4.1.4 We notice that, in light of the above discussion, a PI controller stabiliz-ingA(s) might also stabilize the system in the presence of the nonlinear uncertaintyf (t ,y(t)).However, the transient performance cannot be quantified in the presence of unknown A(s).The L1 adaptive controller instead will generate different control signals u(t) (always inthe low-frequency range) for different unknown systems to ensure uniform transient per-formance for y(t).

In the simulation example below, we demonstrate the application of the L1 adaptivecontroller for an unknown nonminimum phase system in the presence of unknown nonlinearuncertainties.


As an illustrative example, consider the system in (4.1) with

A(s) = s2 −0.5s+0.5

s3 − s2 −2s+8.

We note thatA(s) has both poles and zeros in the right half plane and hence it is an unstablenonminimum phase system. We consider the L1 adaptive controller, defined via (4.19),(4.20), and (4.23), with m= 3, ω = 10, � = 500. We set �= 100. First, we consider theresponse of the closed-loop system to a step-reference signal for d(t) ≡ 0. The simulationresults are shown in Figure 4.2. Next, we consider

f (t ,y(t)) = sin(0.1t)y(t)+2sin(0.1t)

and apply the same controller without retuning. The system response and the control signalare plotted in Figures 4.3(a) and 4.3(b). Further, we consider a time-varying referenceinput r(t) = 0.5sin(0.3t) and notice that, without any retuning of the controller, the system

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0 10 20 30 40−1

0

1

2

time [s]

y(t)r(t)


0 10 20 30 40−10

0

10

20

30

time [s]


Figure 4.2: Performance of the L1 controller for f (t ,y(t)) ≡ 0.

0 10 20 30 40−1

0

1

2

time [s]

y(t)r(t)


0 10 20 30 40−10

0

10

20

30

time [s]


0 10 20 30 40 50−0.5

0

0.5

1

time [s]

y(t)r(t)

(c) y(t) (solid) and r(t) (dashed)

0 10 20 30 40 50−20

−10

0

10

20

time [s]


Figure 4.3: Performance of the L1 controller for f (t ,y) = sin(0.1t)y+2sin(0.1t).

response and the control signal behave as expected (Figures 4.3(c) and 4.3(d)). Finally,Figure 4.4 presents the closed-loop system response and the control signal for a differentuncertainty,

f (t ,y(t)) = sin(0.1t)y(t)+2sin(0.4t),

without any retuning of the controller.

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0 10 20 30 40 50−1

−0.5

0

0.5

1

time [s]

y(t)r(t)


0 10 20 30 40 50−15

−10

−5

0

5

10

time [s]


Figure 4.4: Performance of the L1 controller for f (t ,y) = sin(0.1t)y+2sin(0.4t).

4.2 L1 Adaptive Output Feedback Controller forNon-SPR Reference Systems

This section presents an extension of the L1 adaptive output feedback controller, whichachieves performance specifications defined by a non-SPR reference system. This exten-sion is possible by invoking the piecewise-constant adaptive law, discussed in Section 3.3.The performance bounds between the closed-loop reference system and the closed-loopL1 adaptive system can be rendered arbitrarily small by reducing the step size of integra-tion. The sampling time of the adaptive law can be set according to the available samplingrate of the CPU [33]. This solution has been tested in a midfidelity model of a genericflexible Crew Launch Vehicle provided by NASA [88].


Consider the following SISO system:

y(s) = A(s)(u(s)+d(s)) , y(0) = 0, (4.61)

where u(t) ∈ R is the input; y(t) ∈ R is the system output;A(s) is a strictly proper unknowntransfer function of unknown relative degree nr , for which only a known lower bound1 < dr ≤ nr is available; d(s) is the Laplace transform of the time-varying uncertaintiesand disturbances d(t) = f (t ,y(t)); and f : R×R → R is an unknown map, subject to thefollowing assumption.

Assumption 4.2.1 (Lipschitz continuity) There exist constants L > 0 and L0 > 0, suchthat

|f (t ,y1)−f (t ,y2)| ≤ L|y1 −y2| , |f (t ,y)| ≤ L|y|+L0

hold uniformly in t ≥ 0, where the numbers L and L0 can be arbitrarily large.

Let r(t) be a given bounded continuous reference input signal. The control objectiveis to design an adaptive output feedback controller u(t) such that the system output y(t)tracks the reference input r(t) following a desired reference model M(s), where M(s) is aminimum-phase stable transfer function of relative degree dr > 1.

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4.2. L1 Adaptive Output Feedback Controller for Non-SPR Reference Systems 193



Similar to Section 4.1.2, we can rewrite the system in (4.61) as

y(s) =M(s) (u(s)+σ (s)) , y(0) = 0, (4.62)

σ (s) = (A(s)−M(s))u(s)+A(s)d(s)

M(s). (4.63)

Let (Am, bm, c�m) be a minimal realization of M(s), i.e., (Am, bm, c�m) is controllable andobservable, and Am is Hurwitz. The system in (4.62) can be rewritten as

x(t) = Amx(t)+bm(u(t)+σ (t)) , x(0) = 0,

y(t) = c�mx(t) .(4.64)

The design of the L1 adaptive controller proceeds by considering a strictly propersystem C(s) of relative degree dr , with C(0) = 1. Further, similar to Section 4.1.2, theselection of C(s) andM(s) must ensure that

H (s) � A(s)M(s)

C(s)A(s)+ (1−C(s))M(s)is stable (4.65)

and that the following L1-norm condition holds:

‖G(s)‖L1L < 1, (4.66)

where G(s) �H (s)(1−C(s)).Letting

A(s) = An(s)

Ad (s), C(s) = Cn(s)

Cd (s), M(s) = Mn(s)

Md (s), (4.67)

where the numerators and the denominators are all polynomials of s, it follows from (4.65)that

H (s) = Cd (s)Mn(s)An(s)

Hd (s), (4.68)

where

Hd (s) � Cn(s)An(s)Md (s)+Mn(s)Ad (s)(Cd (s)−Cn(s)) . (4.69)

A strictly properC(s) implies that the orders ofCd (s)−Cn(s) andCd (s) are the same. Sincethe order of Ad (s) is higher than the order of An(s), the transfer function H (s) is strictlyproper.

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Next, let

H0(s) � A(s)

C(s)A(s)+ (1−C(s))M(s), (4.70)

H1(s) � (A(s)−M(s))C(s)

C(s)A(s)+ (1−C(s))M(s), (4.71)

H2(s) � H (s)C(s)

M(s), (4.72)

H3(s) � − M(s)C(s)

(C(s)A(s)+ (1−C(s))M(s)). (4.73)

Using the expressions from (4.67) and (4.69), we can rewrite the equations for H0(s) andH1(s) as

H0(s) = Cd (s)An(s)Md (s)

Hd (s),

H1(s) = Cn(s)An(s)Md (s)−Cn(s)Ad (s)Mn(s)

Hd (s). (4.74)

Since deg(Cd (s) −Cn(s)) is larger than deg(Cn(s)) by dr , then deg(Mn(s)Ad (s)(Cd (s)−Cn(s))) is larger than deg(Cn(s)Ad (s)Mn(s)) by dr . Since the deg(Ad (s)) is largerthan deg(An(s)) by nr ≥ dr , while deg(Mn(s)) is larger than deg(Md (s)) by dr , thendeg(Mn(s)Ad (s)(Cd (s) −Cn(s))) is larger than deg(Cn(s)An(s)Md (s)). Therefore, H1(s)is strictly proper with relative degree dr . We notice from (4.68) and (4.74) that H1(s) hasthe same denominator as H (s), and therefore it follows from (4.66) that H1(s) is stable.Using similar arguments, it can be verified that H0(s) is proper and stable. Similarly, H2(s)is strictly proper and stable.

Also, let

�� ‖H1(s)‖L1‖r‖L∞ +‖H0(s)‖L1 (Lρr +L0)

+(∥∥∥∥H1(s)

M(s)

∥∥∥∥L1

+‖H0(s)‖L1

‖H2(s)‖L1

1−‖G(s)‖L1LL

)γ0 ,

(4.75)

where γ0 > 0 is an arbitrary constant. Since both H1(s) and M(s) are stable and strictlyproper with relative degree dr , and M(s) is minimum phase, H1(s)/M(s) is stable andproper. Hence, ‖H1(s)/M(s)‖L1 is bounded. Therefore � is also bounded. Further, sinceAm is Hurwitz, there exists P = P� > 0 that satisfies the algebraic Lyapunov equation

A�mP +PAm = −Q , for arbitrary Q=Q� > 0.

From the properties of P , it follows that there exits nonsingular√P such that

P = (√P )�

√P .

Given the vector c�m(√P )−1, let D be a (n− 1) ×n matrix that contains the null space of

c�m(√P )−1, i.e.,

D(c�m(√P )−1)� = 0, (4.76)

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and further let

��[

c�mD

√P

].

From the definition of the null space, it follows that

�(√P )−1 =

[c�m(

√P )−1

D

]

is full rank, and hence �−1 exists.

Lemma 4.2.1 For arbitrary ξ � [yz ] ∈ Rn, where y ∈ R and z ∈ R

n−1, there exist p1 > 0and positive definite P2 ∈ R

(n−1)×(n−1) such that

ξ�(�−1)�P�−1ξ = p1y2 + z�P2z .

Proof. Using P = (√P )�

√P , one can write

ξ�(�−1)�P�−1ξ = ξ�(√P�−1)�(

√P�−1)ξ .

We notice that

�(√P )−1 =

[c�m(

√P )−1

D

].

Next, let

q1 = (c�m(√P )−1)(c�m(

√P )−1)� , Q2 =DD� .

From the expression in (4.76) we have

(�(√P )−1)(�(

√P )−1)� =

[q1 00 Q2

].

Nonsingularity of� and√P implies that (�(

√P )−1)(�(

√P )−1)� is nonsingular as well,

and thereforeQ2 is also nonsingular. Hence,

(√P�−1)�(

√P�−1) =

((�(

√P )−1)(�(

√P )−1)�

)−1

=[q−1

1 00 Q−1

2

].

Denoting p1 � q−11 and P2 �Q−1

2 completes the proof. �

Let Ts be an arbitrary positive constant, which can be associated with the samplingrate of the available CPU, and 11 = [1, 0, . . . , 0]� ∈ R

n. Let

1�1 e�Am�

−1t = [η1(t), η�2 (t)] , (4.77)

where η1(t) ∈ R and η2(t) ∈ Rn−1 contain the first and the 2-to-n elements of the row vector

1�1 e�Am�

−1t , and let

κ(Ts) �∫ Ts

0|1�

1 e�Am�−1(Ts−τ )�bm|dτ . (4.78)

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Also, let ς (Ts) be defined as

ς (Ts) � ‖η2(Ts)‖√

α

λmax(P2)+κ(Ts)� ,

α � λmax(�−�P�−1)

(2�‖�−�Pbm‖λmin(�−�Q�−1)

)2

. (4.79)

Further, let �(Ts) be the n×n matrix

�(Ts) �∫ Ts

0e�Am�

−1(Ts−τ )�dτ . (4.80)

Next, we introduce the functions

β1(Ts) � maxt∈[0, Ts ]

|η1(t)| , β2(Ts) � maxt∈[0, Ts ]

‖η2(t)‖ , (4.81)

and also

β3(Ts) � maxt∈[0, Ts ]

η3(t) , β4(Ts) = maxt∈[0, Ts ]

η4(t) , (4.82)

where

η3(t) �∫ t

0|1�

1 e�Am�−1(t−τ )��−1(Ts)e

�Am�−1Ts11|dτ ,

η4(t) �∫ t

0|1�

1 e�Am�−1(t−τ )�bm|dτ .

Finally, let

γ0(Ts) � β1(Ts)ς (Ts)+β2(Ts)√

α

λmax(P2)+β3(Ts)ς (Ts)+β4(Ts)� . (4.83)

Lemma 4.2.2 The following limiting relationship is true:

limTs→0

γ0(Ts) = 0.

Proof. Notice that since β1(Ts),β3(Ts), α, and � are bounded, it is sufficient to prove that

limTs→0

ς (Ts) = 0, (4.84)

limTs→0

β2(Ts) = 0, (4.85)

limTs→0

β4(Ts) = 0. (4.86)

Since

limTs→0

1�1 e�Am�

−1Ts = 1�1 ,

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then

limTs→0

η2(Ts) = 0n−1 ,

which implies

limTs→0

‖η2(Ts)‖ = 0.

Further, it follows from the definition of κ(Ts) in (4.78) that

limTs→0

κ(Ts) = 0.

Since α and � are bounded, we have

limTs→0

ς (Ts) = 0,

which proves (4.84). Since η2(t) is continuous, it follows from (4.81) that

limTs→0

β2(Ts) = limt→0

‖η2(t)‖ .

Since

limt→0

1�1 e�Am�

−1t = 1�1 ,

we have

limt→0

‖η2(t)‖ = 0,

which proves (4.85). Similarly

limTs→0

β4(Ts) = limt→0

‖η4(t)‖ = 0,

which proves (4.86). Boundedness of α, β3(Ts), and � implies

limTs→0

(β1(Ts)ς (Ts)+β2(Ts)

√α

λmax(P2)+β3(Ts)ς (Ts)+β4(Ts)�

)= 0,


The elements of L1 adaptive controller are introduced next.

Output Predictor

We consider the following output predictor:

˙x(t) = Amx(t)+bmu(t)+ σ (t) , x(0) = 0,

y(t) = c�mx(t) ,(4.87)

where σ (t) ∈ Rn is the vector of adaptive parameters. Notice that while σ (t) ∈ R in (4.64)

is matched, the uncertainty estimation σ (t) ∈ Rn in (4.87) is unmatched.

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Adaptation Laws

Letting y(t) � y(t)−y(t), the update law for σ (t) is given by

σ (t) = σ (iTs) , t ∈ [iTs , (i+1)Ts) ,

σ (iTs) = −�−1(Ts)µ(iTs) , i = 0,1,2, . . . ,(4.88)

where �(Ts) was defined in (4.80) and

µ(iTs) = e�Am�−1Ts11y(iTs) , i = 0,1,2, . . . .

Control Law

The control signal is defined as

u(s) = C(s)r(s)− C(s)

c�m(sI−Am)−1bmc�m(sI−Am)−1σ (s) , (4.89)

where C(s) was first introduced in (4.65). The L1 adaptive controller consists of (4.87),(4.88), and (4.89), subject to the L1-norm condition in (4.66).




yref (s) =M(s)(uref (s)+σref (s)) , (4.90)

uref (s) = C(s)(r(s)−σref (s)) , (4.91)

where

σref (s) = (A(s)−M(s))uref (s)+A(s)dref (s)

M(s), (4.92)

and dref (t) � f (t ,yref (t)).

Lemma 4.2.3 LetC(s) andM(s) verify the L1-norm condition in (4.66). Then, the closed-loop reference system in (4.90)–(4.91) is BIBO stable.

Proof. It follows from (4.92) and (4.91) that

uref (s) = C(s)M(s)r(s)−C(s)A(s)dref (s)

C(s)A(s)+ (1−C(s))M(s), (4.93)

while from (4.90)–(4.92) one can derive

yref (s) =H (s)(C(s)r(s)+ (1−C(s))dref (s)

). (4.94)

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Since H (s) is strictly proper and stable, G(s) = H (s)(1 −C(s)) is also strictly proper andstable, and therefore

‖yref ‖L∞ ≤ ‖H (s)C(s)‖L1‖r‖L∞ +‖G(s)‖L1 (L‖yref ‖L∞ +L0) .

Using the L1-norm condition in (4.66), it follows that

‖yref t‖L∞ ≤ ρr � ‖H (s)C(s)‖L1‖r‖L∞ +‖G(s)‖L1L0

1−‖G(s)‖L1L<∞ , (4.95)

which holds uniformly, and hence, ‖yref ‖L∞ is bounded. This completes the proof. �


We will now proceed with the derivation of the performance bounds. Toward this end, letx(t) � x(t)−x(t). Then, the error dynamics between (4.64) and (4.87) are given by

˙x(t) = Amx(t)+ σ (t)−bmσ (t) , x(0) = 0,

y(t) = c�mx(t) .(4.96)

Lemma 4.2.4 Consider the system in (4.61) with the L1 adaptive controller and the closed-loop reference system in (4.90)–(4.91). The following upper bound holds:

‖(yref −y)t‖L∞ ≤ ‖H2(s)‖L1

1−‖G(s)‖L1L‖yt‖L∞ .

Proof. Let

σ (s) � C(s)

c�m(sI−Am)−1bmc�m(sI−Am)−1σ (s)−C(s)σ (s) . (4.97)


u(s) = C(s)r(s)−C(s)σ (s)− σ (s) , (4.98)


y(s) =M(s)(C(s)r(s)+ (1−C(s))σ (s)− σ (s)

). (4.99)

Substituting u(s) from (4.98) into (4.63) gives

σ (s) =(A(s)−M(s)

)(C(s)r(s)−C(s)σ (s)− σ (s)

)+A(s)d(s)

M(s),

and hence

σ (s) =(A(s)−M(s)

)(C(s)r(s)− σ (s)

)+A(s)d(s)

M(s)+C(s)(A(s)−M(s)). (4.100)

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Using the definitions of H0(s) and H1(s) in (4.70) and (4.71), we can write

σ (s) =H1(s)r(s)− H1(s)

C(s)σ (s)+H0(s)d(s) . (4.101)

Substitution into (4.99) leads to

y(s) =M(s)(C(s)+H1(s)(1−C(s))

)(r(s)− σ (s)

C(s)

)+H0(s)M(s)

(1−C(s)

)d(s) .

Recalling the definition of H (s) from (4.65), one can verify that

M(s)(C(s)+H1(s)(1−C(s))) =H (s)C(s) and H (s) =H0(s)M(s) ,

which implies that

y(s) =H (s)(C(s)r(s)− σ (s)

)+H (s)(1−C(s))d(s) .

Letting e(t) � y(t)ref −y(t) and denoting byde(s) the Laplace transform ofde(t) � f (t ,yref (t))−f (t ,y(t)), we can use the expression for yref (s) in (4.94) to derive

e(s) =H (s) ((1−C(s))de(s)+ σ (s)) .

Lemma A.7.1 and Assumption 4.2.1 give the following upper bound:

‖et‖L∞ ≤ ‖H (s)(1−C(s))‖L1L‖et‖L∞ +‖ησt‖L∞ , (4.102)

where ησ (t) is the signal with Laplace transform ησ (s) � H (s)σ (s). Using the expressionfor σ (s) from (4.97), along with the expression for y(s) from (4.62), and taking into con-sideration that

y(s) =M(s)u(s)+ c�m(sI−Am)−1σ (s) ,

it follows that

y(s) = c�m(sI−Am)−1σ (s)−M(s)σ (s)

= M(s)

C(s)

C(s)

M(s)c�m(sI−Am)−1σ (s)−M(s)

C(s)C(s)σ (s) = M(s)

C(s)σ (s) .

(4.103)

This implies that ησ (s) can be rewritten as

ησ (s) = C(s)H (s)

M(s)

M(s)

C(s)σ (s) =H2(s)y(s) ,

and hence

‖ησt ‖L∞ ≤ ‖H2(s)‖L1‖yt‖L∞ .

Substituting this upper bound back into (4.102) completes the proof. �

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Next, notice that using the definitions from (4.67), the transfer functionH3(s) in (4.73)can be rewritten as

H3(s) = −Cn(s)Ad (s)Mn(s)

Hd (s), (4.104)

where Hd (s) was introduced in (4.69). Since deg(Cd (s)−Cn(s))−deg(Cn(s)) = dr , it canbe checked straightforwardly thatH3(s) is strictly proper. We notice from (4.68) and (4.104)thatH3(s) has the same denominator asH (s), and therefore it follows from (4.66) thatH3(s)is stable. Since H3(s) is strictly proper and stable with relative degree dr , H3(s)/M(s) isstable and proper and therefore its L1-norm is bounded.

Finally, consider the state transformation

ξ =�x .


˙ξ (t) =�Am�−1ξ (t)+�σ (t)−�bmσ (t) , ξ (0) = 0, (4.105)

y(t) = ξ1(t) , (4.106)

where ξ1(t) is the first element of ξ (t).

Theorem 4.2.1 Consider the system in (4.61) and the L1 adaptive controller in (4.87),(4.88), and (4.89) subject to the L1-norm condition in (4.66). If we choose Ts to ensure

γ0(Ts)< γ0 , (4.107)

where γ0 is an arbitrary positive constant introduced in (4.75), then

‖y‖L∞ < γ0, (4.108)

‖yref −y‖L∞ ≤ γ1, ‖uref −u‖L∞ ≤ γ2 (4.109)

with

γ1 �‖H2(s)‖L1

1−‖G(s)‖L1Lγ0 , γ2 � ‖H2(s)‖L1Lγ1 +

∥∥∥∥H3(s)

M(s)

∥∥∥∥L1

γ0 .

Proof. First, we prove the bound in (4.108) by a contradiction argument. Since y(0) = 0and y(t) is continuous, then assuming the opposite implies that there exists t ′ such that

|y(t)|< γ0 , ∀ 0 ≤ t < t ′ ,|y(t ′)| = γ0 ,

which leads to

‖yt ′ ‖L∞ = γ0 . (4.110)

Since y(t) = yref (t) − e(t), the upper bound in (4.95) can be used to derive the followingbound:

‖yt ′ ‖L∞ ≤ ‖yref t ′ ‖L∞ +‖et ′ ‖L∞

≤ ρr +‖C(s)H (s)/M(s)‖L1

1−‖G(s)‖L1Lγ0 .

(4.111)

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Also, it follows from (4.101) and (4.103) that

σ (s) =H1(s)r(s)−H1(s)y(s)/M(s)+H0(s)d(s) ,

and hence, the equality in (4.110) implies that

‖σt ′ ‖L∞ ≤ ‖H1(s)‖L1‖r‖L∞ +‖H1(s)/M(s)‖L1 γ0 +‖H0(s)‖L1 (L‖yt ′ ‖L∞ +L0) .

This, along with (4.111), leads to

‖σt ′ ‖L∞ ≤� . (4.112)


ξ (iTs + t) = e�Am�−1t ξ (iTs)+

∫ iTs+t

iTs

e�Am�−1(iTs+t−τ )�σ (iTs)dτ

−∫ iTs+t

iTs

e�Am�−1(iTs+t−τ )�bmσ (τ )dτ

= e�Am�−1t ξ (iTs)+

∫ t

0e�Am�

−1(t−τ )�σ (iTs)dτ

−∫ t

0e�Am�

−1(t−τ )�bmσ (iTs + τ )dτ .

(4.113)

Since

ξ (iTs + t) =[y(iTs + t)

0

]+[

0z(iTs + t)

],

where z(t) � [ξ2(t), ξ3(t), . . . , ξn(t)], it follows from (4.113) that ξ (·) can be decomposed as

ξ (iTs + t) = χ (iTs + t)+ ζ (iTs + t) , (4.114)

where

χ (iTs + t) � e�Am�−1t

[y(iTs)

0

]+

∫ t

0e�Am�

−1(t−τ )�σ (iTs)dτ ,

ζ (iTs + t) � e�Am�−1t

[0

z(iTs)

]−∫ t

0e�Am�

−1(t−τ )�bmσ (iTs + τ )dτ . (4.115)

Next we prove that for all iTs ≤ t ′ one has

|y(iTs)| ≤ ς(Ts) , (4.116)

z�(iTs)P2z(iTs) ≤ α , (4.117)

where ς (Ts) and α were defined in (4.79).

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We start by noting that, since ξ (0) = 0, it is straightforward to show that |y(0)| ≤ ς (Ts),z�(0)P2z(0) ≤ α. Next, for arbitrary (j +1)Ts ≤ t ′, we prove that if

|y(jTs)| ≤ ς(Ts) , (4.118)

z�(jTs)P2z(jTs) ≤ α , (4.119)

then the inequalities (4.118)–(4.119) hold for j + 1 as well, which would imply that thebounds in (4.116)–(4.117) hold for all iTs ≤ t ′.

To this end, assume that (4.118)–(4.119) hold for j and, in addition, that (j+1)Ts ≤ t ′.Then, it follows from (4.114) that

ξ ((j +1)Ts) = χ ((j +1)Ts)+ ζ ((j +1)Ts) ,

where

χ ((j +1)Ts) = e�Am�−1Ts

[y(jTs)

0

]+∫ Ts

0e�Am�

−1(Ts−τ )�σ (jTs)dτ , (4.120)

ζ ((j +1)Ts) = e�Am�−1Ts

[0

z(jTs)

]−∫ Ts

0e�Am�

−1(Ts−τ )�bmσ (jTs + τ )dτ . (4.121)

Substituting the adaptive law from (4.88) in (4.120), we have

χ ((j +1)Ts) = 0. (4.122)

It follows from (4.115) that ζ (t) is the solution to the following dynamics:

ζ (t) =�Am�−1ζ (t)−�bmσ (t) , (4.123)

ζ (jTs) =[

0z(jTs)

], t ∈ [jTs , (j +1)Ts] . (4.124)

Consider now the function

V (t) = ζ�(t)�−�P�−1ζ (t)

over t ∈ [jTs , (j +1)Ts]. Since � is nonsingular and P is positive definite, �−�P�−1 ispositive definite and, hence,V (t) is a positive-definite function. It follows from Lemma 4.2.1and the relationship in (4.124) that

V (ζ (jTs)) = z�(jTs)P2z(jTs) ,

which, along with the upper bound in (4.119), leads to

V (ζ (jTs)) ≤ α . (4.125)

It follows from (4.123) that over t ∈ [jTs , (j +1)Ts]

V (t) = ζ�(t)�−�P�−1�Am�−1ζ (t)+ ζ�(t)�−�A�

m��−�P��−1ζ (t)

−2ζ�(t)�−�P�−1�bmσ (t)

= −ζ�(t)�−�Q�−1ζ (t)−2ζ�(t)�−�Pbmσ (t) .

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Using the upper bound from (4.112), one can derive over t ∈ [jTs , (j +1)Ts]

V (t) ≤ −λmin(�−�Q�−1)‖ζ (t)‖2 +2‖ζ (t)‖‖�−�Pbm‖� . (4.126)

Notice that for all t ∈ [jTs , (j +1)Ts], if

V (t)> α , (4.127)

we have

‖ζ (t)‖>√

α

λmax(�−�P�−1)= 2�‖�−�Pbm‖λmin(�−�Q�−1)

,

and the upper bound in (4.126) yields

V (t)< 0. (4.128)

It follows from (4.125), (4.127), and (4.128) that

V (t) ≤ α , ∀ t ∈ [jTs , (j +1)Ts] ,

and therefore

V ((j +1)Ts) = ζ�((j +1)Ts)(�−�P�−1)ζ ((j +1)Ts) ≤ α . (4.129)

Since

ξ ((j +1)Ts) = χ ((j +1)Ts)+ ζ ((j +1)Ts) , (4.130)

the equality in (4.122) and the upper bound in (4.129) lead to the following inequality:

ξ�((j +1)Ts)(�−�P�−1)ξ ((j +1)Ts) ≤ α .

Using the result of Lemma 4.2.1, one can derive

z�((j +1)Ts)P2z((j +1)Ts) ≤ ξ�((j +1)Ts)(�−�P�−1)ξ ((j +1)Ts) ≤ α ,

which implies that the upper bound in (4.119) holds for j +1.Next, it follows from (4.106), (4.122), and (4.130) that

y((j +1)Ts) = 1�1 ζ ((j +1)Ts) ,

and the definition of ζ ((j +1)Ts) in (4.121) leads to the following expression:

y((j +1)Ts) = 1�1 e�Am�

−1Ts

[0

z(jTs)

]

−1�1

∫ Ts

0e�Am�

−1(Ts−τ )�bmσ (jTs + τ )dτ .

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The upper bounds in (4.112) and (4.119) yield the following upper bound:

|y((j +1)Ts)| ≤ ‖η2(Ts)‖ ‖z(jTs)‖+∫ Ts

0|1�

1 e�Am�−1(Ts−τ )�bm| |σ (jTs + τ )|dτ

≤ ‖η2(Ts)‖√

α

λmax(P2)+κ(Ts)�= ς (Ts) ,

where η2(Ts) and κ(Ts) were defined in (4.77) and (4.78), and ς (Ts) was defined in (4.79).This confirms the upper bound in (4.118) for j + 1. Hence, (4.116)–(4.117) hold for alliTs ≤ t ′.

For all iTs+ t ≤ t ′, where 0 ≤ t ≤ Ts , using the expression from (4.113) we can writethat

y(iTs + t) = 1�1 e�Am�

−1t ξ (iTs)+1�1

∫ t

0e�Am�

−1(t−τ )�σ (iTs)dτ

−1�1

∫ t

0e�Am�

−1(t−τ )�bmσ (iTs + τ )dτ .

The upper bound in (4.112) and the definitions of η1(t), η2(t), η3(t), and η4(t) lead to thefollowing upper bound:

|y(iTs + t)| ≤ |η1(t)| |y(iTs)|+‖η2(t)‖ ‖z(iTs)‖+η3(t)|y(iTs)|+η4(t)� .

Taking into consideration (4.116)–(4.117), and recalling the definitions of β1(Ts), β2(Ts),β3(Ts), β4(Ts) in (4.81)–(4.82), for all 0 ≤ t ≤ Ts and for arbitrary nonnegative integer isubject to iTs + t ≤ t ′, we have

|y(iTs + t)| ≤ β1(Ts)ς (Ts)+β2(Ts)√

α

λmax(P2)+β3(Ts)ς (Ts)+β4(Ts)� .

Since the right-hand side coincides with the definition of γ0(Ts) in (4.83), for all t ∈ [0, t ′]we have the bound

|y(t)| ≤ γ0(Ts) ,

which along with the design constraint on Ts introduced in (4.107) yields

‖yt ′ ‖L∞ < γ0 .

This clearly contradicts the statement in (4.110). Therefore, ‖y‖L∞ < γ0, which proves(4.108). Further, it follows from Lemma 4.2.4 that

‖et‖L∞ ≤ ‖H2(s)‖L1

1−‖G(s)‖L1Lγ0 ,

which holds uniformly for all t ≥ 0 and therefore leads to the first upper bound in (4.109).To prove the second bound in (4.109), we note that from (4.98) and (4.100), it follows

that

u(s) = M(s)C(s)r(s)−M(s)σ (s)−C(s)A(s)d(s)

C(s)A(s)+ (1−C(s))M(s),

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which can be used along with the expression of uref (s) in (4.93) to derive

u(s)ref −u(s) = −H2(s)de(s)− H3(s)

C(s)σ (s)

= −H2(s)de(s)− H3(s)

M(s)

M(s)

C(s)σ (s) .

(4.131)

Hence, it follows from (4.103) and (4.131) that

‖uref −u‖L∞ ≤ L‖H2(s)‖L1‖yref −y‖L∞ +‖H3(s)/M(s)‖L1‖y‖L∞ ,

which, along with the bound in (4.108) and the first bound in (4.109), leads to the secondbound in (4.109). The proof is complete. �

Thus, the tracking error between y(t) and yref (t), as well as between u(t) and uref (t),is uniformly bounded by a constant proportional to Ts . This implies that during the tran-sient phase, one can achieve arbitrary improvement of tracking performance by uniformlyreducing Ts .

Remark 4.2.1 Notice that the parameterTs is the fixed time step in the definition of the adap-tive law. The adaptive parameters in σ (t) ∈ R

n take constant values during [iTs , (i+1)Ts) forevery i = 0,1, . . . . Reducing Ts imposes hardware (CPU) requirements, and Theorem 4.2.1further implies that the performance limitations are consistent with the hardware limitations.This in turn is consistent with the results in Chapter 2, where improvement of the transientperformance was achieved by increasing the adaptation rate in the projection-based adap-tive laws.

Remark 4.2.2 We notice that the ideal control signal

uid(t) = r(t)−σref (t)

is the one that leads to the desired system response

yid(s) =M(s)r(s)

by canceling the uncertainties exactly. Thus, the reference system in (4.90)–(4.91) has adifferent response as compared to the ideal one. It cancels only the uncertainties within thebandwidth of C(s), which can be selected compatible with the control channel specifica-tions. This is exactly what one can hope to achieve with any feedback in the presence ofuncertainties.

Remark 4.2.3 We notice that stability of H (s) is equivalent to stabilization of A(s) by

C(s)

M(s)(1−C(s)). (4.132)

Indeed, consider the closed-loop system, comprised of the system A(s) and the negativefeedback of (4.132). The closed-loop transfer function is

A(s)

1+A(s) C(s)M(s)(1−C(s))

. (4.133)

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m1 m2

k1 k2

b1 b2

x1(t) x2(t) d(t)

u(t)e−sτ

Figure 4.5: The two–cart MSD system.

Incorporating (4.67), one can verify that the denominator of the system in (4.133) is exactlyHd (s). Hence, stability of H (s) is equivalent to the stability of the closed-loop systemin (4.133). This implies that the class of systems A(s) that can be stabilized by the L1adaptive output feedback controller, given by (4.87), (4.88), and (4.89), is not empty. More-over, we note that methods from disturbance observer literature can be effectively used forparametrization of C(s) that would achieve stabilization of H (s) for a sufficiently broadclass of systems A(s) [155]. Research in this direction is underway.

Remark 4.2.4 We also notice that, while the feedback in (4.132) may stabilize the systemin (4.61) for some classes of unknown nonlinearities, it will not ensure uniform transientperformance in the presence of unknown A(s). On the contrary, the L1 adaptive controllerensures uniform transient performance for a system’s signals, both input and output, inde-pendent of the unknown nonlinearity and independent of A(s).

Remark 4.2.5 Finally, it is important to mention that the output predictor of the L1 adaptiveoutput feedback controller presented in this section can be modified similar to the statepredictor of the full-state feedback L1 controllers introduced in Sections 2.1.6 and 3.3.This modification can be used to tune the frequency response and robustness margins of theclosed-loop adaptive system. In general, this modification may require an increase in theadaptation sampling rate.

4.2.4 Simulation Example: Two-Cart Benchmark Problem

The two-cart mass-spring-damper example was originally proposed as a benchmark problemfor robust control design. In a paper published in 2006, Fekri,Athans, and Pascoal [52] used aslightly modified version of the original two-cart system to illustrate the design methodologyand performance of the Robust Multiple-Model Adaptive Control. Next, we will revisit thetwo-cart example with the L1 output feedback adaptive controller presented in this section.The reader will find additional explanations and simulations in [178].

The two-cart system is shown in Figure 4.5. The states x1(t) and x2(t) represent theabsolute position of the two carts, whose masses are m1 and m2, respectively (only x2(t)is measured), d(t) is a random colored disturbance force acting on the mass m2, and u(t)is the control force, which acts upon the mass m1. The disturbance force d(t) is modeledas a first-order (colored) stochastic process generated by driving a low-pass filter withcontinuous-time white noise ξ (s), with zero-mean and unit intensity, i.e., = 1, as follows:

d(s) = α

s+α ξ (s) , α > 0.

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The overall state-space representation is

x(t) = Ax(t)+Bu(t)+Lξ (t) ,

y(t) = Cx(t)+ θ (t) ,

where the state vector is

x�(t) = [x1(t) x2(t) x1(t) x2(t) d(t)

]� ,

and

A=

0 0 1 0 00 0 0 1 0

− k1m1

k1m1

− b1m1

b1m1

0k1m2

− k1+k2m2

b2m2

− b1+b2m2

1m2

0 0 0 0 −α

,

B� =[

0 0 1m1

0 0]

, L� = [0 0 0 0 α

],

C = [0 1 0 0 0

],

(4.134)

while θ (t) is additive sensor noise affecting the only measurement, and it is modeled aswhite noise, independent of ξ (t), and defined by

E{θ (t)} = 0, E{θ (t)θ (τ )} = 10−6δ(t− τ ) .

The following parameters in (4.134) are fixed and known:

m1 =m2 = 1, k2 = 0.15 , b1 = b2 = 0.1 , α = 0.1 ,

while the spring constant, k1, is unknown with known upper and lower bounds

0.25 ≤k1≤ 1.75 .

In addition to the uncertain spring stiffness, an unmodeled time delay τ , whose maximumpossible value is 0.05 s, is assumed to be present in the control channel.

The control objective is to design a control law u(t) so that the mass m2 tracks areference step signal r(t) following a desired model, while minimizing the effects of thedisturbance d(t) and the sensor noise θ (t).

For application of the L1 adaptive output feedback controller to the two-cart example,the design procedure described in [169] leads to

M(s) = 1

s3 +1.4s2 +0.17s+0.052,

C(s) = 0.18s+0.19

s5 +2.8s4 +3.3s2 +2.0s2 +0.66s+0.19,

while the sample time for adaptation is set to Ts = 1 ms.Figures 4.6 to 4.8 show the response of the closed-loop system with the L1 adaptive

output feedback controller to step-reference inputs of different amplitudes and for different

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0 100 200 300 400 500−0.2

0

0.2

0.4

0.6

0.8

1

1.2

time [s]

x 2 [m]

r(t)x

2(t)

(a) system output x2(t)

0 100 200 300 400 500−0.05

0

0.05

0.1

0.15

0.2

time [s]

u [N

]

(b) control signal u(t)

Figure 4.6: Closed-loop response to a step input of 1 m with k1 = 0.25 and τ = 0.05 s.

0 100 200 300 400 500−0.2

0

0.2

0.4

0.6

0.8

1

1.2

time [s]

x 2 [m]

r(t)x

2(t)


0 100 200 300 400 500

0

0.05

0.1

0.15

time [s]

u [N

]


Figure 4.7: Closed-loop response to a step input of 0.5 m with k1 = 0.25 and τ = 0.05 s.

0 100 200 300 400 500−0.2

0

0.2

0.4

0.6

0.8

1

1.2

time [s]

x 2 [m]

r(t)k

1=1.25;τ =0.05

k1=0.35;τ =0.05

k1=0.35;τ =0.02

k1=1.75;τ =0.04


0 100 200 300 400 500−0.05

0

0.05

0.1

0.15

0.2

time [s]

u [N

]


Figure 4.8: Closed-loop response to a step input of 1 m for different values of the unknownsk1 and τ .

values of the unknown parameters k1 and τ . As one can see, the L1 adaptive controllerdrives the massm2 to the desired position in about 15 s for arbitrary values of the unknownparameters while minimizing the effects of both the disturbance and the sensor noise presentin the system. Also, for a scaled reference input, the closed-loop system achieves scaledtransient response as expected (see Figures 4.6 and 4.7).

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Chapter 5

L1 Adaptive Controller forTime-Varying ReferenceSystems

Because of growing complexity of engineering systems, the closed-loop system may needto meet different performance specifications at different points of the operational envelope.This implies that the desired reference system behavior is time varying. A classical exampleto support this statement is flight control system design, which relies on gain scheduling ofthe control parameters across the flight envelope to meet different performance specificationsfor different flight regimes. This section presents the L1 adaptive control architecture fortime-varying reference systems. The presented solution ensures that in the presence offast adaptation, a single design of the L1 adaptive controller leads to uniform performancebounds with respect to a time-varying reference system, without the need for gain scheduling.

5.1 L1 Adaptive Controller for LinearTime-Varying Systems

In this section, we derive the L1 adaptive controller for LTV systems with its performancespecifications given via another LTV system. We prove that the fast adaptation ability ofthe L1 adaptive controller ensures that the signals of the closed-loop system remain closeto the corresponding signals of a bounded LTV reference system. We follow the frameworkof [171], where the methodology was first outlined and was applied to an aerial refuelingproblem for a racetrack maneuver.


Consider the class of systems

x(t) = Am(t)x(t)+b(t)(ωu(t)+ θ�(t)x(t)+σ (t))

), x(0) = x0 ,

y(t) = c�x(t) ,(5.1)

where x(t) ∈ Rn is the system state (measured); Am(t) ∈ R

n×n and b(t) ∈ Rn are a known

time-varying matrix and a known vector, respectively; c ∈ Rn is a known constant vector;

ω ∈ R is a constant unknown parameter representing the uncertainty in the system input

211

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212 Chapter 5. L1 Adaptive Controller for Time-Varying Reference Systems

gain; θ (t) ∈ Rn is the unknown system parameter vector while σ (t) ∈ R is the disturbance;

y(t) ∈ R is the system output; and u(t) ∈ R is the control input.The system above verifies the following assumptions.

Assumption 5.1.1 (Uniform Asymptotic Stability of Desired System) The matrixAm(t)is continuously differentiable and there exist positive constantsµA > 0, dA > 0, andµλ > 0,such that for all t ≥ 0, ‖Am(t)‖∞ ≤ µA, ‖Am(t)‖∞ ≤ dA, and Re[λi(Am(t))] ≤ −µλ ∀ i =1, . . . , n, where λi(Am(t)) is a pointwise eigenvalue of Am(t). Further, for all t ≥ 0, theequilibrium of the state equation

x = Amx(t) , x(t0) = x0

is exponentially stable, and the solution of

A�m(t)P (t)+P (t)Am(t) = −I

satisfies ‖P (t)‖∞ < 1.

Remark 5.1.1 The existence of dA in Assumption 5.1.1 is guaranteed by Lemma A.6.2.

Assumption 5.1.2 (Uniform boundedness of b(t) and its derivative) There exist positiveconstants µb, db > 0, such that ‖b(t)‖ ≤ µb and ‖b(t)‖ ≤ db.Assumption 5.1.3 (Strong controllability of (A(t), b(t))) The pair (A(t), b(t)) is stronglycontrollable.


ω ∈�= [ωl , ωu] , θ (t) ∈� , |σ (t)| ≤� , ∀ t ≥ 0, (5.2)

where 0<ωl <ωu are given known upper and lower bounds,� is a known convex compactset, and � ∈ R

+ is a known (conservative) bound of σ (t).

Assumption 5.1.5 (Uniform boundedness of the rate of variation of parameters) Weassume that θ (t) and σ (t) are continuously differentiable, and their derivatives are uniformlybounded:

‖θ (t)‖ ≤ dθ <∞ , |σ (t)| ≤ dσ <∞ , ∀ t ≥ 0.

In this section we present the L1 adaptive controller, which ensures that the systemoutput y(t) follows a given bounded piecewise-continuous reference signal r(t) ∈ R withuniform and quantifiable transient and steady-state performance bounds.



Let H be the input-to-state map of the system

x(t) = Am(t)x(t)+b(t)u(t) , x(0) = 0.

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5.1. L1 Adaptive Controller for LTV Systems 213

Then, the system in (5.1) can be rewritten as

x = H(ωu+ θ�x+σ )+xin ,

where xin(t) is the solution of

xin(t) = Am(t)xin(t) , xin(0) = x0 .

Notice that from Assumption 5.1.1 and Lemma A.6.2, it follows that ‖xin‖L∞ is bounded.The design of the L1 adaptive controller proceeds by considering a positive feedback

gain k > 0 and a strictly proper stable transfer function D(s), which lead, for all ω ∈�, toa strictly proper stable

C(s) � ωkD(s)

1+ωkD(s)(5.3)

with DC gain C(0) = 1. Let C denote the input-output map for the transfer function C(s).For the proofs of stability and performance bounds, the choice of k and D(s) needs

to ensure that the following L1-norm condition holds:

‖G‖L1L < 1, G � H (1−C) , (5.4)

where

L� maxθ∈� ‖θ‖1 , (5.5)

with � being the compact set defined in (5.2).

The elements of the L1 adaptive controller are introduced next.

State Predictor

We consider the following state predictor with time-varying Am(t) and b(t):

˙x(t) = Am(t)x(t)+b(t)(ω(t)u(t)+ θ�(t)x(t)+ σ (t)

), x(0) = x0 ,

y(t) = c�x(t) ,(5.6)

where x(t) ∈ Rn is the state vector of the predictor, while ω(t), σ (t) ∈ R and θ (t) ∈ R

n arethe adaptive estimates.

Adaptation Laws

The adaptive laws are given by

˙ω(t) = �Proj(ω(t),−x�(t)P (t)b(t)u(t)

), ω(0) = ω0 ,

˙θ (t) = �Proj

(θ (t),−x�(t)P (t)b(t)x(t)

), θ (0) = θ0 ,

˙σ (t) = �Proj(σ (t),−x�(t)P (t)b(t)

), σ (0) = σ0 ,

(5.7)

where x(t) � x(t) − x(t), � ∈ R+ is the adaptation gain, Proj(·, ·) denotes the projection

operator defined inAppendix B, and the symmetric positive definite matrixP (t) =P�(t)> 0

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was defined in Assumption 5.1.1. The projection operator ensures that ω(t) ∈�, θ (t) ∈�,|σ (t)| ≤�.

Control Law

The control law is generated as the output of the (feedback) system

u(s) = −kD(s)(η(s)− rg(s)) , (5.8)

where η(s) is the Laplace transform of η(t) � ω(t)u(t)+ θ�(t)x(t)+ σ (t), while rg(s) is theLaplace transform of

rg(t) � kg(t)r(t) , kg(t) � −1/(c�A−1m (t)b(t)) .

The L1 adaptive controller is defined via (5.6), (5.7), and (5.8), subject to the L1-normcondition in (5.4).

5.1.3 Analysis of L1 Adaptive Controller



xref (t) = Am(t)xref (t)+b(t)(ωuref (t)+ θ�(t)xref (t)+σ (t)

),

uref (s) = C(s)

ω(rg(s)−ηref (s)) ,

yref (t) = c�xref (t) , xref (0) = x0 ,

(5.9)

where xref (t) ∈ Rn is the reference system state vector and ηref (s) is the Laplace transform

of ηref (t) � θ�(t)xref (t) +σ (t). Notice that this reference system is not implementable asit depends upon the unknown quantities ω, θ (t), and σ (t). Therefore, it is used only foranalysis purposes. In particular, it helps to establish stability and performance bounds ofthe closed-loop adaptive system. The next lemma proves the stability of this closed-loopreference system.

Lemma 5.1.1 If the choice of k andD(s) verifies the L1-condition in (5.4), then the closed-loop reference system in (5.9) is BIBS stable with respect to r(t) and x0.

Proof. The closed-loop reference system in (5.9) can be rewritten as

xref = Gηref +HCrg+xin . (5.10)

Lemmas A.6.2 and A.7.5 imply

‖xref τ‖L∞ ≤ ‖G‖L1‖ηref τ‖L∞ +‖HC‖L1‖rg‖L∞ +‖xin‖L∞ . (5.11)

The definition of ηref (t) along with (5.5) gives

‖ηref τ‖L∞ ≤ L‖xref τ‖L∞ +‖στ‖L∞ .

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Substituting it into (5.11) and solving with respect to ‖xref τ‖L∞ leads to

‖xref τ‖L∞ ≤ ‖HC‖L1‖rg‖L∞ +‖G‖L1�+‖xin‖L∞1−‖G‖L1L

.

Since k and D(s) verify the L1-norm condition in (5.4), ‖xref ‖L∞ is uniformly bounded,and hence, the closed-loop system in (5.9) is BIBS stable. �


The error dynamics between the state predictor and the plant are given by

˙x(t) = Am(t)x(t)+b(t)(ω(t)u(t)+ θ�(t)x(t)+ σ (t)

), x(0) = 0, (5.12)


Lemma 5.1.2 The prediction error x(t) in (5.12) is bounded,

‖x‖L∞ ≤√

θm

λPmin�, (5.13)

where

θm � (ωu−ωl)2 +4maxθ∈� ‖θ‖2 +4�2 +4

λPmax

1− εP (maxθ∈� ‖θ‖dθ +�dσ ) , (5.14)

andλPmin � inf

t ∈ [0,∞),i = 1 . . .n

λi(P (t)) , λPmax � supt ∈ [0,∞),i = 1 . . .n

λi(P (t)) .


V (x(t), ω(t), θ (t), σ (t)) = x�(t)P (t)x(t)+ 1

�(ω2(t)+ θ�(t)θ (t)+ σ 2(t)) . (5.15)

Using the adaptation laws in (5.7) and Property B.2 of the Proj(·, ·) operator, we can derivethe following upper bound on the derivative of the Lyapunov function:

V (t) = x�(t)(P (t)Am(t)+A�

m(t)P (t)+ P (t))x�(t)+2x�(t)P (t)b(t)ω(t)u(t)

+2x�(t)P (t)b(t)θ�(t)x(t)+2x�(t)P (t)b(t)σ (t)

+ 2

�(ω(t) ˙ω(t)+ θ�(t) ˙

θ (t)+ σ (t) ˙σ (t))− 2

�

(θ�(t)θ (t)+ σ (t)σ (t)

)= −x�(t)

(I− P (t)

)x�(t)− 2

�

(θ�(t)θ (t)+ σ (t)σ (t)

)+2ω(t)

(x�(t)P (t)b(t)u(t)+Proj

(ω(t),−x�(t)P (t)b(t)u(t)

))+2θ�(t)

(x�(t)P (t)b(t)x(t)+Proj

(θ (t),−x�(t)P (t)b(t)x(t)

))+2σ (t)

(x�(t)P (t)b(t)+Proj

(σ (t),−x�(t)P (t)b(t)

))≤ −x�(t)(I− P (t))x(t)+ 2

�|θ�(t)θ (t)+ σ (t)σ (t)| .

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Projection ensures that θ (t) ∈�, ω(t) ∈�, and |σ (t)| ≤�, for all t ≥ 0, and therefore

maxt≥0

(1

�

(θ�(t)θ (t)+ ω2(t)+ σ 2(t)

))≤ 1

�

(4maxθ∈� ‖θ‖2 +4�2 + (ωu−ωl)2

)for all t ≥ 0. If at some t1 > 0 one has V (t1)> θm/�, then it follows from (5.14) and (5.15)that

x�(t1)P (t1)x(t1)> 4λPmax

�(1− εP )(maxθ∈� ‖θ‖dθ +�dσ ) .

Then, Lemma A.6.2 implies

x�(t1)(I− P (t1))x(t1) ≥ 1− εPλPmax

x�(t1)P (t1)x(t1)>4

�(maxθ∈� ‖θ‖dθ +�dσ ) . (5.16)

Notice that1

�|θ (t)θ (t)+ σ (t)σ (t)| ≤ 2

�(maxθ∈� ‖θ‖dθ +�dσ ) , ∀ t ≥ 0,

which, along with the bound in (5.16), leads to

V (t1)< 0.

Since x(0) = x(0), we can verify that

V (0) ≤ 1

�

(4maxθ∈� ‖θ‖2 +4�2 + (ωu−ωl)2

)<θm

�.

Therefore, we have

V (t) ≤ θm

�, ∀ t ≥ 0.

Since λPmin‖x(t)‖2 ≤ x�(t)P x(t) ≤ V (t), then

‖x(t)‖ ≤√

θm

λPmin�.

The result in (5.13) follows from the fact that this bound holds uniformly for all t ≥ 0. �Theorem 5.1.1 Given the system in (5.1) and the L1 adaptive controller defined via (5.6),(5.7), and (5.8), subject to the L1-norm condition in (5.4), we have

‖xref −x‖L∞ ≤ γ1 , (5.17)

‖uref −u‖L∞ ≤ γ2 , (5.18)

where

γ1 � ‖H‖L1κ0

1−‖G‖L1L

√θm

λPmin�, (5.19)

γ2 �∥∥∥∥C(s)

ω

∥∥∥∥L1

Lγ1 + κ0

ω

√θm

λPmin�, (5.20)

κ0 �(n−1∑i=0

‖Cai+1‖L1

∥∥∥∥ si

cnsn−1 +·· ·+ c1

∥∥∥∥L1

+∥∥∥∥ C(s)sn

cnsn−1 +·· ·+ c1

∥∥∥∥L1

)‖c�T ‖L∞ ,

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and ci , i = 1, . . . ,n, are the coefficients of an arbitrary Hurwitz polynomial p(s) � cnsn−1 +·· · + c1, while T (t) and ai(t) are, respectively, the transformation matrix and the coeffi-cients of the characteristic polynomial for the system given in (5.1), defined according toLemma A.11.2.

Proof. Let

η(t) � θ�(t)x(t)+σ (t) , η(t) � ω(t)u(t)+ θ�(t)x(t)+ σ (t) .


u(s) = −KD(s)(ωu(s)+η(s)− rg(s)+ η(s)) ,


u(s) = −C(s)

ω(η(s)+ η(s)− rg(s)) . (5.21)

Then, the system in (5.1) takes the form

x = Gη+HC(rg− η)+xin .

The expression above, together with (5.10), leads to

xref −x = Gηe+HCη , ηe(t) � θ�(t)(xref (t)−x(t)) . (5.22)

Lemma A.7.5 gives the following upper bound:

‖(xref −x)τ‖L∞ ≤ ‖G‖L1‖ηeτ‖L∞ +‖H‖L1‖(Cη)τ‖L∞ . (5.23)

From the definition of L in (5.5), it follows that ‖θ�(xref − x)τ‖L∞ ≤ L‖(xref − x)τ‖L∞ ,and hence ‖ηeτ‖L∞ ≤L‖(xref −x)τ‖L∞ . Substituting this back into (5.23), and solving for‖(xref −x)τ‖L∞ , one gets

‖(xref −x)τ‖L∞ ≤ ‖H‖L1

1−‖G‖L1L‖(Cη)τ‖L∞ .

Applying Lemma A.12.2 to the last term of this bound, we obtain

‖(xref −x)τ‖L∞ ≤ ‖H‖L1

1−‖G‖L1Lκ0‖xτ‖L∞ .

Taking into account the upper bound from Lemma 5.1.2, one gets

‖(xref −x)τ‖L∞ ≤ ‖H‖L1κ0

1−‖G‖L1L

√θm

λPmin�,

which holds uniformly for all t ≥ 0, leading to the bound in (5.17).To prove the bound in (5.18), we notice that from (5.9) and (5.21) one can derive


ω(ηe(s)− η(s)) .

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It follows from Lemma A.7.5 that

‖(uref −u)τ‖L∞ ≤∥∥∥∥C(s)

ω

∥∥∥∥L1

L‖(xref −x)τ‖L∞ + 1

ω

∥∥(Cη)τ∥∥

L∞ .

Using the upper bound on ‖(xref −x)τ‖L∞ and applying Lemma A.12.2 to the last term ofthis bound, we obtain

‖(uref −u)τ‖L∞ ≤∥∥∥∥C(s)

ω

∥∥∥∥L1

Lγ1 + κ0

ω

√θm

λPmin�,

which leads to the bound in (5.18) and completes the proof. �

Remark 5.1.2 It follows from the definition of γ1 and γ2 in (5.19) and (5.20) that one canachieve arbitrary desired performance bounds for a system’s signals, both input and output,simultaneously by increasing the adaptive gain.


To verify numerically the results proved in this section, we consider the following second-order dynamics:

x(t) = Am(t)x(t)+b(t)(ωu(t)+ θ (t)x(t)+σ (t)) , x(0) = x0 ,

y(t) = c�x(t) ,(5.24)

where

Am(t) =[

0 1−ω2

m(t) −2ζωm(t)

], b(t) =

[0

ω2m(t)

], c =

[10

]�,

with ζ = 0.7, and

ωm(t) = 1+0.4sin( π

40t)

.

In the simulations, we consider the system uncertainties ω and θ (t),

ω1 = 0.8 , θ1(t) = [2, 1]� ,

ω2 = 1.2 , θ2(t) = [2+ sin(0.3t), 0.5sin(0.5t)]� ,

and the disturbances σ (t),

σ1(t) = 1+3cos(0.5t) ,

σ2(t) = 3+ sin(0.5t)+0.5sin(t) ,

so that the compact sets can be conservatively chosen as�= [0.1, 3],�= {ϑ = [ϑ1, ϑ2]� ∈R

2 : ϑi ∈ [−4, 4], for all i = 1, 2}, �= 50.We implement the L1 adaptive controller according to (5.6), (5.7), and (5.8). In the

implementation of the control law we use the filter and feedback gain

k = 60, D(s) = 1

s,

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and we set the adaptation gain to � = 105. The time-varying desired system is given by

xid(t) = Am(t)xid(t)+b(t)kg(t)r(t) ,

yid(t) = c�xid(t) .

First, we verify the assumptions and the L1-norm upper bound from (5.4). Fromthe problem formulation in (5.24), one can easily see that Assumptions 5.1.2, 5.1.3, 5.1.4,and 5.1.5 are satisfied for the class of uncertainties and disturbances, introduced above. Toverify Assumption 5.1.1, we solve the Lyapunov equationA�

m(t)P (t)+P (t)Am(t) = −I forP (t) and obtain

P (t) = ζ

ωm(t) + ωm(t)4ζ + 1

4ζωm(t)1

2ω2m(t)

12ω2m(t)

14ζωm(t)

(1+ 1

ω2m(t)

) ,

which leads to

P (t) = ωm(t)

[ 14ζ − ζ

ω2m(t)

− 14ζω2

m(t)− 1ω3m(t)

− 1ω3m(t)

− 34ζω4

m(t)− 1

4ζω2m(t)

].

Notice that ωm(t) = π80 cos( π40 t) ≤ π

80 , which leads to maxt≥0(‖P (t)‖∞) = 0.55< 1. Thus,Assumption 5.1.1 is satisfied. We also compute L= 8, using the conservatively chosen �,and numerically compute the L1-norm of the map G using Definition A.7.6. This results in

‖G‖L1L= 0.9< 1.

Thus, the L1-norm condition holds for our choice of control design parameters.Figure 5.1 shows the simulation results for both ω1 and ω2 with θ (t) = θ1(t) and

σ (t) = σ1(t). Figure 5.2 shows the simulation results for both θ1(t) and θ2(t) with ω = ω1and σ (t) = σ1(t), and Figure 5.3 shows the simulation results for both disturbances σ1(t) andσ2(t) with ω = ω1 and θ (t) = θ1(t). From these results one can see that the fast adaptationability of the L1 adaptive controller ensures uniform transient performance for differentuncertainties and disturbances. We notice that while the system’s output remains close tothe desired reference signal in the presence of different uncertainties and disturbances, thecontrol signal changes significantly to ensure adequate compensation for the uncertaintiesand the disturbances.

Next, we test the tracking performance of the closed-loop adaptive system. We setthe reference signal to r(t) = sin

(π5 t)

and let ω = ω1, θ (t) = θ1(t), and σ (t) = σ1(t). Thesimulation results are shown in Figure 5.4. One can see that the closed-loop adaptive systemsystem has satisfactory tracking performance. It compensates for the uncertainties in thesystem and rejects the disturbance within the bandwidth of the control channel specified viaC(s), given in (5.3).

Figure 5.5 shows the simulation results for step-reference signals of different ampli-tudes. We observe that the system response is close to scaled response, similar to linearsystems.

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0 5 10 15 20−0.5

0

0.5

1

1.5

time [s]

r(t)y(t) for ω1

y(t) for ω2

yid(t)

(a) r(t), y(t), and yid(t)

0 5 10 15 20−6

−4

−2

0

2

4

time [s]

ω1ω2


Figure 5.1: Performance of the L1 adaptive controller for ω1 and ω2 with fixed θ1(t)and σ1(t).

0 5 10 15 20−0.5

0

0.5

1

1.5

time [s]

r(t)y(t) for θ1

y(t) for θ2

yid(t)


0 5 10 15 20−6

−4

−2

0

2

4

time [s]

θ1θ2


Figure 5.2: Performance of the L1 adaptive controller for θ1(t) and θ2(t) with fixed ω1and σ1(t).

0 5 10 15 20−0.5

0

0.5

1

1.5

time [s]

r(t)y(t) for σ1

y(t) for σ2

yid(t)


0 5 10 15 20−10

−5

0

5

time [s]

σ1σ2


Figure 5.3: Performance of the L1 adaptive controller for σ1(t) and σ2(t) with fixed ω1and θ1(t).

Next, we test the system performance in the presence of nonzero initialization error.We set the initial conditions of the system different from the initial conditions of the statepredictor: x0 = [0.5, 1]�, x0 = [1.5, 0.1]�. The simulation results in Figure 5.6 verifythe performance of the L1 adaptive controller in the presence of nonzero initializationerrors.

L1book2010/7/22page 221

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0 5 10 15 20−1.5

−1

−0.5

0

0.5

1

time [s]

r(t)y(t)

(a) r(t), y(t)

0 5 10 15 20−10

−5

0

5

time [s]


Figure 5.4: Performance of the L1 adaptive controller for r(t) = sin(π5 t).

0 5 10 15 20−1

0

1

2

3

time [s]

r1(t)r2(t)

(a) r(t), y(t)

0 5 10 15 20−10

−5

0

5

time [s]

r1(t)r2(t)


Figure 5.5: Performance of the L1 adaptive controller for step-reference signals.

0 5 10 15 20�0.5

0

0.5

1

1.5

2

time [s]

r(t)y(t)y(t)

(a) r(t), y(t), and y(t)

0 5 10 15 20−15

−10

−5

0

5

time [s]


Figure 5.6: Performance of the L1 adaptive controller in the presence of nonzero initializa-tion error.

Finally, we numerically test the robustness of the L1 adaptive controller to timedelays. Figure 5.7 shows the simulation results in the presence of time delay of 20 msfor the uncertainties considered above. One can see that the system has some expecteddegradation in the performance but remains stable. Moreover, the system output in thepresence of time delay remains close to the one in the absence of time delay for both typesof the above considered uncertainties.

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0 5 10 15 20−0.5

0

0.5

1

1.5

time [s]

r(t)y(t)

(a) ω1, θ1(t), σ1(t)

0 5 10 15 20−6

−4

−2

0

2

4

time [s]

(b) ω1, θ1(t), σ1(t)

0 5 10 15 20−0.5

0

0.5

1

1.5

time [s]

r(t)y(t)

(c) ω2, θ1(t), σ1(t)

0 5 10 15 20−6

−4

−2

0

2

time [s]

(d) ω2, θ1(t), σ1(t)

0 5 10 15 20−0.5

0

0.5

1

1.5

time [s]

r(t)y(t)

(e) ω1, θ2(t), σ1(t)

0 5 10 15 20−6

−4

−2

0

2

4

time [s]

(f) ω1, θ2(t), σ1(t)

0 5 10 15 20−0.5

0

0.5

1

1.5

time [s]

r(t)y(t)

(g) ω1, θ1(t), σ2(t)

0 5 10 15 20−8

−6

−4

−2

0

time [s]

(h) ω1, θ1(t), σ2(t)

Figure 5.7: Performance of the L1 adaptive controller with time delay of 20 ms.

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5.2. Controller for Nonlinear Systems with Unmodeled Dynamics 223

It is important to emphasize that in the simulations above there is no retuning ofthe L1 adaptive controller from one scenario to another, and the same constant controlparameters are used for every simulation. The time-varying nature of the desired referencesystem is reflected in the state predictor, which uses Am(t) and b(t).

5.2 L1 Adaptive Controller for Nonlinear Systems in thePresence of Unmodeled Dynamics

This section considers the class of uncertain systems with time- and state-dependent un-known nonlinearities and unmodeled dynamics. The L1 adaptive controller yields uniformperformance bounds with respect to a bounded LTV reference system, which hold semiglob-ally [91].



x(t) = Am(t)x(t)+b(t)(µ(t)+f (t ,x(t),z(t))

), x(0) = x0 ,

xz(t) = g(t ,xz(t),x(t)) , xz(0) = xz0 ,

z(t) = go(t ,xz(t)) ,

y(t) = c�x(t) ,

(5.25)

where x(t) ∈ Rn is the system state; Am(t) ∈ R

n×n and b(t) ∈ Rn are a known time-varying

matrix and a vector, respectively; c ∈ Rn is a known constant vector; y(t) ∈ R is the system

output; f : R × Rn × R

l → R is an unknown nonlinear map, which represents systemnonlinearities; xz(t) ∈ R

m and z(t) ∈ Rl are the state and the output of the unmodeled

nonlinear dynamics; g : R × Rm× R

n → Rm, g0 : R × R

m → Rl are unknown nonlinear

maps continuous in their arguments; and µ(t) ∈ R is the output of the following system:

µ(s) = F (s)u(s) ,

whereu(t) ∈ R is the control signal, andF (s) is an unknown BIBO-stable and proper transferfunction with known sign of its DC gain. The initial condition x0 is assumed to be inside anarbitrarily large known set, so that ‖x0‖∞ ≤ ρ0 <∞ with known ρ0 > 0.

LetX� [x�, z�]�, and with a slight abuse of notation let f (t ,X) � f (t ,x,z). Similarto the previous section, Assumptions 5.1.1, 5.1.2, and 5.1.3 hold. In addition, we imposethe following assumptions.

Assumption 5.2.1 (Uniform boundedness of f (t ,0,0)) There exists B > 0, such that|f (t ,0)| ≤ B holds for all t ≥ 0.

Assumption 5.2.2 (Semiglobal uniform boundedness of partial derivatives) For arbit-rary δ > 0, there exist positive constants dfx (δ) > 0 and dft (δ) > 0 independent of time,such that for all ‖X‖∞ ≤ δ the partial derivatives of f (t ,X) are piecewise-continuous andbounded, ∥∥∥∥∂f (t ,X)

∂X

∥∥∥∥1≤ dfx (δ),

∣∣∣∣∂f (t ,X)

∂t

∣∣∣∣≤ dft (δ) .

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Assumption 5.2.3 (Stability of unmodeled dynamics) The xz-dynamics are BIBO stablewith respect to both initial conditions xz0 and input x(t), i.e., for arbitrary initial condition xz0there exist L1, L2 > 0 such that for all t ≥ 0

‖zt‖L∞ ≤ L1‖xt‖L∞ +L2 .

Assumption 5.2.4 (Partial knowledge of actuator dynamics) There exists LF > 0 ver-ifying ‖F (s)‖L1 ≤ LF . Also, we assume that there exist known constants ωl , ωu ∈ R

satisfying

0< ωl ≤ F (0) ≤ ωu ,


Next, we present the L1 adaptive controller that ensures that the system output y(t)tracks a given bounded piecewise-continuous reference signal r(t) ∈ R with uniform andquantifiable performance bounds.



Let H be the input-to-state map of the system

x(t) = Am(t)x(t)+b(t)u(t) , x(0) = 0.

Then, the system in (5.25) can be rewritten as

x = Hµ+Hf +xin ,

where f denotes f (t ,x(t),z(t)), and xin(t) is the solution of

xin(t) = Am(t)xin(t) , xin(0) = x0 .

Notice that from Assumption 5.1.1 and Lemma A.6.2 it follows that ‖xin‖L∞ is bounded,and ‖xin‖L∞ ≤ ρin, where

ρin � max‖x0‖∞≤ρ0‖xin‖L∞ .

Further, let

Lδ � δ(δ)

δdfx (δ(δ)) , δ(δ) � max{δ+ γ1, L1(δ+ γ1)+L2} , (5.26)

wheredfx (·) was introduced inAssumption 5.2.2, and γ1> 0 is an arbitrary positive constant.Similar to Section 5.1.2, the design of L1 adaptive controller proceeds by considering

a positive feedback gain k > 0 and a strictly proper stable transfer function D(s), whichlead, for all F (s) ∈ F�, to a strictly proper stable

C(s) � kF (s)D(s)

1+kF (s)D(s), (5.27)

with DC gain C(0) = 1. Also, let C denote the input-output map for C(s).

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For the proofs of stability and performance bounds, the choice of k andD(s) needs toensure that, for a given ρ0, there exists ρr > ρin, such that the following L1-norm conditioncan be verified:

‖G‖L1 <ρr −‖HCkg‖L1‖r‖L∞ −ρin

Lρr ρr +B, (5.28)

where G � H (1−C), and kg(t) � −1/(c�A−1m (t)b(t)) is the feedforward gain required for

tracking the reference signal r(t).To streamline the subsequent analysis of stability and performance bounds, we in-

troduce the following notation. Similar to the previous section, let rg(t) � kg(t)r(t). Alsolet Cu(s) � C(s)/F (s), and let Cu be the input-output map for Cu(s). Note that Cu(s) is astrictly proper BIBO-stable transfer function. Next, let ci , i = 1, . . . ,n, be the coefficients ofan arbitrary Hurwitz polynomial cnsn−1 +·· ·+ c1. Let T (t) be the transformation matrix,reducing (Am(t),b(t)) to its controllable canonical form, and let ai , i = 1, . . . ,n, be thecoefficients of the characteristic polynomial, as discussed in Lemma A.11.2.

Let

ρ � ρr + γ1

and

γ1 � ‖HF ‖L1κ0

1−‖G‖L1Lρrγ0 +β , (5.29)

where F is the input-to-state map of the system F (s) and κ0 is defined as

κ0 �(n−1∑i=0

‖Cuai+1‖L1

∥∥∥∥ si

cnsn−1 +·· ·+ c1

∥∥∥∥L1

+∥∥∥∥ Cu(s)sn

cnsn−1 +·· ·+ c1

∥∥∥∥L1

)‖c�T ‖L∞ ,

while β and γ0 are arbitrary small positive constants such that γ1 ≤ γ1. Moreover, let

ρu � ρur +γ2 ,


ρur � ‖Cu(s)‖L1 (‖kg‖L∞‖r‖L∞ +Lρr ρr +B) ,

γ2 � ‖Cu(s)‖L1Lρr γ1 +κ0γ0 . (5.30)

Finally, using the conservative knowledge of F (s), let

�1 � LρL2 +B+ ε , (5.31)

�2 � maxF (s)∈F�

‖F (s)− (ωl+ωu)/2‖L1ρu , (5.32)

��1 +�2 , (5.33)

ρu � ‖ksD(s)‖L1 (ρuωu+Lρρ+�+‖kg‖L∞‖r‖L∞ ) ,

where ε > 0 is an arbitrarily small positive constant.


The elements of the L1 adaptive controller are introduced next.

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State Predictor

The following state predictor is used for derivation of the adaptive laws:

˙x(t) = Am(t)x(t)+b(t)(ω(t)u(t)+ θ (t)‖xt‖L∞ + σ (t)

), x(0) = x0 ,

y(t) = c�x(t) ,(5.34)

where x(t) ∈ Rn is the state of the predictor, while ω(t), θ (t), σ (t) ∈ R are the adaptive

estimates.

Adaptation Laws

The adaptive laws are defined via the projection operator as follows:

˙ω(t) = �Proj(ω(t),−x�(t)P (t)b(t)u(t)

), ω(0) = ω0 ,

˙θ (t) = �Proj

(θ (t),−x�(t)P (t)b(t)‖xt‖L∞

), θ (0) = θ0 ,

˙σ (t) = �Proj(σ (t),−x�(t)P (t)b(t)

), σ (0) = σ0 ,

(5.35)

where x(t) � x(t)−x(t), � ∈ R+ is the adaptation gain, and the symmetric positive definite

matrix P (t) = P�(t)> 0 solves the Lyapunov equationA�m(t)P (t)+P (t)Am(t) = −I. The

projection operator ensures that ω(t) ∈�� [ωl , ωu], θ (t) ∈�� [−Lρ , Lρ] and |σ (t)| ≤�.

Control Law

The control law is generated as the output of the following (feedback) system:

u(s) = −kD(s)(η(s)− rg(s)) , (5.36)

where rg(s) and η(s) are the Laplace transforms of rg(t) and η(t) � ω(t)u(t)+ θ (t)‖xt‖L∞ +σ (t), respectively, while k and D(s) were introduced in (5.27).

The L1 adaptive controller is defined via (5.34), (5.35), and (5.36), subject to theL1-norm condition in (5.28).




xref (t) = Am(t)xref (t)+b(t)(µref (t)+f (t ,xref (t),z(t))

), xref (0) = x0 ,


uref (s) = −Cu(s)(ηref (s)− rg(s)) ,


(5.37)

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where xref (t) ∈ Rn is the reference system state vector, ηref (s) is the Laplace transform

of ηref (t) � f (t ,xref (t),z(t)). Notice that this reference system is not implementable, asit contains the unknowns f (t ,xref (t),z(t)) and F (s). This system is used only for analysispurposes. The next lemma proves the stability of this closed-loop reference system subject toan assumption on z(t), which will be verified later in the proof of stability and performancebounds of the closed-loop adaptive system (Theorem 5.2.1).

Lemma 5.2.1 For the closed-loop reference system given in (5.37), subject to the L1-normcondition in (5.28), if for some τ ≥ 0

‖zτ‖L∞ ≤ L1(‖xref τ‖L∞ +γ1)+L2 , (5.38)

then the following bounds hold:

‖xref τ‖L∞ < ρr , ‖uref τ‖L∞ < ρur . (5.39)

Proof. The response of the closed-loop reference system in (5.37) can be rewritten as

xref = Gηref +HCkgr+xin . (5.40)

Lemmas A.6.2 and A.7.5, along with the fact ‖xin‖L∞ ≤ ρin, imply that

‖xref τ‖L∞ ≤ ‖G‖L1‖ηref τ‖L∞ +‖HCkg‖L1‖r‖L∞ +ρin . (5.41)

Next, we use a contradictive argument. Assume that the first bound in (5.39) does not hold.Then, since ‖xref (0)‖∞ = ‖x0‖∞ ≤ ρ0 < ρr and xref (t) is continuous, there exists a timeinstant τ1 ∈ (0, τ ], such that

‖xref (t)‖∞ < ρr , ∀ t ∈ [0,τ1) , and ‖xref (τ1)‖∞ = ρr ,

which implies that

‖xref τ1‖L∞ = ρr . (5.42)

It follows from (5.38) that ‖zτ1‖L∞ ≤ L1(ρr +γ1)+L2, which implies

‖Xref τ1‖L∞ =∥∥∥∥∥[xrefz

]τ1

∥∥∥∥∥L∞

≤ max{ρr +γ1, L1(ρr +γ1)+L2} ≤ ρr (ρr ) .

Assumption 5.2.2 further implies that, for all ‖Xref ‖∞ ≤ ρr (ρr ), the following inequalityholds:

|f (t ,Xref )−f (t ,0)| ≤ dfx (ρr (ρr ))‖Xref ‖∞ , ∀ t ∈ [0, τ ] .

Further, Assumption 5.2.1 and the definition of Lδ in (5.26) lead to

‖ηref τ1‖L∞ ≤ dfx (ρr (ρr ))ρr (ρr )+B = Lρr ρr +B . (5.43)

Thus, from (5.41) we get

‖xref τ1‖L∞ ≤ ‖G‖L1 (Lρr ρr +B)+‖HCkg‖L1‖r‖L∞ +ρin .

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From the L1-norm condition in (5.28) one has

‖G‖L1 (Lρr ρr +B)+‖HCkg‖L1‖r‖L∞ +ρin < ρr ,

which implies ‖xref τ1‖L∞ < ρr . This contradicts (5.42), which proves the first boundin (5.39).

Because this bound is strict and holds uniformly for all τ1 ∈ (0, τ ], one can rewrite(5.43) as a strict inequality,

‖ηref τ‖L∞ < Lρr ρr +B .

Then, from (5.37) it follows that

‖uref τ‖L∞ ≤ ‖Cu(s)‖L1 (‖rg‖L∞ +‖ηref τ‖L∞ )

< ‖Cu(s)‖L1 (‖kg‖L∞‖r‖L∞ +Lρr ρr +B) = ρur ,



In this section, we transform the original nonlinear system with unmodeled dynamicsin (5.25) into an equivalent (semi-)linear time-varying system with unknown time-varyingparameters and disturbances. This transformation requires us to impose the following as-sumptions on the signals of the system: the control signal u(t) is continuous, and moreoverthe following bounds hold

‖xτ‖L∞ ≤ ρ , ‖uτ‖L∞ ≤ ρu , ‖uτ‖L∞ ≤ ρu; (5.44)

these will be verified later in the proof of Theorem 5.2.1. Next we construct the equivalentLTV system in two steps.

First Equivalent System Consider the system in (5.25). From (5.25) and the first twobounds in (5.44), it follows that ‖xτ‖L∞ is bounded for all τ ∈ [0, ∞). Thus, Lemma A.9.1implies that there exist continuous θ (t) and σ1(t) with (piecewise)-continuous derivative,defined over t ∈ [0, τ ], such that

|θ (t)|< Lρ , |θ (t)| ≤ dθ , (5.45)

|σ1(t)|<�1 , |σ1(t)| ≤ dσ1 ,

and

f (t ,x(t),z(t)) = θ (t)‖xt‖L∞ +σ1(t) ,

where Lρ and �1 are as defined in (5.26) and (5.31), while the algorithm for computingdθ > 0, dσ1 > 0 is derived in the proof of Lemma A.9.1. Thus, the system in (5.25) can berewritten over t ∈ [0, τ ] as

x(t) = Am(t)x(t)+b(t)(µ(t)+ θ (t)‖xt‖L∞ +σ1(t)

), x(0) = x0 ,

y(t) = c�x(t) .(5.46)

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Second Equivalent System Keeping in mind the assumptions on u(t) and its derivativein (5.44), and using Lemma A.10.1, we can rewrite the signal µ(t) as

µ(t) = ωu(t)+σ2(t) ,

whereω ∈ (ωl , ωu) is an unknown constant and σ2(t) is a continuous signal with (piecewise)-continuous derivative, defined over t ∈ [0, τ ], such that

|σ2(t)| ≤�2 , |σ2(t)| ≤ dσ2 ,

with �2 as introduced in (5.32), and dσ2 � ‖F (s) − (ωl +ωu)/2‖L1ρu. This implies thatone can rewrite the system in (5.46) over t ∈ [0, τ ] as

x(t) = Am(t)x(t)+b(t)(ωu(t)+ θ (t)‖xt‖L∞ +σ (t)

), x(0) = x0 ,

y(t) = c�x(t) ,(5.47)

where σ (t) � σ1(t) +σ2(t) is an unknown time-varying signal subject to |σ (t)| <�, with� as introduced in (5.33), |σ (t)|< dσ , dσ � dσ1 +dσ2 , and θ (t) as introduced in (5.45).


Using (5.47), one can write the error dynamics over t ∈ [0, τ ],

˙x(t) = Am(t)x(t)+b(t)(ω(t)u(t)+ θ (t)‖xt‖L∞ + σ (t)

), x(0) = 0, (5.48)


Lemma 5.2.2 If

‖xτ‖L∞ ≤ ρ , ‖uτ‖L∞ ≤ ρu , ‖uτ‖L∞ ≤ ρu ,

then

‖xτ‖L∞ ≤√θm(ρ,ρu,ρu)

λPmin�, (5.49)

where

θm(ρ,ρu,ρu) � (ωu−ωl)2 +4L2ρ +4�2 +4

λPmax

1− εP (Lρdθ +�dσ ),

and

λPmin � inft ∈ [0,∞),i = 1 . . .n

λi(P (t)) , λPmax � supt ∈ [0,∞),i = 1 . . .n

λi(P (t)) .


V (x(t), ω(t), θ (t), σ (t)) = x�(t)P (t)x(t)+ 1

�(ω2(t)+ θ2(t)+ σ 2(t)) . (5.50)

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Using the adaptation laws in (5.35) and Property B.2 for the Proj(·, ·) operator, we com-pute the upper bound on the derivative of the Lyapunov function similar to the proof ofLemma 5.1.2:

V (t) ≤ −x�(t)(I− P (t))x(t)+ 2

�|θ (t)θ (t)+ σ (t)σ (t)| .

Let t1 ∈ (0, τ ] be the time instant when the first discontinuity of θ (t) or σ (t) occurs, or t1 = τif there are no discontinuities. Consider the Lyapunov function candidate given in (5.50).Notice that

maxt∈[0, t1]

1

�

(ω2(t)+ θ2(t)+ σ 2(t)

)≤ (ωu−ωl)2 +4L2

ρ +4�2

�<θm(ρ,ρu,ρu)

�.

This inequality, along with the fact that x(0) = 0, leads to

V (0)<θm(ρ,ρu,ρu)

�.

If at arbitrary time t2 ∈ [0, t1] we have

V (t2)>θm(ρ,ρu,ρu)

�,

then it follows that

x�(t2)P (t2)x(t2)> 4λPmax

�(1− εP )(Lρdθ +�dσ ) .

Further, from Lemma A.6.2, one can write

x�(t2)(I− P (t2))x(t2) ≥ 1− εPλPmax

x�(t2)P (t2)x(t2)>4

�(Lρdθ +�dσ ) . (5.51)

Moreover, notice that

1

�|θ (t)θ (t)+ σ (t)σ (t)| ≤ 2

�(Lρdθ +�dσ ) ,


V (t2)< 0.

Therefore, we have

V (t2) ≤ θm(ρ,ρu,ρu)

�.

The continuity of V (t) allows for repeating these derivations for all points of discontinuityof θ (t) or σ (t), which leads to the following uniform bound:

V (t) ≤ θm(ρ,ρu,ρu)

�, ∀ t ∈ [0,τ ] .

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This further implies

‖x(t)‖ ≤√θm(ρ,ρu,ρu)

λPmin�, ∀ t ∈ [0,τ ] ,

which leads to the bound in (5.49). �

Theorem 5.2.1 If the adaptive gain verifies the lower bound

� ≥ θm(ρ,ρu,ρu)

λPminγ20

, (5.52)

where γ0 > 0 is an arbitrary constant introduced in (5.30), then the following bounds hold:

‖uref ‖L∞ ≤ ρu , (5.53)

‖xref ‖L∞ ≤ ρr , (5.54)

‖x‖L∞ ≤ γ0 , (5.55)

‖xref −x‖L∞ ≤ γ1 , (5.56)

‖uref −u‖L∞ ≤ γ2 . (5.57)

Proof. We prove (5.56) and (5.57) following a contradicting argument. Assume that (5.56)and (5.57) do not hold. Then, since

‖xref (0)−x(0)‖∞ = 0, ‖uref (0)−u(0)‖∞ = 0,

continuity of xref (t), x(t), uref (t), u(t) implies that there exists time τ > 0 for which

‖xref (t)−x(t)‖∞ < γ1 , ‖uref (t)−u(t)‖∞ < γ2 , ∀ t ∈ [0,τ ),

and

‖xref (τ )−x(τ )‖∞ = γ1 , or ‖uref (τ )−u(τ )‖∞ = γ2 .

This implies that at least one of the following equalities holds:


The first equality above leads to

‖zτ‖L∞ ≤ L1(‖xref τ‖L∞ +γ1)+L2 .

Then, since all the conditions of Lemma 5.2.1 hold, the following bounds are valid:


which in turn leads to

‖xτ‖L∞ ≤ ρr +γ1 ≤ ρ , ‖uτ‖L∞ ≤ ρur +γ2 = ρu . (5.60)

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Further, consider the control law in (5.36). From the properties of the projection operator,we have

‖ητ‖L∞ ≤ ωuρu+Lρρ+� ,

and consequently

‖uτ‖ ≤ ‖ksD(s)‖L1 (ωuρu+Lρρ+�+‖kg‖L∞‖r‖L∞ ) = ρu . (5.61)

Then, since all the conditions of Lemma 5.2.2 hold, selection of � following (5.52) gives

‖xτ‖L∞ ≤ γ0 . (5.62)

Next, let η(t) and η(t) be defined as

η(t) � θ (t)‖ x t ‖L∞ +σ (t) , η(t) � ω(t)u(t)+ θ (t)‖xt‖L∞ + σ (t) .

From the upper bounds in (5.60) and the bound in (5.61), it follows that, for all t ∈ [0,τ ],the following equality holds:

η(t) = f (t ,x(t),z(t)) .

Then, notice that, for all t ∈ [0, τ ], we have

η(t) = µ(t)+ η(t)+η(t) ,

which implies that the control law in (5.36) can be written as

u(s) = −Cu(s)(η(s)+η(s)− rg(s)) . (5.63)

Further, the system in (5.25) can be rewritten as

x = Gη−HCη+HCrg+xin .

Recall that in (5.40) the reference system is presented as

xref = Gηref +HCrg+xin .

The two expressions above yield

xref −x = G(ηref −η)+HCη . (5.64)

Notice now that, from Assumption 5.2.3 and (5.60), it follows that

‖zτ‖L∞ ≤ L1(ρr +γ1)+L2 ,

which, along with (5.60), leads to

‖Xτ‖L∞ ≤ max{ρr +γ1, L1(ρr +γ1)+L2} ≤ ρr (ρr ) .

Similarly, one can show that

‖Xref τ‖L∞ ≤ ρr (ρr ) .

L1book2010/7/22page 233

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�

�

�

�

�


Thus, using Assumption 5.2.2, one can write

‖(ηref −η)τ‖L∞ ≤ dfx (ρr (ρr ))‖(xref −x)τ‖L∞ ,

and since dfx (ρr (ρr ))< Lρr , it follows that


From (5.64), it follows that

‖(xref −x)τ‖L∞ ≤ ‖G‖L1Lρr‖(xref −x)τ‖L∞ +‖HF ‖L1‖(Cuη)τ‖L∞ .

The L1-norm condition in (5.28) implies ‖G‖L1Lρr < 1, which allows for the followingderivation:

‖(xref −x)τ‖L∞ ≤ ‖HF ‖L1

1−‖G‖L1Lρr‖(Cuη)τ‖L∞ .

Since Cu(s) is a strictly proper BIBO-stable transfer function, application of Lemma A.40to the linear time-varying prediction error dynamics in (5.48) yields

‖(Cuη)τ‖L∞ ≤ κ0‖xτ‖L∞ ,

which implies that

‖(xref −x)τ‖L∞ ≤ ‖HF ‖L1κ0

1−‖G‖L1Lρr‖xτ‖L∞ .

This bound, along with (5.62), leads to

‖(xref −x)τ‖L∞ ≤ ‖HF ‖L1κ0

1−‖G‖L1Lρrγ0 = γ1 −β < γ1 . (5.65)

Thus, we obtain a contradiction to the first equality in (5.58).To show that the second equation in (5.58) also cannot hold, consider (5.37) and

(5.63), which lead to

uref (s)−u(s) = −Cu(s)(ηref (s)−η(s))+Cu(s)η(s) .

Using the bounds on ‖(ηref −η)τ‖L∞ and ‖(Cuη)τ‖L∞ , one can write

‖(uref −u)τ‖L∞ ≤ ‖Cu(s)‖L1Lρr‖(xref −x)τ‖L∞ +κ0‖xτ‖L∞ .

Finally, the upper bounds on ‖xτ‖L∞ and ‖(xref − x)τ‖L∞ in (5.62) and (5.65), togetherwith the above inequality, lead to

‖(uref −u)τ‖L∞ ≤ ‖Cu(s)‖L1Lρr (γ1 −β)+κ0γ0 < γ2 , (5.66)

which contradicts the second equality in (5.58), which implies that the upper bounds in (5.65)and (5.66) hold uniformly. The upper bound in (5.55) follows from (5.62) directly. The upperbounds in (5.53) and (5.54) follow from (5.59). �

Remark 5.2.2 It follows from (5.52) that one can prescribe an arbitrary desired performancebound γ0 by increasing the adaptive gain, which further implies, from (5.29) and (5.30), that

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0 5 10 15 20 25 301

1.5

2

2.5

3

3.5

4

4.5

time [s]

(a) ωm(t)

0 5 10 15 20 25 300

5

10

15

20

25

time [s]

(b) bm(t)

Figure 5.8: Plots of ωm(t) and bm(t).

one can achieve arbitrarily small γ1 and γ2 for the system’s signals, both input and output,simultaneously.


To illustrate the results presented in this chapter, consider the system dynamics in (5.25),with

Am(t) =[

0 1−ω2

m(t) −2ζmωm(t)

], b(t) =

[0

bm(t)

],

c = [1 0

],

ωm(t) =

1+20(1− e−0.01t ) , t ∈ [0, 15] ,21−20e−0.15 +0.5sin(0.4e−0.15(t−15)) , t ∈ (15, 15+ e0.155π/4] ,21.5−20e−0.15 , t > 15+ e0.155π/4,

and ζm = 0.7, bm(t) = ω2m(t)+0.2ω2

m(t) sin(0.2πt). The plots of ωm(t) and bm(t) are givenin Figure 5.8. One can see that the natural frequency of the system is continuously growingwith time and is changing from 1 to ≈ 4.5. Also, bm(t) is significantly changing within theinterval [1, 20] s of the simulation time.

The unmodeled dynamics are given by the Lorenz attractor:

g(t ,xz,x) = ts σl(xz2 −xz1 )rlxz1 −xz2 −xz1xz3 +klx1

xz1xz2 −blxz3

,

go(t ,xz) = c�l xz,where ts = 0.1 is a time-scaling coefficient; σl = 10, rl = 28, bl = 8/3 are the Lorenzsystem parameters; and kl = 50 and c�l = [0, 1/15, 0] are the input and the output gains,respectively. To illustrate the behavior of the unmodeled dynamics, we consider it separatelyfrom the system in (5.25) and excite it with the step-impulse signal of the form x(t) =[u(t)−u(t−1), 0], where u(t) is a unit step function. The response is given in Figure 5.9.One can see that it has a stable limit cycle, biased from the equilibrium. Moreover, theresponse to the input signal is aggressive and has a relatively high derivative.

L1book2010/7/22page 235

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0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

1.5

2

time [s]

x1(t)z(t)

Figure 5.9: Response of the unmodeled dynamics to x = [u(t)−u(t−1), 0]�.

To illustrate the performance of L1 adaptive controller, we consider two scenarioswith different unmodeled actuator dynamics and uncertainties:

– Scenario 1: Let

F1(s) = 6400

s2 +112s+6400

and

f (t ,x,z) = x21 + sin(x1)+ z+d(t),

where d(t) = 0.5sin(0.3πt)+0.3sin(0.2πt).

– Scenario 2: Let

F2(s) = 10000

s2 +100s+10000

and

f (t ,x,z) = 2x21 +x1x2 + z+d(t),

where d(t) = 0.7sin(0.1πt)+0.1sin(0.2πt).

For the implementation of the L1 adaptive controller, we set

k = 25, D(s) = 100

s(s2 +140s+100).

Further, we set the adaptive gain and the projection bounds to be�= 105,�= [−100, 100],�= [0.1, 10], �= 100.

The simulation results of the L1 adaptive controller are shown in Figures 5.10–5.14.Figure 5.10 shows the simulations results for Scenario 1 for a series of step-reference signals,introduced at t1 = 0 s, t2 = 10 s, t3 = 15 s. One can see that y(t) = x1(t) tracks r(t), and theclosed-loop system is behaving close to the time-varying ideal system xid(t), given by

xid(t) = Am(t)xid(t)+b(t)kg(t)r(t) ,

yid(t) = c�xid(t) .

L1book2010/7/22page 236

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0 5 10 15 20−0.2

0

0.2

0.4

0.6

0.8

1

1.2

time [s]

r(t)x1(t)xid1(t)

(a) Time response of first state

0 5 10 15 20−4

−3

−2

−1

0

1

2

time [s]

(b) Control history

0 5 10 15 20−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

time [s]

(c) Time response of second state

0 5 10 15 20−30

−20

−10

0

10

20

30

time [s]

(d) Derivative of the control signal

0 5 10 15 20−1

0

1

2

3

4

5

time [s]

z(t)f (t, x(t), z(t))

(e) Output of the unmodeled dynamics and timehistory of the nonlinearities contribution

0 5 10 15 20�3

�2

�1

0

1

2

3

4

time [s]

ω(t)

σ(t)

θ(t)

(f) Estimations

Figure 5.10: Performance of the L1 adaptive controller for Scenario 1.

Notice that the desired time constant changes its value from 1 s to ≈ 1/4 s, and the closed-loop system response reflects this change by decreasing the settling time for step commandsat t2 and t3.

Figure 5.11 shows the results for Scenario 2 for two series of step signals with differentamplitudes r1(t) ≡ 1, r2(t) ≡ 0.5. One can see that the transient for both commands is scaled,similar to linear systems response. Figure 5.12 shows the tracking results for Scenarios 1and 2 with the reference signal r(t) = sin(t). One can see that the closed-loop system hastracking performance very close to ideal.

Next we simulate the closed-loop system with nonzero initialization error. We setthe initial condition of the system to x0 = [0.5, 0.5]� and the initial condition of the statepredictor to x0 = [1.5, − 0.2]�. We let the initial conditions of unmodeled dynamics be

L1book2010/7/22page 237

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0 5 10 15 20−0.2

0

0.2

0.4

0.6

0.8

1

1.2

time [s]

r(t)x1(t)xid1(t)


0 5 10 15 20−5

−4

−3

−2

−1

0

1

2

3

time [s]

r1r2

(b) Control history

0 5 10 15 20−2

−1

0

1

2

3

time [s]

r1r2


0 5 10 15 20−30

−20

−10

0

10

20

30

time [s]

r1r2


Figure 5.11: Response of the L1 adaptive controller for Scenario 2 to the reference com-mands r1(t) and r2(t).

0 5 10 15 20−1.5

−1

−0.5

0

0.5

1

1.5

time [s]

r(t)x1(t), sc. 1xid1(t)x1(t), sc.2


0 5 10 15 20−4

−3

−2

−1

0

1

2

time [s]

scenario 1scenario 2

(b) Control history

0 5 10 15 20−1.5

−1

−0.5

0

0.5

1

1.5

time [s]



0 5 10 15 20−80

−60

−40

−20

0

20

40

time [s]



Figure 5.12: Tracking performance of the L1 adaptive controller for r(t) = sin(t) for Sce-narios 1 and 2.

L1book2010/7/22page 238

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�

�


0 5 10 15 20�0.5

0

0.5

1

1.5

time [s]

r(t)x1(t)x1(t)


0 5 10 15 20−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

time [s]

(b) Control history

0 5 10 15 20�2

�1.5

�1

�0.5

0

0.5

1

1.5

2

time [s]

x2(t)x2(t)


0 5 10 15 20−30

−20

−10

0

10

20

30

time [s]


Figure 5.13: System response in the presence of nonzero initialization error.

0 5 10 15 20−0.2

0

0.2

0.4

0.6

0.8

1

1.2

time [s]

r(t)

x1(t), sc. 1

xid1(t)

x1(t), sc.2


0 5 10 15 20−5

−4

−3

−2

−1

0

1

2

3

time [s]


(b) Control history

0 5 10 15 20−2

−1

0

1

2

3

time [s]



0 5 10 15 20−30

−20

−10

0

10

20

30

40

time [s]



Figure 5.14: System response in the presence of time delay of 13 ms for Scenarios 1 and 2.

L1book2010/7/22page 239

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xz0 = [3, 7, 10]�. The results are shown in Figure 5.13. One can see that the closed-loopsystem is stable and the state of the predictor rapidly converges to the state of the system. Weobserve that the nonzero initialization error does not deteriorate significantly the system’stransient response.

Further, we verify the robustness of the L1 adaptive controller to time delay in thecontrol channel. Figure 5.14 shows the simulation results in the presence of time delay of13 ms for Scenarios 1 and 2. One can see that system has insignificant degradation in theperformance and remains stable.

We note that the L1 adaptive controller guarantees smooth and uniform transientresponse for time-varying reference systems without any retuning of the controller in thepresence of different types of nonlinear uncertainties, unmodeled actuator dynamics, anddisturbances.

L1book2010/7/22page 241

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Chapter 6

Applications, Conclusions, andOpen Problems

In this book we attempted to present a unified treatment of the L1 adaptive control theorywith detailed proofs of the main results. The key feature of its architectures is the guaranteedrobustness in the presence of fast adaptation. We considered a broad class of determinis-tic systems and presented state-feedback and output-feedback architectures, summarizingthe main assumptions and proofs for the uniform guaranteed performance bounds. Thischapter presents preliminary results on the development and application of L1 adaptivecontrol architectures to the design of inner-loop flight control systems and then gives a briefdescription of the results not covered in the book, draws some concluding remarks, andsummarizes open problems for future research.

6.1 L1 Adaptive Control in FlightInner-loop adaptive flight control systems may provide the opportunity to improve aircraftperformance and reduce pilot compensation in challenging flight envelope conditions or inthe event of control surface failures and vehicle damage. However, implementing adaptivecontrol technologies can increase the complexity of the flight control systems beyond thecapability of current Verification and Validation (V&V) processes [175]. This fact, combinedwith the criticality of inner-loop flight control systems, leads to high certification costs andrenders difficult the transition of these technologies to military and commercial applications.Programs like NASA’s Integrated Resilient Aircraft Control (IRAC) program and Wright-Patterson AFRL’s Certification Techniques for Advanced Flight Critical Systems representan effort to advance the state of the art in adaptive control technology, to analyze thedeficiencies of current V&V practices, and to advance airworthiness certification of adaptiveflight control systems.

These two programs have significantly contributed to the ongoing efforts in the de-velopment, flight verification and validation, and transition of L1 adaptive control from atheoretical research field into a viable and reliable technology toward improving the robust-ness and performance of advanced flight control systems. The main goal of implementingan L1 adaptive controller onboard is to guarantee that an aircraft, suddenly experiencing anadverse flight regime or an unexpected failure, will not “escape” its α-β wind-tunnel dataenvelope (see Figure 6.1), provided that some control redundancy remains. It is important

241

L1book2010/7/22page 242

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242 Chapter 6. Applications, Conclusions, and Open Problems

Figure 6.1: Loss-of-control accident data relative to angle of attack and angle of sideslip.This figure appears here with the permission of NASA [53] .

to notice that, outside this wind-tunnel envelope, the aerodynamic models available areusually obtained by extrapolation of wind-tunnel test data, implying that these models arehighly uncertain. This fact suggests that pilots might not be correctly trained to fly the air-craft in these regimes (or, in the case of unmanned vehicles, the guidance loops might notbe properly designed for safe recovery). Moreover, it does not seem reasonable to rely ona flight control system to compensate for the uncertainties in these flight conditions, asaircraft controllability is not even guaranteed in such regimes. In this sense, flight controlsystems with only asymptotic guarantees might not prevent the aircraft from entering theseadverse flight conditions with unusual attitudes, and therefore inner-loop control architec-tures ensuring transient response with desired specifications and guaranteed robustnessappear to be imperative for safe operation of manned (and unmanned) aircraft in the pres-ence of anomalies. In particular, it is important to note that successful recovery from afailure, if possible at all, can be achieved only during the first few seconds after it occurs,in which the airplane is still in a regime with some controllability guarantees (Figure 6.1).Hence, the guaranteed fast and robust adaptation of L1 adaptive control architecturesmakes this control theory ideally suited for such eventualities. In fact, the L1 adaptiveflight control system has been already shown to be capable of compensating for sudden,unknown, severe failure events, while delivering predictable performance across the flightenvelope without resorting to gain scheduling of the control parameters, persistency of ex-citation, or control reconfiguration (see Sections 6.1.1 and 6.1.2 for details). Also, a gracefuldegradation in performance and handling qualities has been observed as the failures andstructural damage impose increasingly severe limitations on the controllability of the air-craft.

The scope of this section is to demonstrate the advantages of L1 adaptive control as averifiable robust adaptive control architecture with the potentiality of reducing flight controldesign costs and facilitating the transition of adaptive control into advanced flight controlsystems.

L1book2010/7/22page 243

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6.1. L1 Adaptive Control in Flight 243

6.1.1 Flight Validation of L1 Adaptive Control at NavalPostgraduate School

Recognizing the value of experimental V&V of advanced flight control algorithms, theNaval Postgraduate School (NPS) team has developed the so-called Rapid Flight Test Pro-totyping System (RFTPS) [47]. The RFTPS consists of a testbed unmanned aerial vehicle(UAV) equipped with a commercial autopilot (AP), an embedded computer running theresearch algorithms in real time, and a ground control station for flight management anddata monitoring and collection. This system facilitates the real-time onboard integration ofadvanced control algorithms and provides the opportunity to design and conduct compre-hensive flight test programs to evaluate the robustness and performance characteristics ofthese algorithms.

In order to demonstrate the benefits of L1 adaptive control, the commercial au-topilot of the RFTPS was augmented with the L1 adaptive output feedback architecturespresented in Chapter 4. The L1 augmentation loop is introduced to enhance the angular-rate tracking capabilities of the autopilot across the flight envelope in the event of controlsurface failures and in the presence of significant environmental disturbances. The inner-loop L1 adaptive flight control architecture implemented on the RFTPS is represented inFigure 6.2.

In the following sections, we present an overview of the main results from the extensiveflight test program conducted by NPS since 2006 in Camp Roberts, CA. The reader will finddetailed explanations and further hardware-in-the-loop simulations and flight test resultsin [3, 46, 82, 94, 110, 117].

Aggressive Path Following

Conventional autopilots are normally designed to provide only guidance loops for waypointnavigation. In order to extend the range of possible applications of (small) UAVs equippedwith traditional autopilots, a solution to the problem of three-dimensional (3D) path-following control was presented in [82]. The solution proposed exhibits a multiloop controlstructure, with (i) an outer-loop path-following control law that relies on a nonlinear controlstrategy derived at the kinematic level, and (ii) an inner-loop consisting of the commercialautopilot augmented with the L1 adaptive controller. The overall closed-loop system withthe L1 adaptive augmentation loop is presented in Figure 6.3.

Flight test results comparing the performance of the path-following algorithm withand without L1 adaptation are shown in Figure 6.4. The flight test data include the two-dimensional (2D) horizontal projection of the commanded and the actual paths, the com-manded rc(t) and the measured r(t) turn rate responses, and the path-tracking errors yF (t)and zF (t). The results show that the UAV is able to follow the path, keeping the path-following tracking errors reasonably small during the whole experiment. The plots alsodemonstrate the improved path-following performance when the L1 augmentation loop isenabled. One can observe that the nominal outer-loop path-following controller exhibitssignificant oscillatory behavior, with rate commands going up to 0.35 rad/s and with maxi-mum path-tracking errors around 18 m, while meantime the L1 augmentation loop is able toimprove the angular-rate tracking capabilities of the inner-loop controller, which results inrate commands not exceeding 0.15 rad/s and path-tracking errors below 8 m. Furthermore,it is important to note that the adaptive controller does not introduce any high-frequency

L1book2010/7/22page 244

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UAVAP

Gp

L1Augmentation

radrcmd

r

(a) Inner-loop structure with L1 adaptive augmentation

UAV

& APControl

Law

AdaptiveLaw

StatePredictor

L1 Augmentation

−

rcmdrad

r

r

σr

r

(b) L1 adaptive controller for turn-rate control

Figure 6.2: Inner-loop L1 adaptive augmentation loop tested by NPS.

content into the commanded turn-rate signal, as can be seen by comparing Figures 6.4(d)and 6.4(c).

Finally, it is important to mention that, in the derivations in [82], the uniform perfor-mance bounds that the L1 adaptive controller guarantees in both transient and steady-stateare critical to prove stability of the path-following closed-loop system, which takes intoaccount the dynamics of the UAV with its autopilot.

L1book2010/7/22page 245

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UAVAP

GpPath-Following

KinematicsGe

Path-FollowingControl

Algorithm

L1 AdaptiveAugmentation

x

u

y

yc

Figure 6.3: Closed-loop path-following system with L1 adaptive augmentation.

L1 Adaptation in the Presence of Control Surface Failures

The path-following flight test setup introduced in the previous section was used in [46]to demonstrate that the L1 augmentation loop provides fast recovery to sudden locked-in-place failures in either one of the ailerons or in the rudder of the RFTPS, while thenominal unaugmented system goes unstable. In these experiments, we took advantage of thecapability of the RFTPS to instantaneously deflect and hold a preprogrammed combinationof control surfaces at a predefined position without notifying or reconfiguring the nominalautopilot.

While the flight experiments considered failures in the left aileron covering the rangefrom 0 deg to −12 deg (with respect to a trim value −2.34 deg), and rudder failures from0 deg to 2 deg, in this section we present only an extract from these results. In particular,Figures 6.5 and 6.6 illustrate the performance of the path-following system with two levelsof sudden left-aileron locked-in-place failures at −2 deg and −10 deg (with respect to trimvalue −2.34 deg):

(i) Analysis of the −2 deg case showed that even such a small deflection pushes the UAVaway to the right from the desired path (see Figure 6.5(a)), resulting in almost 25 mof lateral miss distance (see Figure 6.5(b)). After the failure is introduced, the UAVconverges to a 5-m lateral error boundary in about 20 s.

(ii) The results of the −10 deg left-aileron failure (Figure 6.6) are similar to the onesfor the previous case. Naturally, the errors and the control efforts increase due to theincreased severity of the failure, and the impaired UAV converges to the 5-m boundaryin approximately 30 s.

Moreover, analysis of the entire series of results with left-aileron failures (coveringthe range from 0 deg to −12 deg) shows a graceful and predictable degradation in the path-following performance. It is important to mention that the predictability in the responseprovided by the L1 adaptive controller is especially critical for manned aircraft.

L1book2010/7/22page 246

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400 500 600 700 800 900 10000

200

400

600

800

1000

1200

1400

1600

1800UAV1 I.C.

UAV2 Fin.C.

East, m

Nor

th, m

UAVCmd

(a) L1 OFF: 2D projection

400 500 600 700 800 900 10000

200

400

600

800

1000

1200

1400

1600

1800UAV1 I.C.

UAV2 Fin.C.

East, m

Nor

th, m

UAVCmd

(b) L1 ON: 2D projection

0 10 20 30 40 50 60−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

time [s]

r c, r [r

ad/s

]

rc

r

(c) L1 OFF: Commanded and measuredturn rate

0 10 20 30 40 50 60−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

time [s]

r c, r [r

ad/s

]

rc

r

(d) L1 ON: Commanded and measured turn rate

0 10 20 30 40 50 60−15

−10

−5

0

5

10

15

20

time [s]

y f, zf [m

]

yf

zf

(e) L1 OFF: Path-following errors

0 10 20 30 40 50 60−15

−10

−5

0

5

10

15

20

time [s]

y f, zf [m

]

yf

zf

(f) L1 ON: Path-following errors

Figure 6.4: Fight test. Path-following performance with and without L1 adaptive augmen-tation.

Finally, we notice that the L1 adaptive controller automatically readjusts the controlsignals in order to stabilize the impaired airplane and uses the remaining control authority tosteer the airplane along the path, without resorting to fault detection and isolation methodsor reconfiguration of the existing inner-loop control structure. Therefore, integration ofL1 adaptation onboard increases the fault tolerance of the system.

L1book2010/7/22page 247

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−1000100200300400500600

100

200

300

400

500

600

700

800

900

1,000

1,100

1,200

1,300

1,400

I.C.

Y East, m

X N

orth

, m

UAVCmd

Transient

CSF enabled

Fin.C.

(a) trajectories

0 10 20 30 40 50 600

20

40

60

Err

, m

t, s

yerr

, m

(b) lateral error

Figure 6.5: Flight test. 2 deg locked-in-placeleft-aileron failure.

−400−300−200−1000100200300400

0

100

200

300

400

500

600

700

800

900

1000

I.C.

Fin.C.

Y East, m

X N

orth

, m

UAVCmd

CSF enabled

Transient

(a) trajectories

0 10 20 30 40 500

20

40

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Figure 6.6: Flight test. 10 deg locked-in-place left-aileron failure.

Rohrs’ Example in Flight

Motivated by Rohrs’ example, which was analyzed in detail in Section 2.3, the paper in [94]extends the setup introduced by Rohrs in [147] to the flight test environment, in which thefirst-order nominal plant is replaced by the UAV with its commercial autopilot. In [94], differ-ent unmodeled dynamics were considered, ranging from very lightly damped second-ordersystems (representing, for example, a flexible body mode of an airplane) to control surfacefailures. Figure 6.7 shows the block diagrams with the implementation of the unmodeleddynamics for two particular cases of second-order transfer functions at the output of thenominal plant.

Initially, a conventional output feedback MRAC algorithm from [75] with propertiessimilar to the adaptive controller used in [147] was used as an augmentation loop. Thiswas done to verify correctness of the flight test set-up—the results obtained in flight shouldbe similar to the ones obtained in [147]. The same scenario was then used to evaluatethe performance of the L1 adaptive augmentation. In [94], the authors also consideredthe implementation of some of the adaptive law modifications developed to overcome theproblem of parameter drift in conventional MRAC. We note that the determination of thephase crossover frequency, necessary to reproduce Rohrs’example, required identification ofthe frequency response of the nominal plant consisting of the UAV with the autopilot. Next,we present an extract of the results presented in [94], in which the second-order unmodeleddynamics introduced artificially at the output of the nominal plant were characterized by anatural frequency of 1.5 rad/s and a damping ratio of 0.45.

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A/P

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Figure 6.7: System with two cases of unmodeled dynamics and output disturbance.

First, Figure 6.8 shows the response of the closed-loop MRAC adaptive system inthe presence of the unmodeled dynamics to a biased sinusoidal reference signal rcmd atthe phase-crossover frequency. One can see that the parameters drift slowly, generatingcommand signals rad larger than ±100 deg

s , which are sent to the autopilot. It is important tonote that for safety reasons, in the actual implementation the autopilot limits the commandsreceived from the adaptive controller to avoid undesirable attitudes that might lead to theloss of the UAV. The same test was performed considering three different modificationsof the adaptive laws (σ -modification, e-modification, and projection operator) to verifythat, in fact, the closed-loop adaptive system remains stable. Flight test results for thee-modification, which are not included in this book, can be found in [94]. The results obtainedshow that under appropriate (trial-and-error-based) tuning, the stability of the closed-loopsystem can be preserved, although the resulting performance is in general poor, and, similarto pure MRAC architectures (without adaptive law modifications), the transient responsecharacteristics of the system are highly unpredictable.

The same experiment was conducted for the L1 adaptive augmentation; see Figure 6.9.The system maintains stability during the whole flight, and the control signal rad remainsinside reasonable bounds during the experiment. As one would expect, since the frequencyof the reference signal is well beyond the bandwidth of the low-pass filter in the control law,the L1 adaptive controller is not able to recover desired performance of the closed-loop

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Figure 6.8: MRAC augmentation. Closed-loop response in the presence of second-order un-modeled dynamics to a biased sinusoidal reference signal at the phase-crossover frequency.

adaptive system. The response with L1 adaptive controller is consistent during the entireflight and does not exhibit undesirable characteristics like bursting. The setup above wasalso used to illustrate the degradation of performance as the frequency of the reference signalincreases beyond the bandwidth of the low-pass filter. To this end, the closed-loop systemwith the second-order unmodeled dynamics and the L1 controller implemented on board wasdriven with a set of biased sinusoidal reference signals at different frequencies. Figure 6.10shows the results of these experiments. It can be seen that the output r of the closed-loop adaptive system is able to track the output of the reference system rm for referencesignals at low frequencies (ω = 0.3 rad

s ), and, as the frequency of the reference signalincreases, the performance degrades slowly and progressively. This graceful degradation inthe performance of the system is consistent with the theoretical claims of the L1 adaptivecontrol theory, which predict the response of the closed-loop adaptive system and ensuregraceful degradation of it outside the bandwidth of the design.

Finally, a combined experiment is presented in Figure 6.11. In this experiment, thesecond-order unmodeled dynamics were first injected to the nominal plant (unaugmented au-topilot), and then the adaptive algorithms were enabled, first the MRAC adaptive algorithmand then the L1 adaptive controller. The figure shows that with the unmodeled dynamics

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Figure 6.9: L1 augmentation. Closed-loop response in the presence of second-order un-modeled dynamics to a biased sinusoidal reference signal at the phase-crossover frequency.

injected, the closed-loop MRAC system becomes unstable, and the control command fromthe adaptive algorithm eventually hits the saturation limits of the autopilot. Then, at t = 56 s,the adaptive augmentation loop was switched from MRAC to L1 control, and the UAV re-covered stability in around 1.5 s, which confirms the theoretical claims of the L1 adaptivecontrol architectures regarding their fast adaptation with guaranteed robustness.

Implementation Details

Because the performance bounds and the stability margins of the L1 adaptive controller canbe systematically improved by increasing the adaptation gain, it is critical to ensure that theimplementation of the fast estimation scheme does not lead to numerical instabilities andthat the onboard CPU has enough computational power to robustly execute the fast integra-tion. Therefore, in this section we summarize implementation details of the L1 controllerin the RFTPS, providing some intuitive guidelines in the required hardware specifications.

We start by noting that the entire control system implemented onboard the smallUAV is a multirate algorithm consisting of two primary subsystems: (i) the hardware in-terfacing modules executing at the rate allowed by the sensors or actuators, and (ii) the

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s .

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control subsystem that utilizes sensor data to produce a control signal to be sent back to theactuators.

In most of the cases, the hardware interfacing subsystem is executed at a significantlylower rate (10–100 Hz) than the control algorithm (100–1000 Hz), therefore demandinginsignificant CPU power. This provides a suitable framework for the implementation ofL1 adaptive control architectures, as the computational power can be effectively used for

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Figure 6.11: Combined experiment: switching from MRAC to L1 adaptive controller in thepresence of second-order unmodeled dynamics.

fast adaptation. Furthermore, as soon as the base sampling time (fastest execution rate) ofthe multirate system is chosen to satisfy the performance requirements of the L1 adaptivecontroller (see equations (4.31) and (4.107)), and the multirate transitions are matched,the task execution time (TET) of the entire code (including the hardware interfacing loop)can be precisely predicted. Finally, knowing the overhead required by the real-time oper-ating system (5–10% of TET), the performance (CPU frequency, bus speed, memory) ofthe required processing board can be calculated. More details on the real-time schedul-ing of custom developed algorithms can be found in the technical documentation to thexPC Target.1

Figure 6.2(b) shows implementation of the L1 adaptive output feedback controller fora single turn rate control channel (SISO). The solution looks rather trivial with an integratoras the most computationally demanding elementary block (see equations in Sections 4.1.2and 4.2.2). Being a part of the control subsystem, the L1 controller execution is scheduledat the highest available rate. Clearly, the “price” associated with the implementation of theL1 adaptive controller lies primarily in the computational power required to accommodatethe high adaptation rate (� = 30000 in the current NPS setup), resulting in significant stiff-ness of the underlying differential equation. From a numerical point of view, this translatesinto the boundedness of the integration error during the entire execution time. While offline,one can use, for example, the advances of MATLAB/Simulink that allow for iterative ormultistep integration algorithms. However, in almost every real-time implementation thatuses discretized algorithms running at a fixed sampling time, the conflict between numericalaccuracy and fixed sampling interval might result in the loss of precision during integra-tion [40,41]. When a dynamically adjustable integration step is not available, this may leadto the cessation of execution and system failure. Therefore, finding the optimal trade-offbetween the complexity of the integration algorithm, a feasible (fastest available) samplingtime, and the numerical stability is of paramount importance.

1xPC Target: Perform Real-Time Rapid Prototyping and Hardware-in-the-Loop Simulation Using PCHardware. http://www.mathworks.com/products/xpctarget/

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140 160 180 200 220 240 260

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Figure 6.12: Hardware in the loop. TET: negligible increase in CPU load.

The complexity of numerical real-time implementation can be resolved via the devel-opment of accurate and stable numerical integration algorithms and by utilizing the latestadvancements of the automatic code generation. Besides verifying and generating a highlyoptimized executable code targeted for almost all existing CPU architectures, these toolsprovide powerful profiling methods that suggest possible bottlenecks in the code execution.Based on the detailed reports provided by these tools, the correct numerical algorithms, thesampling time, and the appropriate scheduling can be chosen to enable optimal multiratesystems execution, providing full utilization of the available CPU power.

As an illustration of the feasibility of the above-described system implementation,we have analyzed the computational load of the algorithms. Figure 6.12 shows the compu-tational power required for implementation of the outer-loop path-following controller inreal time with and without L1 adaptive augmentation. The parameter chosen to representthe computational load is the average TET of the entire code (including hardware interfac-ing and control) during one sampling interval (10 ms). The control code with the standardODE3 Bogacki–Shampine solver was implemented onboard an MSM900BEV2 industrialPC104 computer using the xPC/RTW Target development environment. This figure high-lights two important points. First, the average TET (≈ 1 ms) is an order of magnitude lessthan the base sampling time of the real-time code (10 ms), which implies that the samplingtime of the code implementation was chosen quite conservatively and could be reduced toimprove the closed-loop performance. Second, the difference in the CPU load when theL1 adaptive controller is enabled or disabled is negligible (an additional 0.052% with re-spect to the nominal controller), which supports the ease of implementation in almost anyplatform. With the current pace of evolution of new processors and advances in automaticcode generation, validation, and verification tools, the resources available for embeddedL1 adaptive control implementation are practically unlimited.

2Advanced Digital Logic (ADL) | PC104+ : AMD Geode LX900 CPU 500MHz, MSM900BEV, http://www.adl-usa.com/products/cpu/index.php

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6.1.2 L1 Adaptive Control Design for the NASA AirSTAR FlightTest Vehicle

The research control law developed for the GTM aircraft has as its primary objective achiev-ing tracking for a variety of tasks with guaranteed stability and robustness in the presenceof uncertain dynamics, such as changes due to rapidly varying flight conditions duringstandard maneuvers, and unexpected failures. All of this must be achieved while providingLevel I [39] handling qualities under nominal as well as adverse flight conditions. In partic-ular, one essential objective for safe flight under adverse conditions is for the aircraft neverto leave the extended α-β flight envelope; once outside the boundary and in uncontrollablespace, no guarantees for recovery can be made (see Figure 6.1). Consequently, the adaptivecontroller should learn fast enough to keep the aircraft within the extended flight envelope.This implies that the control law action in the initial 2 to 3 seconds after initiation of anadverse condition is the key to safe flight.

The L1 control system used for this application is a three-axes angle of attack (α),roll rate (p)-sideslip angle (β) command augmentation system, and is based on the theorydeveloped in Section 3.3, which compensates for both matched and unmatched dynamicuncertainties. For inner-loop flight control system design, the effects of slow outer-loopvariables (e.g., airspeed, pitch angle, bank angle) may appear as unmatched uncertainties inthe dynamics of the fast inner-loop variables we are trying to control (e.g., angle of attack,sideslip angle, roll rate). Also, unmodeled nonlinearities, cross coupling between channels,and dynamic asymmetries may introduce unmatched uncertainties in the inner-loop systemdynamics. If the design of the inner-loop flight control system does not account for theseuncertainties, their effect in the inner-loop dynamics will require continuous compensationby the pilot, thereby increasing the pilot’s workload. Therefore, automatic compensationfor the undesirable effects of these unmatched uncertainties on the output of the systemis important to achieve desired performance, reduce pilot’s workload, and improve theaircraft’s handling qualities.

It is important to note that the L1 adaptive flight control system provides a systematicframework for adaptive controller design that allows for explicit enforcement of MIL-Standard requirements [1] and significantly reduces the tuning effort required to achievedesired closed-loop performance, which in turn reduces the design cycle time and develop-ment costs. In particular, the design of the L1 adaptive flight control system for the GTM isbased on the linearized dynamics of the aircraft at an (equivalent) airspeed of 80 knots andat an altitude of 1000 ft. Since the airplane is Level I at this flight condition, the nominaldesired dynamics of the (linear) state predictor were chosen to be similar to those of the ac-tual airplane; only some additional damping was added to both longitudinal and directionaldynamics, while the lateral dynamics were set to be slightly faster than the original ones inorder to satisfy performance specifications. The state predictor was scheduled to specify dif-ferent performance requirements at special flight regimes (high-speed regimes and high-αregimes). In order to improve the handling qualities of the airplane, a linear prefilter wasadded to the adaptive flight control system to ensure desired decoupling properties as wellas desired command tracking performance. Overdamped second-order low-pass filters withunity DC gain were used in all the control channels, while their bandwidths were set toensure (at least) a time-delay margin of 130 ms and a gain margin of 6 dB. Finally, the adap-tation sampling time was set to 1

600 s, which corresponds to the maximum integration stepallowed in the AirSTAR flight control computer. We notice that the same control parametersfor the prefilter, the low-pass filters, and the adaptation rate were used across the entire

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flight envelope with no scheduling or reconfiguration. Further details about the design ofthe L1 adaptive controller for the GTM can be found in [177].

This section presents the preliminary results of a piloted-simulation evaluation on theGTM aircraft high-fidelity simulator, which includes full nonlinear, asymmetric aerodynam-ics, actuator dynamics, sensor dynamics including nonlinearities, noise, biases, and scalingfactors, and other nonlinear elements typical of these simulators. This piloted-simulationevaluation was part of development and profile planning of flight test tasks, and the taskswere flown with no training or repeatability and thus are considered to be a preliminaryevaluation. At the time of this evaluation, the GTM aircraft had three flight control modes.Mode 1 was the revisionary stick-to-surface control (the aircraft is Level I in this configu-ration); Mode 2 was referred to as the baseline control law and was an α-command, p-βstability augmentation system; Mode 3 was the research control law with the L1 adaptive α,p-β command control. The baseline α-command was an LQR-PI based and was designed toan operational limit of α ≤ 10 deg. It served as the operational baseline feedback control lawfor the GTM. Hence, comparison between the L1 controller and the baseline α-commandcould be performed only for α ≤ 10 deg; comparisons for higher α range were performedwith the stick-to-surface control law (Mode 1).

The main results and conclusions of this evaluation are presented in [59]. Flight testresults obtained during the AirSTAR deployments in March and June 2010 can be foundin [60]. The reader can also find in [37] a study on the handling qualities and possible adversepilot interactions in the GTM equipped with the L1 adaptive flight control system.

Angle of Attack Captures in the Presence of Static Stability Degradation

The task is trim at V = 80 knots end air speed (KEAS) and Alt = 1200 ft and then captureα = 8 deg within 1 s, hold for 2 s, with α-desired ±1 deg, α-adequate ±2 deg. This taskwas repeated for various levels of static stability expressed as�Cmα and ranging from 0 to100%, i.e., from nominal stable aircraft to neutral static longitudinal stability. The changein�Cmα is achieved by using both inboard elevator sections, scheduled with α, to producea destabilizing effect. These two elevator sections also become unavailable to the controllaw. In a sense, it is a double fault—a destabilized aircraft and a reduction in control powerin the affected axis.

This longitudinal task was evaluated for the L1 adaptive control law and the base-line, both of which are α-command response type. The performance of both control lawsis provided in Figure 6.13. For nominal GTM aircraft, performance of both control laws isvery similar, as illustrated in Figures 6.13(a) and 6.13(b), and solid Level I flying qualities(FQ) according to pilot comments. However, as the static stability is decreased by 50%(Figures 6.13(c) and 6.13(d)), the performance of the baseline controller degrades to highLevel II (Cooper–Harper rating (CHR) 4), while the L1 adaptive controller remains solidLevel I FQ [39]. With �Cmα = 75% (Figures 6.13(e) and 6.13(f)), the L1 adaptive con-troller remains predictable and Level I, while the baseline performance degrades to achievingonly adequate performance (Level II, CHR 5). At the point of neutral static stability (Fig-ures 6.13(g) and 6.13(h)), the L1 adaptive controller is still described as predictable butdoes experience some oscillations, and its performance is reduced to high Level II (CHR 4),while the baseline controller is described as pilot-induced oscillation prone and FQ are re-duced to Level 3 bordering on uncontrollable CHR 10. We would like to emphasize that theperformance of the L1 adaptive control law was found predictable by the pilot for all levelsof static stability.

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(a) �Cmα = 0% – L1 adaptive (b) �Cmα = 0%, baseline

(c) �Cmα = 50% – L1 adaptive (d) �Cmα = 50%, baseline

(e) �Cmα = 75% – L1 adaptive (f) �Cmα = 75%, baseline

(g) �Cmα = 100% – L1 adaptive (h) �Cmα = 100%, baseline

Figure 6.13: Angle-of-attack capture task with variable static stability.

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High Angle of Attack Captures

One of the several research objectives for the AirSTAR facility is to identify high angle-of-attack dynamics and verify these against obtained wind-tunnel and CFD data. In order to doso, the GTM aircraft flight must be able to safely fly at the very edges of the attainable flightenvelope. Part of the scheduled flight test is the high α envelope expansion. In addition,the GTM exhibits highly nonlinear pitch break phenomena for 12 ≤ α ≤ 18 deg. In otherwords, if in open loop the aircraft reaches α= 12 deg, it will pitch up and could be recoveredonly once it reaches α = 18 deg. Thus the envelope expansion is flown in the revisionarystick-to-surface control mode with 2 deg-α increments from α = 18 deg to α = 28 deg. Onthe other hand, the L1 adaptive controller is expected to perform α capture task for entirepoststall region starting at α = 12 deg.

The aerodynamics in the poststall region are nonlinear and increasingly asymmetricwith increased α. For 12 ≤ α ≤ 20 deg the aerodynamics are expected to be asymmetric,the roll damping (Clp) is expected to be low, and nose roll-off is also expected. Beyondα = 28 deg, in addition to aerodynamic asymmetry and low roll damping, a pronouncednose-slice due Cnβ is expected. The α capture performance for the L1 adaptive controlleris illustrated in Figure 6.14. The task is, starting in trim at V = 80 KEAS, to capture theindicated α at the rate of 3 deg

s , hold for 4 s, with α-desired ±1 deg, α-adequate ±2 deg.Note that the desired and the adequate criteria are the same as for the α capture in thelinear region; additionally, holding for 4 s would expose any control law to instabilityin this “pitch break” region. Due to the nature of expected dynamic behavior in the highα region, β and φ are additional variables of interest plotted in Figure 6.14. The L1 adaptivecontroller performance was judged as close to nominal α capture task. For α = 20 deg case(Figure 6.14(b)), the approach was at high pitch rate with Vmin getting into the 30s (KEAS);this created some nascent small oscillations asα approached 20 deg. Theα= 26 deg case wasperformed less rapidly and the slight oscillations are no longer present (Figure 6.14(c)). Forcomparison purposes, the same α case for the revisionary stick-to-surface mode is shownin Figure 6.14(d). Note the oscillatory nature of α as it follows the ramping αcmd; alsonote the bank and sideslip angle excursions, which illustrate the expected roll-off (φ) andnose-slice (β) dynamics. Figure 6.15 provides another way of looking at the high α capturetask by illustrating the coupling between these variables. Ideally, an α excursion wouldbe completely decoupled from β and would produce a straight vertical line on the plot. TheL1 adaptive controller produces |β| ≤ 1 deg excursions, as shown in Figure 6.15(a). On theother hand, the revisionary stick-to-surface control law shows significantly greater degreeof coupling between the α-β axis, as illustrated in Figure 6.15(b).

From Figure 6.15, and recalling the α-β flight envelope in Figure 6.1, it is evidentthat the L1 adaptive controller keeps the airplane inside the normal flight envelope duringany of the high-α maneuvers, thus ensuring controllability of the airplane during the wholetask. The same cannot be said about the revisionary stick-to-surface control law, for whichthe airplane experiences high-α/high-β excursions.

Sudden Asymmetric Engine Failure

This maneuver was performed unrehearsed once for each control law—reversion stick-to-surface mode and L1 adaptive controller. The results of this maneuver are used for qualitativecomparison between the two control law responses with pilot in the loop and no training.

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(a) αcmd = 14 deg – L1 adaptive (b) αcmd = 20 deg – L1 adaptive

(c) αcmd = 26 deg – L1 adaptive (d) αcmd = 26 deg, stick-to-surface

Figure 6.14: High α capture task.

(a) αcmd = 14/20/26 degL1 adaptive

(b) αcmd = 26 degStick-to-surface vs. L1 adaptive

Figure 6.15: α-β excursions for high-α capture task.

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(a) Stick-to-surface (b) L1 adaptive control

Figure 6.16: Asymmetric engine failure.

The task starts the airplane climbing at ≈ 30 deg attitude and throttles at full power, then atsome time the left throttle is reduced from 100% to 0% thrust in less than 0.5 s, as illustratedin Figure 6.16. This is primarily a lateral-directional task since the sudden change in thrustinduces a rolling moment affecting p, φ and a side force affecting β. These variables arecoplotted with throttle activity in Figure 6.16. The loss of control for the revisionary stick-to-surface control law is evident from Figure 6.16(a). From Figure 6.16(b) it is also evidentthat this sudden asymmetric thrust is a nonevent for the L1 adaptive controller, especiallyfrom the stability perspective.

We would like to emphasize that the L1 adaptive controller was not redesigned orretuned in all these scenarios and that a single set of control parameters was used for all thepiloted tasks and throughout the whole flight envelope. As stated earlier, only the referencemodel (state predictor) is scheduled in order to specify different performance requirementsat different flight regimes.

6.1.3 Other Applications

The L1 adaptive controller presented in Section 3.3 and used for the development of theL1 adaptive flight control system on GTM has also been validated for the X-48B air-craft [106] as an augmentation of a dynamic inversion baseline controller, and for theX-29 aircraft [63] as an augmentation of an LQR-PI baseline. The same L1 adaptive archi-tecture has been applied to the longitudinal control of a flexible fixed-wing aircraft [143].Also, the L1 adaptive output feedback adaptive augmentation architecture that is beingflight tested at NPS was used in [124] for the control of indoor autonomous quadrotors anda fixed-wing aerobatic aircraft. The output feedback architecture presented in Section 4.2has been applied in a MIMO setting to the ascent control of a generic flexible crew launchvehicle [88]. In [64], the authors design a high-bandwidth inner-loop controller to provideattitude and velocity stabilization of an autonomous small-scale rotorcraft in the presenceof wind disturbances. Also, the L1 adaptive control architecture presented in Section 2.2has been used on a vision-based tracking and motion estimation system as an augmentationloop for the control of a gimbaled pan-tilt camera onboard a UAV [110].

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The L1 adaptive controller has been applied to other areas outside aerospace appli-cations. Reference [111] explored application of an integrated estimator and L1 adaptivecontroller for pressure control in well-drilling systems. In [51], the authors explored theapplication of the L1 adaptive controller to compensate for undesired hysteresis and con-stitutive nonlinearities present in smart-material-based transducers. Also, the applicationof L1 adaptive control in a nuclear power plant to improve recovery of the system fromunexpected faults and emergency situations was studied in [78].

6.2 Key Features, Extensions, and Open Problems

6.2.1 Main Features of the L1 Adaptive Control Theory

The main features of the L1 adaptive control theory, proved in theory and consistentlyverified in experiments, can thus be summarized as follows:

• Guaranteed robustness in the presence of fast adaptation;

• Separation (decoupling) between adaptation and robustness;

• Guaranteed transient response, without resorting to

persistency of excitation type assumptions,

high-gain feedback, or

gain-scheduling of the controller parameters;

• Guaranteed (bounded-away-from-zero) time-delay margin;

• Uniform scaled transient response dependent on changes in

initial conditions,

unknown parameters, and

reference inputs.

With these features the architectures of the L1 adaptive control theory reduce theperformance limitations to hardware limitations and provide a suitable framework for de-velopment of theoretically justified tools for V&V of feedback systems. The next sectionsummarizes the open problems in this direction.

6.2.2 Extensions Not Covered in the Book

The L1 adaptive control theory has been developed for a broader class of systems, thepresentation of which is outside the limits of this book. Some of the important extensionsare presented in [107, 116, 179, 183, 188].

Robustness features of the L1 adaptive controller have also been verified using theframework for robustness analysis of nonlinear systems in the gap metric [109]. By an ap-propriate extension of a classical result [55], the L1 adaptive controller has been provedto have guaranteed robust stability margin. The computation of the robust stability marginin the gap metric confirmed that in the absence of the low-pass filter one loses the robust-ness guarantees of this feedback structure (similar to Theorem 2.2.4). The computational

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6.2. Key Features, Extensions, and Open Problems 261

tractability of this method leads, in turn, to explicit derivation of the margins for a wideclass of system uncertainties such as time delay and multiplicative unmodeled dynamics.

The results in [116] summarize the extension to nonaffine-in-control systems. Follow-ing Lemma A.8.1, the linear parametrization of the control-dependent nonlinearity leads totime-dependent control input gain ω(t). This further implies that the low-pass filter, usedfor definition of the control signal, can no longer be effectively employed. An appropriateextension is developed, which considers an LTV system for definition of the control signal,and its analysis is pursued by elaborating the tools used in the proofs of Chapter 5 for LTVreference systems. These new mathematical developments also supported the extension tononlinear systems in the presence of input hysteresis [188].

Reference [107] analyzes the performance bounds of the L1 adaptive controller inthe presence of input saturation. Following the approach in [85], appropriate modificationof the state predictor is considered to remove the effect of the control deficiency fromthe adaptation process. The uniform performance bounds are computed with respect to abounded reference system, which can be designed to meet the performance specificationsfor the given input constraints.

The results in [183] extended the L1 adaptive controller to decentralized setup byconsidering large-scale interconnected nonlinear systems in the presence of unmodeleddynamics. The decentralized local L1 adaptive controllers compensate for the effect ofunmodeled dynamics and nonlinearities on the system output without having access to thestates or outputs of other subsystems.

Reference [162] analyzed the performance of the L1 adaptive controller in the pres-ence of input quantization. The resulting performance bounds are shown to be decoupledinto two terms, one of which is the standard term that one has in the absence of quantizationand can be systematically improved by increasing the rate of adaptation, while the otherterm can be reduced by improving the quality of the quantizer. This decoupled nature of theperformance bounds allows for independent design of the quantizers and can broaden theapplication domain of quantized control.

Parallel to this, reference [173] considered implementation of the L1 adaptive con-troller over real-time networks using event triggering. Event-triggering schedules the datatransmission dependent upon errors exceeding certain threshold. Similar to [162], with theproposed event-triggering schemes and with the L1 adaptive controller in the feedbackloop, the performance bounds of the networked system are decoupled into two terms, oneof which is the standard term that one has in the absence of networking and event-triggeringand can be systematically improved by increasing the rate of adaptation, while the otherterm can be reduced by increasing the data transmission frequency. This further implies thatthe performance limitations of the L1 adaptive closed-loop systems are consistent with thehardware limitations.

6.2.3 Open Problems

The main challenge of the L1 adaptive control theory is the optimal design of the bandwidth-limited filter. Because the filter defines the trade-off between performance and robustness, itsoptimal design is a problem of constrained optimization. Moreover, the L1 norm conditionfor minimization of the performance bounds renders the optimization problem nonconvexand hence more challenging. The design of the filter involves consideration of its order andparametrization. While we have provided partial design guidelines in Section 2.6 for thefull state-feedback architecture, the problem is still largely open and hard to address.

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The design problem is especially challenging in output feedback due to the stabilitycondition for the transfer function H (s) defined in (4.5) and (4.65) as

H (s) � A(s)M(s)

C(s)A(s)+ (1−C(s))M(s).

As shown in Chapter 4, the definition ofH (s) involves the uncertain plantA(s), and thereforethe uncertainty is not decoupled in the L1-norm condition as it is in the state-feedbacksolution. Hence, stability ofH (s) adds an additional constraint to the choice of the filter andthe desired reference system.

In [169], tools from robust control were invoked to address the problem for the case ofreference systems of higher order, which do not verify the SPR property for their input-outputtransfer functions. In this case, partial results have been obtained by resorting the analysisto Kharitonov’s theorem from robust control [169]. We note that the disturbance observerliterature has methods on offer for parametrization ofC(s) that would achieve stabilization ofH (s) for a sufficiently broad class of systemsA(s) [155]. The current research is focused onappropriate extension of these methods to capture non-minimum phase systems amongA(s).

Special attention is deserved in the case of MIMO systems in the presence of un-matched uncertainties, analyzed in Sections 3.2 and 3.3. The sufficient condition for stabil-ity and performance, given in terms of L1 norm bound, involves a restriction on the rateof variation of uncertainties in both matched and unmatched channels, in addition to thedesired performance specifications, given byAm and Bm matrices. From (3.71) and (3.126)it follows that the cross-coupling, expressed inGum(s), will directly affect the choice ofAmand Bm and also the invariant set where the solutions lie. While the sufficient conditionsare intuitive, their complete analytical investigation is largely an open area of research. Inparticular, in the case of the design of the L1 flight control system for the GTM, which wasdiscussed in Section 6.1.2, the selection of Am and Bm followed from (MIL-standard [1])performance requirements at different flight regimes, while the (second-order) low-passfilters in the control laws were designed to guarantee stability of the closed-loop adaptivesystem and achieve a satisfactory level of robustness. Currently there exists no generalmethodology that one could formalize for this purpose, and we expect this analysis to bedone on case-by-case basis dependent upon the nature of the application.

An important direction for future research is the extension of the time-delay mar-gin proof to more complex classes of systems. The proof in Section 2.2.5 is pursued forLTI (open-loop) systems with time-varying disturbance. The lower bound for the time-delaymargin is provided via an LTI system, given in (2.80). Notice that this system depends uponthe original LTI system, given by H (s). Obviously, if the original system were not LTI, thenthe explicit computation of the time-delay margin could not be reduced to an expressionsimilar to the one in (2.80). More elaborate tools are needed from nonlinear systems theoryto address the problem for more complex classes of systems.

While a complete answer to the above described problems may not be obtained inthe near future, L1 adaptive control, with its key feature of decoupling adaptation fromrobustness, has already facilitated new opportunities in the area of networked systems andevent-driven adaptation [3,162,173]. We expect rapid developments in these directions overthe next couple of years.

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Appendix A

Systems Theory

In this chapter we provide a brief review of some basic facts from stability theory and robustcontrol, which are used throughout the book.

A.1 Vector and Matrix NormsThe norm of a vector or a matrix is a real valued function ‖ · ‖, defined on the vector or thematrix space, verifying the following properties for all vectors or matrices u, v, and λ ∈ R:

• ‖u‖> 0 if u = 0, and ‖u‖ = 0 if and only if u= 0;

• ‖u+v‖ ≤ ‖u‖+‖v‖ ;

• ‖λu‖ = |λ|‖u‖ .

Obviously, the norm is not uniquely defined. Below we introduce the most frequentlyused norms.

A.1.1 Vector Norms

1. The 111-norm of a vector u= [u1, . . . ,um]� ∈ Rm is defined as

‖u‖1 �m∑i=1

|ui | .

2. The 222-norm of a vector u= [u1, . . . ,um]� ∈ Rm is defined as

‖u‖2 �√u�u .

3. The ppp-norm of a vector u= [u1, . . . ,um]� ∈ Rm for 1 ≤ p <∞ is defined as

‖u‖p �(m∑i=1

|ui |p)1/p

.

263

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264 Appendix A. Systems Theory

4. The ∞∞∞-norm of a vector u= [u1, . . . ,um]� ∈ Rm is defined as

‖u‖∞ � max1≤i≤m |ui | .

Throughout the book, when the type of norm is not explicitly specified, the 2-norm isassumed.

A.1.2 Induced Norms of Matrices

The matrix A ∈ Rn×m can be viewed as an operator that maps R

m, the space of m-dimensional vectors, into R

n, the space of n-dimensional vectors. The operator norm, or theinduced p-norm, of a matrix is defined as

‖A‖p � supx =0

‖Ax‖p‖x‖p = sup

‖x‖p=1‖Ax‖p .

The proof of the last equality is straightforward. Indeed, if x = 0, then

‖Ax‖p‖x‖p =

∥∥∥∥A x

‖x‖p∥∥∥∥p

.

Taking the sup of both sides proves the last equality above. This definition leads to thefollowing matrix norms:

1. The induced 111-norm of the matrix A ∈ Rn×m is defined as

‖A‖1 � max1≤j≤m

n∑i=1

|aij | (column sum) .

2. The induced 222-norm of the matrix A ∈ Rn×m is defined as

‖A‖2 �√λmax(A�A) ,

where λmax(·) denotes the maximum eigenvalue.

3. The induced ∞∞∞-norm of the matrix A ∈ Rn×m is defined as

‖A‖∞ � max1≤i≤n

m∑j=1

|aij | (row sum) .

Throughout the book, and similar to vector norms, when the type of norm is notexplicitly specified, the 2-norm is assumed. The following properties are straightforward toverify for all the norms:

1. ‖A‖ = ‖A�‖.

2. For arbitrary vector x and arbitrary matrix A with appropriate dimensions, the fol-lowing inequality holds:

‖Ax‖ ≤ ‖A‖‖x‖ .

3. All the norms are equivalent, i.e., if ‖ ·‖p and ‖ ·‖q are two different norms, then, forarbitrary vector or matrix X, there exist two constants c1 > 0 and c2 > 0 such that

c1‖X‖p ≤ ‖X‖q ≤ c2‖X‖p .

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A.2. Symmetric and Positive Definite Matrices 265

A.2 Symmetric and Positive Definite MatricesWe recall definitions of symmetric and positive definite matrices plus one key property oftenused in this book.

Definition A.2.1 The matrixM ∈ Rn×n is symmetric ifM =M�.

Definition A.2.2 The matrixM ∈ Rn×n is

• positive definite if, for an arbitrary nonzero vector x ∈ Rn, the following inequality

holds:

x�Mx > 0 and x�Mx = 0 only for x = 0;

• positive semidefinite if, for arbitrary vector x ∈ Rn,

x�Mx ≥ 0;

• negative definite if, for arbitrary nonzero vector x ∈ Rn,

x�Mx < 0 and x�Mx = 0 only for x = 0;

• negative semidefinite if, for arbitrary vector x ∈ Rn,

x�Mx ≤ 0.

For arbitrary vectorx ∈ Rn and arbitrary positive definite symmetric matrixM ∈ R

n×n,the following inequalities hold:

λmin(M)‖x‖2 ≤ x�Mx ≤ λmax(M)‖x‖2 ,

where λmin(M) and λmax(M) denote, respectively, the minimum and maximum eigenval-ues ofM .

Definition A.2.3 The matrix A is Hurwitz if all its eigenvalues have negative real part:

�(λi)< 0, det|λiI−A| = 0, i = 1, . . . ,n .

Lemma A.2.1 The matrix A is Hurwitz if and only if given an arbitrary positive definitesymmetric matrix Q, there exists a positive definite symmetric matrix P solving the alge-braic Lyapunov equation

A�P +PA= −Q .

A.3 L-spaces and L-normsUsing the definition for the vector norms, we define norms of functions f : [0,+∞) → R

n

as follows:

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• L1L1L1-norm and L1L1L1-space: The space of piecewise-continuous integrable functionswith bounded L1-norm

‖f ‖L1 �∫ ∞

0‖f (τ )‖dτ <∞

is denoted Ln1, where any of the vector norms can be used for ‖f (τ )‖.

• LpLpLp-norm and LpLpLp-space: The space of piecewise-continuous integrable functionswith bounded Lp-norm

‖f ‖Lp �(∫ ∞

0‖f (τ )‖pdτ

)1/p

<∞

is denoted Lnp. As above, any of the vector norms can be used for ‖f (τ )‖. However,

for the L2-space it is conventional to use the 2-norm of the vector.

• L∞L∞L∞-norm: The space of piecewise-continuous bounded functions with L∞-norm

‖f ‖L∞ � max1≤i≤n

{supτ≥0

|fi(τ )|}<∞

is denoted Ln∞.

Notice that the requirement for the L-norm to be finite restricts the class of functionsthat can belong to the Ln

p space. So as not to be restricted by this, we consider the extendedspace Ln

eLneLne , defined as the space of functions

Lne � {f (t) | fτ (t) ∈ Ln, ∀ τ ∈ [0,∞)} ,

where fτ (t) is the truncation of the function f (t) defined by

fτ (t) ={f (t), 0 ≤ t ≤ τ ,0, t > τ .

Thus, every function that does not have finite escape time belongs to the extended spaceLne . Notice that any of the Lp-norms can be used in the definition of the extended space.

The definitions above can also be extended for functions defined on [t0, ∞), t0 > 0.The L-norm of the function f : [t0, ∞) → R

n and its truncated L-norm are given as theL-norms of f[t0,∞) and f[t0,τ ], respectively, where

f[t0,∞)(t) ={

0, 0 ≤ t < t0 ,f (t), t0 ≤ t , f[t0,τ ](t) =

0, 0 ≤ t < t0 ,f (t), t0 ≤ t ≤ τ ,0, t > τ .

In this book we omit the index t0 from the notation of the norms ‖f[t0,∞)‖L, ‖f[t0,τ ]‖L andsimply write ‖f ‖L, ‖fτ‖L if it is clear from the context which norm has been used.

It is worth mentioning that for function norms the principle of equivalence does nothold (some of the norms can be finite, while some can be unbounded, i.e., not defined).

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A.4. Impulse Response of Linear Time-Invariant Systems 267

Example A.3.1 Consider the piecewise-continuous function

f (t) ={

1/√t , 0< t ≤ 1,

0 , t > 1.

It has bounded L1-norm:

‖f ‖L1 =∫ 1

0

1√tdt = 2.

Its L∞-norm does not exist, since ‖f ‖L∞ = supt≥0 |f (t)| = ∞, and its L2-norm isunbounded because the integral of 1/t is divergent. Thus, f (t) ∈ L1, but f (t) /∈L2 ∪L∞.

Example A.3.2 Next, consider the continuous function

f (t) = 1

1+ t .

It has bounded L∞-norm and bounded L2-norm:

‖f ‖L∞ = supt≥0

∣∣∣ 1

1+ t∣∣∣= 1, ‖f ‖L2 =

(∫ ∞

0

1

(1+ t)2dt)1/2 = 1.

Its L1-norm does not exist, since

‖f ‖L1 =∫ ∞

0

1

1+ t dt = limt→∞ ln(1+ t) = ∞ .

Thus f (t) ∈ L2 ∩L∞, but f (t) /∈ L1.

A.4 Impulse Response of Linear Time-Invariant SystemsA SISO LTI system has the following state-space representation:

x(t) = Ax(t)+bu(t) , x(0) = x0 ,

y(t) = c�x(t) ,(A.1)

where x ∈ Rn is the state vector, u ∈ R is the input, y ∈ R is the output of the system,

and x0 is the initial condition. Further, A ∈ Rn×n is the state matrix, and b, c ∈ R

n arevectors of appropriate dimensions. In the frequency domain, the system is defined by meansof its transfer function y(s) = G(s)u(s), where G(s) � c�(sI−A)−1b. Notice that in thisrepresentation x0 = 0. Obviously, given y(s) =G(s)u(s), the state-space realization in (A.1)is not unique.

Letting u(s) = 1, which corresponds to the Dirac-delta impulse function,

u(t) ={

∞ t = 0,0 t = 0,

and∫ ∞

−∞u(t)dt = 1,

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we have y(s) = G(s). The inverse Laplace transform of the transfer function G(s), givenby g(t) = L−1(G(s)), is called the impulse response of the system.

For MIMO systems, the state-space representation is given by

x(t) = Ax(t)+Bu(t) , x(0) = x0 ,

y(t) = Cx(t) ,(A.2)

where x ∈ Rn, u ∈ R

m, and y ∈ Rl are the state, the input, and the output of the system,

while A ∈ Rn×n, B ∈ R

n×m, and C ∈ Rl×n are the corresponding matrices.

The transfer matrix of the system in (A.2) is given by

G(s) � C(sI−A)−1B .

Let gj : R → Rl be the response of the system to the unit Dirac-delta function applied at the

j th input with all the initial conditions set to zero. The matrix g(t) ∈ Rl×m with its entries

gj (t) is called the impulse response matrix of the system and can be computed as theinverse Laplace transform of the transfer matrix:

g(t) = L−1(G(s)) .

A.5 Impulse Response of Linear Time-Varying SystemsWhen the system matrices in (A.2) depend upon time, i.e.,

x(t) = A(t)x(t)+B(t)u(t) , x(t0) = x0 ,

y(t) = C(t)x(t) ,(A.3)

then the linear system is time varying. For an LTV system, the impulse response is definedvia its state transition matrix.

Definition A.5.1 The state transition matrix �(t , t0)�(t , t0)�(t , t0) of the LTV system in (A.3) is thesolution of the following linear homogeneous matrix equation:

∂�(t , t0)

∂t= A(t)�(t , t0) , �(t0, t0) = I . (A.4)

When u(t) ≡ 0, the state trajectory of (A.3) and the state transition matrix are relatedas follows:

x(t) =�(t , t0)x(t0) .

The impulse response of the LTV system in (A.3) is computed according to the followingequation:

g(t , t0) = C(t)�(t , t0)B(t0) , t ≥ t0 . (A.5)

Notice that in the case of LTI systems, whenA(t) =A,B(t) =B,C(t) =C in (A.3) areindependent of time, assuming t0 = 0, the solution of (A.4) for�(t ,0) =�(t) in frequencydomain can be written as

�(s) = (sI−A)−1 .

Then, (A.5) in the frequency domain takes the form

L(g(t)) = C�(s)B = C(sI−A)−1B =G(s) ,

which recovers the result for LTI systems.

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A.6. Lyapunov Stability 269

A.6 Lyapunov StabilityIn this section we recall several fundamental results of Lyapunov stability theory. We beginby considering the nonlinear autonomous system of the form

x(t) = f (x(t)) , x(0) = x0 , (A.6)

where x(t) ∈ R is the system state, and f (x) : Rn → R

n is a locally Lipschitz nonlinearity.

A.6.1 Autonomous Systems

We assume that the system (A.6) has an equilibrium at x = 0, i.e., f (0) = 0.

Definition A.6.1 The equilibrium point at the origin is

stable if for all ε > 0 there exists δ = δ(ε)> 0, such that, if ‖x0‖< δ, then ‖x(t)‖< ε forall t ≥ 0;

(locally) asymptotically stable if it is stable and δ can be chosen such that, if ‖x0‖ < δ,then limt→∞ ‖x(t)‖ = 0;

globally asymptotically stable if it is stable and for all x0 ∈ Rn, limt→∞ ‖x(t)‖ = 0;

unstable if it is not stable.

The following result, known as Lyapunov’s direct method, gives sufficient conditionsfor stability. For this result we consider an open set D ⊂ R

n, which contains the equilibriumx = 0, and a continuously differentiable function V : D → R with its derivative along thetrajectories given by V (x(t)) = ∂V (x)

∂xf (x(t)).

Definition A.6.2 A function V : D → R with V (0) = 0 is called

positive definite if V (x)> 0, x ∈ D \ {0};positive semidefinite if V (x) ≥ 0, x ∈ D \ {0};negative definite if V (x)< 0, x ∈ D \ {0};negative semidefinite if V (x) ≤ 0, x ∈ D \ {0} .

Theorem A.6.1 Consider the system (A.6) and assume that there exists a continuouslydifferentiable positive definite function V : D → R, such that

V (0) = 0, V (x)> 0 for x ∈ D \ {0} .

Then, the equilibrium x = 0 is (locally) stable if

V (x(t)) ≤ 0, x ∈ D ,

and the equilibrium x = 0 is (locally) asymptotically stable if

V (x(t))< 0, x ∈ D \ {0} .

Additionally, if D = Rn, and V (x) is radially unbounded, i.e., lim‖x‖→∞V (x) = ∞, then

these results hold globally.

The proof of this theorem can be found in [86]. If V (x) verifies the properties above, it iscalled the Lyapunov function for the system.

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A.6.2 Time-Varying Systems

Next we consider nonautonomous systems of the type

x(t) = f (t ,x(t)) , x(t0) = x0 , (A.7)

with f (t ,x) : [t0, ∞) × Rn → R

n being piecewise-continuous in t and locally Lipschitzin x. We assume that f (t ,0) ≡ 0, so that the system has an equilibrium at the origin. A keyfeature of the nonautonomous systems is the uniformity of its convergence properties.

Definition A.6.3 The equilibrium point x = 0 is

stable if for all ε > 0 there exists δ = δ(ε, t0) > 0, such that if ‖x0‖ < δ, then ‖x(t)‖ < εfor all t ≥ t0 ≥ 0;

(locally) asymptotically stable if it is stable and δ can be chosen such that, if ‖x0‖ < δ,then limt→∞ ‖x(t)‖ = 0;

globally asymptotically stable if it is stable and for all x0 ∈ Rn, limt→∞ ‖x(t)‖ = 0;

unstable if it is not stable.

The stability is said to hold uniformly if δ = δ(ε)> 0 is independent of t0.

Definition A.6.4 The equilibrium point x = 0 of (A.7) is (locally) exponentially stable ifthere exist positive constants c > 0, a > 0, λ > 0, such that

‖x(t)‖ ≤ a‖x0‖e−λ(t−t0) , ∀ ‖x0‖ ≤ c .

It is globally exponentially stable if this bound holds for arbitrary initial condition x0.

Example A.6.1 (see [86]) To highlight the significance of uniformity, consider the follow-ing first-order system:

x(t) = (6t sin t−2t)x(t) , x(t0) = x0 .

Its solution is given by

x(t) = x(t0)exp

[∫ t

t0

(6τ sin τ −2τ )dτ

]= x(t0)exp[6sin t−6t cos t− t2 −6sin t0 +6t0 cos t0 + t20 ].

For arbitrary t0, the term −t2 will dominate, which implies that

∃ c(t0) ∈ R such that |x(t)|< |x(t0)|c(t0) , ∀ t ≥ t0 .

For arbitrary ε > 0, the choice of δ = ε/c(t0) shows that the origin is stable. However,assuming that t0 = 2nπ , n= 0, 1, 2, . . . , we get

x(t0 +π ) = x(t0)exp[(4n+1)(6−π )π ] ,

which implies that for x(t0) = 0

limn→∞

x(t0 +π )

x(t0)= ∞ .

Thus, given ε > 0 there is no δ independent of t0 that would verify the stability definitionuniformly in t .

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A.6. Lyapunov Stability 271

In the time-varying caseV : R×D → R may also be time dependent. Thus, its deriva-tive along the trajectories of the system is given by V (t ,x(t)) = ∂V (t ,x)

∂t+ ∂V (t ,x)

∂xf (t ,x(t)).

Theorem A.6.2 Consider the system (A.7), and assume that there exists a continuouslydifferentiable function V (t ,x) such that

W1(x) ≤ V (t ,x) ≤W2(x) , ∀ x ∈ D ,

whereW1(x),W2(x) are continuous and positive definite functions on D . If

• V (t ,x(t)) ≤ 0, for all x ∈ D , then the equilibrium is uniformly stable;

• V (t ,x(t)) ≤ −W3(x), for all x ∈ D , whereW3(x) is a continuous and positive definitefunction on D , then the equilibrium is uniformly asymptotically stable.

Additionally, if D = Rn, andW1(x) is radially unbounded, then these results hold globally.

The proof of this theorem can be found in [86].

Example A.6.2 Consider the second-order system

x1(t) = x2(t) ,

x2(t) = −x1(t)−φ(t)x2(t) ,(A.8)

where φ(t) is a positive definite continuously differentiable function with bounded deriva-tive. Consider the following Lyapunov function candidate:

V (t ,x) = 1

2(x2

1 +x22 ) .

Its derivative along the trajectories of the system is given by

V (t ,x(t)) = x1(t)x2(t)−x2(t)x1(t)−φ(t)x22 (t) = −φ(t)x2

2 (t) ≤ 0.

This allows for concluding uniform stability of the origin. However, using Theorem A.6.2,one cannot conclude asymptotic stability.

Next we show the result known as Barbalat’s lemma, which in some cases can helpto conclude stronger stability properties.

Lemma A.6.1 (Barbalat’s lemma) Let f : R → R be a uniformly continuous function on[0, ∞). Assume that limt→∞

∫ t0 f (τ )dτ exists. Then

limt→∞f (t) = 0.

The proof of this lemma can be found in [86].

Corollary A.6.1 If a scalar function V (t ,x) satisfies the conditions

• V (t ,x) is lower bounded,

• V (t ,x) is negative semidefinite, and

• V (t ,x) is uniformly continuous in time,

then V (t ,x(t)) → 0 as t → ∞.

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Recall Example A.6.2. Notice that the second derivative

V (t ,x(t)) = −2φ(t)x2(t)− φ(t)x22 (t)

is bounded. This implies that V (t ,x(t)) is uniformly continuous, and thus according toCorollary A.6.1,

limt→∞ V (t ,x(t)) = 0,

which leads to

limt→∞x2(t) = 0.

From (A.8) one can see that x2(t) is uniformly continuous, and application of Barbalat’slemma gives us

limt→∞ x2(t) = 0,

which along with (A.8) implies that

limt→∞x1(t) = 0.

Therefore, the origin is asymptotically stable.

Lemma A.6.2 Suppose that for the linear state equation

x(t) = A(t)x(t) , x(0) = x0 ,

with continuously differentiable A(t) ∈ Rn×n there exist positive constants µA, µλ, such

that for all t ≥ 0, ‖A(t)‖∞ ≤ µA, and at each time t , the eigenvalues of A(t) (pointwiseeigenvalue) satisfy �[λ(t)] ≤ −µλ. Then, there exists a positive constant ζ such that, ifthe time derivative of A(t) satisfies ‖A(t)‖∞ ≤ ζ for all t ≥ 0, the equilibrium of the stateequation is exponentially stable and ‖P (t)‖∞ < 1, where P (t) is the solution of

A�(t)P (t)+P (t)A(t) = −I .

The proof is similar to the proof of Theorem 8.7 in [149].

A.7 L-StabilityNext we consider the stability of input-output models of dynamical systems and refer to asystem as

y = Gu ,

where G denotes the map from the input u(t) ∈ Rm to the output y(t) ∈ R

l . We introducetwo special spaces of functions, which will be used in the forthcoming analysis.

Definition A.7.1 A continuous function α : [0, ∞) → [0, ∞) is said to belong to class K ,if it is strictly increasing and α(0) = 0. It is said to belong to class K∞, if in additionlimx→∞α(x) = ∞.

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A.7. L-Stability 273

Definition A.7.2 A continuous function α : [0, ∞)× [0, ∞) → [0, ∞) is said to belong toclass KL, if β(x) = α(t0,x) ∈ K for each fixed t0, and for each fixed x0 it is a decreasingfunction in t , i.e., limt→∞α(t ,x0) = 0.

Definition A.7.3 The map G : Lme → Ll

e is L-stable if there exist a class K function α(·)and a nonnegative constant b such that

‖(Gu)τ‖L ≤ α(‖uτ‖L)+bfor allu(t) ∈ Lm

e and τ ∈ [0, ∞). It is finite-gain L-stable if there exist nonnegative constantsa, b such that

‖(Gu)τ‖L ≤ a‖uτ‖L +bfor all u(t) ∈ Lm

e and τ ∈ [0, ∞).

If Definition A.7.3 holds for the L∞-norm of the signals, the system is called BIBOstable.

Consider the particular case in which the input-output map G is given by the dynamics

x(t) = f (t ,x(t),u(t)) , x(t0) = x0 ,

y(t) = g(t ,x(t),u(t)) ,

where x(t) ∈ Rn, y(t) ∈ R

l are the state and the output of the system, respectively; u(t) ∈ Rm

is the input of the system; f (t ,x,u) : [t0,∞) ×Rn×R

m → Rn is a function satisfying the

sufficient conditions of existence and uniqueness of solution; and g(t ,x,u) : [t0,∞)×Rn×

Rm → R

l is a given function. Let H denote the map from the control input to the state ofthe system: x = Hu. Then if Definition A.7.3 holds for the map H with L∞-norm of thesignals, the system is called BIBS stable.

Next we provide necessary and sufficient conditions for BIBO stability of LTI andLTV systems.

A.7.1 BIBO Stability of LTI Systems

Definition A.7.4 For a givenm-input and l-output LTI systemG(s) with impulse responseg(t) ∈ R

l×m, its L1 norm is defined as

‖g‖L1 � maxi=1,...,l

m∑j=1

‖gij‖L1

.

For the purpose of simplifying the notation in relatively complex derivations, involvingcascades of several systems, in the book ‖G(s)‖L1 is used instead of ‖g‖L1 .

Lemma A.7.1 Assume that g(t) ∈ L1, i.e., ‖g‖L1 <∞. Then for arbitrary u(t) ∈ L∞e,we have

‖yτ‖L∞ ≤ ‖g‖L1‖uτ‖L∞ ,

and y(t) ∈ L∞e.

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Proof. Let yi(t) be the ith element of y(t) and uj (t) be the j th element of u(t). Then, forarbitrary t ∈ [t0, τ ], we have

yi(t) =∫ t

t0

m∑j=1

gij (t− ξ )uj (ξ )

dξ .

This leads to the following upper bound

|yi(t)| ≤∫ t

t0

m∑j=1

|gij (t− ξ )||uj (ξ )|dξ

≤ maxj=1,...,m

(supt0≤ξ≤t

|uj (ξ )|)∫ t

t0

m∑j=1

|gij (t− ξ )|dξ

= maxj=1,...,m

(supt0≤ξ≤t

|uj (ξ )|)

m∑j=1

∫ t

t0

|gij (ξ )|dξ

≤ ‖ut‖L∞

m∑j=1

‖gij‖L1 , ∀ t ∈ [t0, τ ] .

Hence, it follows that

‖yτ‖L∞ = maxi=1,...,l

‖yiτ‖L∞ ≤ ‖uτ‖L∞ maxi=1,...,l

m∑j=1

‖gij‖L1

= ‖g‖L1‖uτ‖L∞ ,


Lemma A.7.2 A continuous-time LTI (proper) system y(s) = G(s)u(s) with impulse re-sponse matrix g(t) is BIBO stable if and only if its L1-norm is bounded, i.e., ‖g‖L1 <∞,or equivalently g(t) ∈ L1.

Proof. Sufficiency immediately follows from Lemma A.7.1. To show necessity, we willprove that if the L1-norm of g(t) is not bounded, then there exists at least one boundedinput that will force the output y(t) to diverge. If ‖g(t)‖L1 = ∞, then from Definition A.7.4it follows that there exists at least one entry {ij} in the impulse response matrix such that∫ ∞

0|gij (σ )|dσ = ∞ .

For every t let the j th element of the vector u(t−σ ) be

uj (t−σ ) ={

+1 if gij (σ ) ≥ 0,−1 if gij (σ )< 0,

and let other elements of the vector u(t−σ ) be zero. Then gi(σ )u(t−σ ) = |gij (τ )|, wheregi(σ ) denotes the ith row of the matrix g(σ ). This implies that the ith element of the output

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is given by

yi(t) =∫ t

0gi(σ )u(t−σ )dσ =

∫ t

0|gij (σ )|dσ .

Thus, we have limt→∞ yi(t) = ∞, which implies limt→∞ ‖y‖L∞ = ∞ and contradicts theassumption on system’s stability. �

Remark A.7.1 Notice that for a BIBO-stable LTI system with impulse response matrixg(t), if its input u(t) is uniformly bounded, i.e., u(t) ∈ L∞, then one has

‖y‖L∞ ≤ ‖g‖L1‖u‖L∞ .

Lemma A.7.3 The LTI system

x(t) = Ax(t)+Bu(t)

is finite-gain L-stable if and only if A is Hurwitz.

A proof of this lemma can be found in [86].

Lemma A.7.4 For a cascaded systemG(s) =G2(s)G1(s), whereG1(s) andG2(s) are stableproper systems, we have

‖G(s)‖L1 ≤ ‖G2(s)‖L1‖G1(s)‖L1 .

Proof. Let y1(s) =G1(s)u1(s), y2(s) =G2(s)u2(s), and u2(t) ≡ y1(t). Further, let u1(t) ∈L∞. From Lemma A.7.1 it follows that

‖y1‖L∞ ≤ ‖G1(s)‖L1‖u1‖L∞ ,

‖y2‖L∞ ≤ ‖G2(s)‖L1‖u2‖L∞ .

SinceG(s) =G2(s)G1(s), then y2(s) =G(s)u1(s) =G2(s)G1(s)u1(s) =G2(s)y1(s). Hencewe have

‖y2‖L∞ ≤ ‖G2(s)‖L1‖y1‖L∞ ≤ ‖G2(s)‖L1‖G1(s)‖L1‖u1‖L∞ . (A.9)

On the other hand, from BIBO stability of G(s) it follows that

‖y2‖L∞ ≤ ‖G(s)‖L1‖u1‖L∞ .

Next we show that ‖G(s)‖L1 is the least upper bound of ‖y2‖L∞ . This can be doneby contradiction. Without loss of generality, let ‖u1‖L∞ ≤ 1, and assume that there exists alower upper bound η, such that ‖y2‖L∞ ≤ η < ‖G(s)‖L1 . This implies that

supt≥0

‖y2(t)‖∞ ≤ η < ‖G(s)‖L1 .

Then, there exist t1 > 0 and index k such that

m∑j=1

∫ t1

0

∣∣gkj (t1 −σ )∣∣dσ > η ,

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where g(t) is the impulse response matrix for the system G(s). We can choose the controlsignal as

u1(σ ) ={

[sgn(gk1(t1 −σ )), . . . , sgn(gkm(t1 −σ ))]� , σ ∈ [0, t1] ,0 , σ > t1 .

Notice that for this control signal ‖u1‖L∞ ≤ 1. Then we have

(y2)k(t1) =m∑j=1

∫ t1

0gkj (t1 −σ )(u1)j (σ )dσ =

m∑j=1

∫ t1

0

∣∣gkj (t1 −σ )∣∣dσ > η .

This implies ‖y2‖L∞ > η, which contradicts the fact that η is an upper bound for ‖y2‖L∞ .Hence ‖G(s)‖L1 is the least upper bound for ‖y2‖L∞ . This fact, along with (A.9), completesthe proof. �

A.7.2 BIBO Stability for LTV Systems

Consider the LTV system

x(t) = A(t)x(t)+B(t)u(t) , x(t0) = x0 ,

y(t) = C(t)x(t) ,(A.10)

where x ∈ Rn is the system state andA(t) ∈ R

n×n,B(t) ∈ Rn×m,C(t) ∈ R

l×n are piecewise-continuous in time. Let G be the input-output map of this system for given t0 and zero initialcondition x0 = 0.

Definition A.7.5 The system in (A.10) is uniformly BIBO stable if there exists a posi-tive constant a > 0, such that for arbitrary t0 and arbitrary bounded input signal u(t) thecorresponding response for x0 = 0 verifies

supt≥t0

‖y(t)‖∞ ≤ a supt≥t0

‖u(t)‖∞ .

Definition A.7.6 The L1-norm of the m-input l-output LTV system in (A.3) is defined as

‖g‖L1 � max1≤i≤n

l∑j=1

‖gij‖L1

,

where

‖gij‖L1 � supt≥τ ,τ∈R+

∫ t

τ

|gij (t ,σ )|dσ ,

with gij (t ,σ ) being the {ij}th entry of the impulse response matrix.

We will use ‖G‖L1 � ‖g‖L1 to denote the L1-norm of the input-output map G of thesystem with impulse response matrix g(t , t0).

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Lemma A.7.5 Consider the system in (A.10) with zero initial condition x0 = 0. Supposeit has a uniformly asymptotically stable equilibrium at the origin, and there exist positiveconstants b, c > 0 such that for all t ≥ 0

‖B(t)‖ ≤ b , ‖C(t)‖ ≤ c .

Then the system is also uniformly BIBO stable. Further, if u(t) ∈ L∞, then

‖y‖L∞ ≤ ‖G‖L1‖u‖L∞ .


Lemma A.7.6 For a cascaded system G = G2G1, where G1 and G2 are uniformly BIBO-stable systems, we have

‖G‖L1 ≤ ‖G2‖L1‖G1‖L1 .

Proof. Let y1 = G1u1, y2 = G2u2, and u2(t) ≡ y1(t). Further, let u1(t) ∈ L∞. From LemmaA.7.5 it follows that

‖y1‖L∞ ≤ ‖G1‖L1‖u1‖L∞ ,

‖y2‖L∞ ≤ ‖G2‖L1‖u2‖L∞ .

Since G = G2G1, then y2 = Gu1 = G2G1u1 = G2y1. Hence we have

‖y2‖L∞ ≤ ‖G2‖L1‖y1‖L∞ ≤ ‖G2‖L1‖G1‖L1‖u1‖L∞ . (A.11)

On the other hand, from uniform BIBO stability of G it follows that

‖y2‖L∞ ≤ ‖G‖L1‖u1‖L∞ .

Next we show that ‖G‖L1 is the least upper bound of ‖y2‖L∞ . This can be done bycontradiction. Without loss of generality, let ‖u1‖L∞ ≤ 1, and assume that there exists alower upper bound η, such that ‖y2‖L∞ ≤ η < ‖G‖L1 . This implies that

supt≥0

‖y2(t)‖∞ ≤ η < ‖G‖L1 .

Then, there exist t0 and t1, t1 > t0, and index k such that

m∑j=1

∫ t1

t0

∣∣gkj (t1,σ )∣∣dσ > η ,

where g(t , t0) is the impulse response matrix for the system G. We can choose the controlsignal as

u1(σ ) ={

[sgn(gk1(t1,σ )), . . . , sgn(gkm(t1,σ ))]� , σ ∈ [t0, t1] ,0 , σ > t1 .

Notice that for this control signal ‖u1‖L∞ ≤ 1. Then we have

(y2)k(t1) =m∑j=1

∫ t1

t0

gkj (t1,σ )(u1)j (σ )dσ =m∑j=1

∫ t1

t0

∣∣gkj (t1,σ )∣∣dσ > η .

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This implies ‖y2‖L∞ > η, which contradicts the fact that η is an upper bound for ‖y2‖L∞ .Hence ‖G‖L1 is the least upper bound for ‖y2‖L∞ . This fact, along with (A.11), completesthe proof. �

A.8 Linear Parametrization of Nonlinear SystemsConsider the nonlinear mapf (t ,x) : [0, ∞)×R

n→ R, subject to the following assumptions.

Assumption A.8.1 (Uniform boundedness of f (t ,0)) There exists B > 0, such that|f (t ,0)| ≤ B for all t ≥ 0.

Assumption A.8.2 (Semiglobal uniform boundedness of partial derivatives) The mapf (t ,x) is continuous in its arguments, and moreover, for arbitrary δ > 0, there exist dft (δ)> 0and dfx (δ) > 0 independent of time, such that for all ‖x‖∞ ≤ δ the partial derivatives off (t ,x) with respect to t and x are piecewise continuous and bounded,∥∥∥∥∂f (t ,x)

∂x

∥∥∥∥1≤ dfx (δ) ,

∣∣∣∣∂f (t ,x)

∂t

∣∣∣∣≤ dft (δ) .

The next lemma proves that, subject toAssumptionsA.8.1 andA.8.2, on any finite timeinterval the nonlinear function f (t ,x(t)) can be linearly parameterized in two time-varyingparameters using ‖x(t)‖∞ as a regressor.

Lemma A.8.1 (see [30]) Let x(t) be a continuous and (piecewise)-differentiable functionof t for t ≥ 0. If ‖xτ‖L∞ ≤ ρ and ‖xτ‖L∞ ≤ dx for τ ≥ 0, where ρ and dx are some positiveconstants, then there exist continuous θ (t) and σ (t) with (piecewise)-continuous derivative,such that for all t ∈ [0,τ ]

f (t ,x(t)) = θ (t)‖x(t)‖∞ +σ (t) , (A.12)

where

|θ (t)|< θρ , |θ (t)|< dθ ,

|σ (t)|< σb , |σ (t)|< dσ ,

with θρ � dfx (ρ), σb � B + ε, in which ε > 0 is an arbitrary constant, and dθ , dσ arecomputable bounds.

Proof. The semiglobal uniform boundedness of the partial derivatives of f (t ,x) inAssump-tion A.8.2 implies that for arbitrary ‖x‖∞ ≤ ρ

|f (t ,x)−f (t ,0)| ≤ dfx (ρ)‖x‖∞ . (A.13)

Next, from Assumption A.8.1, it follows that if ‖x(0)‖∞ ≤ ρ, the following bound holds:

|f (0,x(0))| ≤ dfx (ρ)‖x(0)‖∞ +B < dfx (ρ)‖x(0)‖∞ +B+ ε ,

where ε > 0 is an arbitrary constant. This implies that there exist θ (0) and σ (0) such that

|θ (0)|< θρ , |σ (0)|< σb , (A.14)

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A.8. Linear Parametrization of Nonlinear Systems 279

and

f (0,x(0)) = θ (0)‖x(0)‖∞ +σ (0) . (A.15)

Notice that the choice of θ (0) and σ (0) is not unique. For the sake of proof we select arbitraryθ (0), σ (0) that verify (A.14) and (A.15).

We construct the trajectories of θ (t) and σ (t) according to the dynamics[θ (t)σ (t)

]= A−1

η (t)

[df (t ,x(t))

dt− θ (t) d‖x(t)‖∞

dt0

], (A.16)

where

Aη(t) =[ ‖x(t)‖∞ 1

−(σb−|σ (t)|) θρ −|θ (t)|]

, (A.17)

with any initial values satisfying (A.15). The determinant of Aη is

det(Aη(t)) = ‖x(t)‖∞(θρ −|θ (t)|)+σb−|σ (t)| . (A.18)

If

|θ (t)|< θρ , |σ (t)|< σb , (A.19)

then it follows from (A.18) that det(Aη(t)) = 0 for all t ∈ [0, τ ), where τ > 0 is an arbitraryconstant or ∞. Hence, it follows from (A.16), (A.17) that

df (t ,x(t))

dt=d(θ (t)‖x(t)‖∞ +σ (t)

)dt

, (A.20)

σ (t)

σb−|σ (t)| = θ (t)

θρ −|θ (t)| (A.21)

for all t ∈ [0, τ ). Using the selected initial condition from (A.15), we can integrate to obtain

f (t ,x(t)) = θ (t)‖x(t)‖∞ +σ (t) , (A.22)∫ τ−

0

σ (t)

σb−|σ (t)|dt =∫ τ−

0

θ (t)

θρ −|θ (t)|dt , (A.23)

where∫ τ−

0 (·)dt � limξ→τ−∫ ξ

0 (·)dt .Next we compute the left-hand side of (A.23) assuming that |σ (t)|< σb:∫ τ−

0

σ (t)

σb−|σ (t)|dt =∫ τ−

0

1

σb−|σ (t)|d(sgn(σ (t))|σ (t)|)

dtdt

= limt→τ

(sgn(σ (t)) ln(σb−|σ (t)|))

−sgn(σ (0)) ln(σb−|σ (0)|) .

Notice that sgn(σ (t)) is not differentiable when σ (t) crosses zero. However, the set of points{ti}, where σ (ti) = 0, is a countable set with Lebesgue measure zero. This allows us to

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exclude these points while taking the integral, and thus we can ensure d(sgn(σ (t))|σ (t)|) =sgn(σ (t))d(|σ (t)|). Similarly, assuming that |θ (t)| < θρ , the right-hand side of (A.23) isgiven by ∫ τ−

0

θ (t)

θρ −|θ (t)|dt = limt→τ

(sgn(θ (t)) ln(θρ −|θ (t)|))

−sgn (θ (0)) ln(θρ −|θ (0)|) .

Using these arguments, we rewrite (A.23):

limt→τ

(sgn(σ (t)) ln(σb−|σ (t)|))− sgn(σ (0)) ln(σb−|σ (0)|)

= limt→τ

(sgn(θ (t)) ln(θρ −|θ (t)|))− sgn(θ (0)) ln(θρ −|θ (0)|) .

(A.24)

In what follows, we prove (A.19) by contradiction. If (A.19) is not true, since θ (t)and σ (t) are continuous, it follows from (A.14) that there exists τ ∈ [0, τ ] such that either

(i) limt→τ

|θ (t)| = θρ or (A.25)

(ii) limt→τ

|σ (t)| = σb , (A.26)

while |θ (t)|< θρ , |σ (t)|< σb for all t ∈ [0, τ ).

(i) In this case we have

| limt→τ

(sgn(θ (t)) ln(θρ −|θ (t)|))| = ∞ .

Since it is obvious that sgn(σ (0)) ln(σb−|σ (0)|) and sgn(θ (0)) ln(θρ−|θ (0)|) are bounded,it follows from (A.24) that

| limt→τ

(sgn(σ (t)) ln(σb−|σ (t)|))| = ∞ ,

and hence

limt→τ

|σ (t)| = σb . (A.27)

Thus, from (A.22) we have

limt→τ

f (t ,x(t)) = limt→τ

(θ (t)‖x(t)‖∞ +σ (t)

),

which along with (A.25) and (A.27) implies that

| limt→τ

f (t ,x(t))| = |f (τ ,x(τ ))| = θρ‖x(τ )‖∞ +σb . (A.28)

From (A.13) and Assumption A.8.1, it follows that

|f (τ ,x(τ ))| ≤ dfx (ρ)‖x(τ )‖∞ +B = θρ‖x(τ )‖∞ +σb− ε ,

which contradicts (A.28), and therefore (A.25) is not true.

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A.9. LTV Representation of Systems with Linear Unmodeled Dynamics 281

(ii) Following the same steps as above, one can derive a contradicting argumentto (A.26).

Since (A.25), (A.26) are not true, then the relationships in (A.19) hold. Equality (A.12)follows from (A.19) and (A.22) directly.

Further, if‖xτ‖L∞ is bounded, then in light ofAssumptionA.8.2, df (t ,x(t))dt

and d‖x(t)‖∞dt

are bounded, although the derivative d‖x(t)‖∞dt

may not be continuous. Since θ (t) is bounded,

then df (t ,x(t))dt

− θ (t) d‖x(t)‖∞dt

is bounded. From (A.19), it follows that detAη(t) = 0, andtherefore we conclude from (A.16) that θ (t) and σ (t) are bounded. This concludes theproof. �


x(t) = Amx(t)+b(ωu(t)+f (t ,x(t)))

, x(0) = x0 ,

y(t) = c�x(t) ,(A.29)

where x(t) ∈ Rn is the system state; u(t) ∈ R is the given bounded system input; y(t) ∈ R

is the system output; Am ∈ Rn×n is a known Hurwitz matrix; b, c ∈ R

n are known vectors;ω ∈ R is an unknown parameter; and f : R×R

n → R is an unknown nonlinear function.Subject toAssumptionsA.8.1 andA.8.2, the nonlinear system in (A.29) can be rewritten overt ∈ [0, τ ] for arbitrary τ ≥ 0 as a semilinear time-varying system with bounded parametersthat have bounded derivatives:

x(t) = Amx(t)+b(ωu(t)+ θ (t)‖x(t)‖∞ +σ (t))

, x(0) = x0 ,

y(t) = c�x(t) .

A.9 Linear Time-Varying Representation of Systems withLinear Unmodeled Dynamics

Consider the following dynamics:

xz(t) = g(t ,xz(t),x(t)) , xz(0) = x0 ,

z(t) = g0(t ,xz(t)) ,

where x(t) ∈ Rn is a bounded differentiable signal with bounded derivative, the functions

g : R×Rm×R

n → Rm, g0 : R×R

m → Rl are unknown nonlinear maps continuous in their

arguments, and xz(t) ∈ Rm and z(t) ∈ R

l represent the state and the output of the system.Further, consider the map f (t ,x,z) : R×R

n×Rl → R on the time interval t ∈ [0, τ )

for arbitrary τ ≥ 0. Let X � [x�, z�]�, and with a slight abuse of language let f (t ,X) �f (t ,x,z), i = 1,2. The function f (·) verifies the following assumptions.

Assumption A.9.1 (Uniform boundedness of f (t ,0)) There exists B > 0, such that|f (t ,0)| ≤ B holds for all t ≥ 0.

Assumption A.9.2 (Semiglobal uniform boundedness of partial derivatives) The mapf (t ,x) is continuous in its arguments, and for arbitrary δ > 0, there exist dft (δ) > 0 and

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dfx (δ)> 0 independent of time, such that for all ‖X‖∞ ≤ δ the partial derivatives of f (t ,X)with respect to t and X are piecewise-continuous and bounded:∥∥∥∥∂f (t ,X)

∂X

∥∥∥∥1≤ dfx (δ) ,

∣∣∣∣∂f (t ,X)

∂t

∣∣∣∣≤ dft (δ) .

Assumption A.9.3 (Stability of unmodeled dynamics) The z-dynamics are BIBO stable;i.e., there exist L1 > 0 and L2 > 0 such that for all t ≥ 0

‖zt‖L∞ ≤ L1‖xt‖L∞ +L2 .

Let δ � max{δ+γ , L1(δ+γ )+L2} for an arbitrary γ > 0, and define Lδ � δδdfx (δ).

The next lemma proves that, subject to Assumptions A.9.1, A.9.2, and A.9.3, on anyfinite time interval the nonlinear function f (t ,x(t),z(t)) can be linearly parameterized intwo time-varying parameters using ‖xt‖L∞ as a regressor.

Lemma A.9.1 (see [31]) If ‖xτ‖L∞ ≤ ρ and ‖xτ‖L∞ ≤ dx for τ ≥ 0, where ρ and dx aresome positive constants, then there exist continuous and (piecewise)-differentiable θ (t) ∈ R

and σ (t) ∈ R with bounded derivatives, such that for all t ∈ [0,τ ]

f (t ,X(t)) = θ (t)‖xt‖L∞ +σ (t) ,

and

|θ (t)|< θρ , |θ (t)|< dθ ,

|σ (t)|< σb , |σ (t)|< dσ ,

with θρ � Lρ , σb � LρL2 +B+ ε, in which ε > 0 is an arbitrary constant, and dθ , dσ arecomputable bounds.

Proof. From Assumption A.9.3, it follows that for all t ∈ [0, τ ), if ‖x(t)‖∞ ≤ ρ, then

‖zt‖L∞ ≤ L1‖xt‖L∞ +L2 < L1(ρ+γ )+L2 ,

where γ > 0 is an arbitrary constant. Let ρ � max{ρ+γ , L1(ρ+γ )+L2}. Then

‖X(t)‖∞ ≤ max{‖xt‖L∞ , L1‖xt‖L∞ +L2}< ρ .

The uniform boundedness of the partial derivatives of f (t ,X) in Assumption A.9.2 impliesthat for arbitrary ‖X‖∞ ≤ ρ

|f (t ,X)−f (t ,0)| ≤ dfx (ρ)‖X‖∞ .

Further, from Assumptions A.9.1 and the definition of Lδ , it follows that

|f (t ,X(t))| ≤ dfx (ρ)‖X(t)‖∞ +B < Lρ‖xt‖L∞ +LρL2 +B+ ε , (A.30)

where ε > 0 is an arbitrary constant.Next, we follow the same steps as in the proof of Lemma A.8.1, replacing the ‖x(t)‖∞

norm with ‖xt‖L∞ norm and keeping in mind the difference in the definition of σb. �

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A.10. LTV Representation of Systems with Linear Unmodeled Actuator Dynamics283

The result of Lemma A.9.1 can be extended to the vector case, when f (t ,x,z) :R×R

n×Rl → R

m. If Assumptions A.9.1, A.9.2 hold for each component of f (t ,x,z) withthe same upper bound, then they can be rewritten as

‖f (t ,0)‖∞ ≤ B ,

∥∥∥∥∂f (t ,X)

∂X

∥∥∥∥∞≤ dfx (δ) , and

∥∥∥∥∂f (t ,X)

∂t

∥∥∥∥∞≤ dft (δ) ,

where the first and the third norms are vector ∞-norm, and the second is a matrix-induced∞-norm. This leads to the result stated in the following lemma.

Lemma A.9.2 If ‖xτ‖L∞ ≤ ρ and ‖xτ‖L∞ ≤ dx for τ ≥ 0, whereρ and dx are some positiveconstants, then there exist differentiable θ (t) ∈ R

m and σ (t) ∈ Rm with bounded derivatives,

such that for all t ∈ [0,τ ]

f (t ,X(t)) = θ (t)‖xt‖L∞ +σ (t) ,

where

‖θ (t)‖∞ < θρ , ‖θ (t)‖∞ < dθ ,

‖σ (t)‖∞ < σb , ‖σ (t)‖∞ < dσ ,

with θρ � Lρ , σb � LρL2 +B+ ε, in which ε > 0 is an arbitrary constant, and dθ , dσ arecomputable bounds.


x(t) = Amx(t)+b(ωu(t)+f (t ,x(t),z(t)))

, x(0) = x0 ,

y(t) = c�x(t) ,

xz(t) = g(t ,xz(t),x(t)) , xz(0) = x0 ,

z(t) = g0(t ,xz(t)) ,

(A.31)

where x(t) ∈ Rn is the system state; u(t) ∈ R is the given bounded system input; y(t) ∈ R

is the system output; Am ∈ Rn×n is a known Hurwitz matrix; b, c ∈ R

n are known vectors;ω ∈ R is an unknown parameter; and f : R × R

n × Rl → R, g : R × R

m × Rn → R

m,g0 : R×R

m → Rl are unknown nonlinear maps. Then, subject to Assumptions A.9.1–A.9.3,

the nonlinear system in (A.31) can be rewritten over t ∈ [0, τ ] for arbitrary τ ≥ 0 as asemilinear time-varying system with bounded parameters that have bounded derivatives:

x(t) = Amx(t)+b(ωu(t)+ θ (t)‖xt‖L∞ +σ (t))

, x(0) = x0 ,

y(t) = c�x(t) .

A.10 Linear Time-Varying Representation of Systems withLinear Unmodeled Actuator Dynamics

Consider the following system given by its transfer function:

µ(s) = F (s)u(s) , (A.32)

whereµ(s), u(s) ∈ R are the Laplace transforms of the system output and input, respectively,and F (s) is an unknown BIBO-stable transfer function with known DC gain and knownupper bound for its L1-norm.

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Assumption A.10.1 There exists LF > 0 verifying ‖F (s)‖L1 ≤ LF . Also, we assume thatthere exist known constants ωl , ωu ∈ R, verifying

0< ωl ≤ F (0) ≤ ωu ,

where, without loss of generality, we have assumed F (0)> 0.

The next lemma proves that the dynamical relationship between u(t) and µ(t) in(A.32) can be equivalently replaced by an algebraic relationship, linear in its structure.

Lemma A.10.1 (see [28]) Consider the system in (A.32). If for some τ > 0

‖uτ‖L∞ ≤ ρu , ‖uτ‖L∞ ≤ du ,

then there exist ω and differentiable σ (t) over t ∈ [0, τ ], such that

µ(t) = ωu(t)+σ (t) ,

where

ω ∈ (ωl , ωu), |σ (t)| ≤ σb , |σ (t)| ≤ dσ ,

with σb � ‖F (s)− (ωl+ωu)/2‖L1ρu, and dσ � ‖F (s)− (ωl+ωu)/2‖L1du.

Proof. Let

ω � ωl+ωu2

, σ (t) � µ(t)−ωu(t) .

The second equation leads to

µ(t) = ωu(t)+σ (t) ,

and further

σ (s) = (F (s)−ω)u(s) , sσ (s) = (F (s)−ω)su(s) .

Lemma A.7.1 implies

‖στ‖L∞ ≤∥∥∥∥F (s)− ωl+ωu

2

∥∥∥∥L1

‖uτ‖L∞ ≤∥∥∥∥F (s)− ωl+ωu

2

∥∥∥∥L1

ρu

and

‖στ‖L∞ ≤∥∥∥∥F (s)− ωl+ωu

2

∥∥∥∥L1

du . �

Consider the system

x(t) = Amx(t)+bµ(t) , x(0) = x0 ,

y(t) = c�x(t) ,

µ(s) = F (s)u(s) ,

(A.33)

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A.11. Properties of Controllable Systems 285

where x(t) ∈ Rn is the state of the system; Am ∈ R

n×n and b ∈ Rn are a known Hurwitz

matrix and a constant vector, respectively; c ∈ Rn is a known constant vector; µ(t) ∈ R is

the actuator output; y(t) ∈ R is the system output; u(t) ∈ R is the control input; and F (s) isan unknown BIBO-stable transfer function.

Using LemmaA.10.1, the system (A.33) with multiplicative uncertainty can be rewrit-ten as the following equivalent LTI system with unknown constant system input gain andadditive disturbance σ (t), which is bounded and has bounded derivative:

x(t) = Amx(t)+b(ωu(t)+σ (t))

, x(0) = x0 ,

y(t) = c�x(t) .

A.11 Properties of Controllable Systems

A.11.1 Linear Time-Invariant Systems

Consider an LTI system given by

x(s) = (sI−A)−1bu(s) , (A.34)

where x(s), u(s) are the Laplace transforms of the system state and the system input, A ∈Rn×n, b ∈ R

n, and assume that

(sI−A)−1b = N (s)

D(s), (A.35)

whereD(s) = det(sI−A), andN (s) is a n×1 vector with its ith element being a polynomialfunction

Ni(s) =n∑j=1

Nij sj−1 . (A.36)

Lemma A.11.1 If (A, b) is controllable, then the matrix N of entries Nij is full rank.

Proof. Controllability of (A, b) implies that given an initial condition x(0) = 0, and arbitraryt1 and xt1 , there exists u(τ ), τ ∈ [0, t1], such that x(t1) = xt1 . IfN is not full rank, then thereexists a µ ∈ R

n, µ = 0, such that µ�N (s) = 0. Thus, for x(0) = 0 one has

µ�x(s) = µ�N (s)

D(s)u(s) = 0, ∀ u(s) ,

which implies that, in particular, x(t) =µ for any t . This contradicts the fact that x(t1) = xt1can be any point in R

n. Thus, N must be full rank. �

Corollary A.11.1 If the pair (A, b) in (A.34) is controllable, then there exists co ∈ Rn, such

that c�o N (s)/D(s) has relative degree one, i.e., deg(D(s))−deg(c�o N (s)) = 1, andN (s) hasall its zeros in the left half plane.

Proof. It follows from (A.35) that for arbitrary vector co ∈ Rn

c�o (sI−A)−1b = c�o N [sn−1 · · · 1]�

D(s),

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where N ∈ Rn×n is the matrix with its ith row j th column entry Nij introduced in (A.36).

Since (A, b) is controllable, it follows from Lemma A.11.1 that N is full rank. Consider anarbitrary vector c ∈ R

n such that c�[sn−1 · · · 1]� is a stable (n−1)-order polynomial, andlet co = (N−1)�c. Then

c�o (sI−A)−1b = c�[sn−1 · · · 1]�

D(s)

has relative degree 1 with all its zeros in the left half plane. �

A.11.2 Linear Time-Varying Systems

Consider the following time-varying system dynamics:

x(t) = A(t)x(t)+b(t)u(t) , x(t0) = x0 , (A.37)

where x(t) ∈ Rn is the state of the system, u(t) ∈ R is the input of the system, and A(t) ∈

Rn×n, b(t) ∈ R

n are piecewise-continuous in time.

Definition A.11.1 (see [158]) The system in (A.37) is uniformly controllable if the control-lability matrix

Qc(t) = [p0(t), p1(t), . . . , pn−1(t)] ∈ Rn×n ,

where pk+1(t) = −A(t)pk(t)+ pk(t) with p0(t) = b(t) is nonsingular for all t ≥ t0. The rep-resentation is strongly controllable if the controllability matrixQc(t) is strongly nonsingular,i.e., there exists a constant q > 0, such that

|det(Qc(t))| ≥ q , ∀ t ≥ t0 .

Lemma A.11.2 (see [156, 157, 167]) Consider the single-input time-varying system in(A.37). There exists a nonsingular continuously differentiable transformation T (t) reducingthe system to its controllable (phase-variable) canonical form given by

˙x(t) = A(t)x(t)+ bu(t) ,

where A(t) = (T (t)A(t)+ T (t))T −1(t), b = T (t)b(t), with

A(t) =

0 1 0 · · · 00 0 1 · · · 0...

......

. . ....

0 0 0 · · · 1−a1(t) −a2(t) −a3(t) · · · −an(t)

, b =

00...01

,

if and only if (A(t), b(t)) is uniformly controllable. Moreover, ifA(t) and b(t) are uniformlybounded and smooth, and if (A(t), b(t)) is strongly controllable, then the transformationT (t)is uniformly bounded, and the resulting entries ai(t) in A(t) (coefficients of the characteristicpolynomial) are also uniformly bounded.

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A.12. Special Case of State-to-Input Stability 287

A.12 Special Case of State-to-Input StabilityConsider the following dynamics:

x(t) = Ax(t)+bu(t) , x(0) = 0, (A.38)

where x(t) ∈ Rn is the system state vector, u(t) ∈ R is the control input (bounded and

piecewise-continuous), A ∈ Rn×n, and b ∈ R

n. Let G(s) � (sI−A)−1b so that

x(s) =G(s)u(s) .

The solution for x(t) from (A.38) can be written explicitly:

x(t) =∫ t

0eA(t−τ )bu(τ )dτ . (A.39)

In Section A.7 we defined the notion of BIBO stability for linear systems. Notice thatthe output of the system in (A.38) is the state x(t) itself. Therefore, the bound on the normof the system state can be written in the following form:

‖xτ‖L∞ ≤ γ ‖uτ‖L∞ , γ = ‖G(s)‖L1 , ∀ τ ∈ [0, ∞) .

Thus, for a BIBO-stable linear system, it is always possible to upper bound the norm of theoutput by a function of the norm of the input. Looking at (A.39) one may ask the oppositequestion, namely, whether it is possible to find an upper bound on the system input in termsof its output without invoking the derivatives. While in general the answer to this question isnegative [112], we prove that a similar upper bound can be derived for the low-pass-filteredinput.

A.12.1 Linear Time-Invariant Systems

Lemma A.12.1 Let the pair (A, b) in (A.38) be controllable. Further, letF (s) be an arbitrarystrictly proper BIBO-stable transfer function. Then, there exists a proper and stable G1(s),given by

G1(s) � F (s)

c�o G(s)c�o ,

where c0 ∈ Rn, and c�o G(s) is a minimum phase transfer function with relative degree one,

such that

F (s)u(s) =G1(s)x(s) .

Proof. From Corollary A.11.1, it follows that there exists co ∈ Rn, such that c�o G(s) has

relative degree one, and c�o G(s) has all its zeros in the left half plane. Hence, we can write

F (s)u(s) = F (s)

c�o G(s)c�o G(s)u(s) =G1(s)x(s) ,

where the properness of G1(s) is ensured by the fact that F (s) is strictly proper, whilestability follows immediately from its definition. �

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Letting µ(s) = F (s)u(s), it follows from Lemma A.7.1 that

‖µ‖L∞ ≤ ‖G1(s)‖L1‖x‖L∞ .

A.12.2 Linear Time-Varying Systems

Consider the system

x(t) = A(t)x(t)+b(t)u(t) , x(0) = x0 , (A.40)

where x(t) ∈ Rn is the system state, u(t) ∈ R is the input, and A(t) ∈ R

n×n, b(t) ∈ Rn are

piecewise-continuous in time.The next lemma shows that, for arbitrary strictly proper BIBO-stable transfer func-

tion F (s), and under appropriate assumptions on the system in (A.40), the output of thesystem F (s) to the input u(t) can be upper bounded in terms of the output x(t).

Lemma A.12.2 Let the system in (A.40) be strongly controllable, and let A(t) and b(t) beuniformly bounded and smooth. Further, let F be the input-output map of F (s). Then forarbitrary τ ≥ 0 we have

‖(F u)τ‖L∞ ≤ γ ‖xτ‖L∞ ,

where

γ �(n−1∑i=0

‖F ai+1‖L1

∥∥∥∥ si

cnsn−1 +·· ·+ c1

∥∥∥∥L1

+∥∥∥∥ F (s)sn

cnsn−1 +·· ·+ c1

∥∥∥∥L1

)‖c�T ‖L∞ ,

and ci , i = 1,2, . . . ,n, are the coefficients of an arbitrary Hurwitz polynomial p(s) �cns

n−1 + ·· ·+ c1, while T (t) and ai(t) are the transformation matrix and the coefficientsof the characteristic polynomial for the system given in (A.40), defined according toLemma A.11.2.

Proof. Lemma A.11.2 implies that for all t ∈ [0, τ ] there exists a uniformly boundedtransformation T (t), such that x(t) � T (t)x(t) transforms the dynamics in (A.40) to itscontrollable canonical form:

˙x(t) = A(t)x(t)+ bu(t) ,

where A(t) and b are given by Lemma A.11.2. Then, the relationship between ϕ(t) � x1(t)and u(t) is described by the following ODE:

dnϕ(t)

dtn+an(t)

dn−1ϕ(t)

dtn−1+·· ·+a1(t)ϕ(t) = u(t) ,

where the time-varying coefficients ai(t) are defined in Lemma A.11.2. Applying the mapF to the both sides of this equation, taking the L∞-norms, and using the triangular norminequality, we obtain the following bound:

‖(F u)τ‖L∞ ≤∥∥∥(F ϕ(n)

)τ

∥∥∥L∞

+∥∥∥(F anϕ(n−1) +·· ·+F a1ϕ

)τ

∥∥∥L∞

. (A.41)

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A.12. Special Case of State-to-Input Stability 289

Let

z(t) � c�x(t) ,

where c � [c1, . . . , cn]�. Then, it follows that

ϕ(i)(s) = si

cnsn−1 +·· ·+ c1z(s) ,

which implies

F (s)ϕ(n)(s) = F (s)sn

cnsn−1 +·· ·+ c1z(s) .

Thus, (A.41) leads to

‖(F u)τ‖L∞ ≤(∥∥∥∥ F (s)sn

cnsn−1 +·· ·+ c1

∥∥∥∥L1

+n−1∑i=0

‖F ai+1‖L1

∥∥∥∥ si

cnsn−1 +·· ·+ c1

∥∥∥∥L1

)‖zτ‖L∞ .

Noticing that z(t) = c�T (t)x(t), one can write ‖zτ‖L∞ ≤ ‖c�T ‖L∞‖xτ‖L∞ . This leads to

‖(F u)τ‖L∞ ≤ γ ‖xτ‖L∞ ,


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Appendix B

Projection Operator forAdaptation Laws

Projection-based adaptation laws are used quite often to prevent parameter drift in adaptationschemes. In this appendix we introduce some definitions and facts from convex analysisand present some properties of the projection operator used throughout the book.

Definition B.1 (see [24]) �⊆ Rn is a convex set if for all x ,y ∈� the following holds:

λx+ (1−λ)y ∈� , ∀ λ ∈ [0, 1] .

The illustrations of convex and nonconvex sets are shown in Figure B.1.

(a) Convex set (b) Nonconvex set

Figure B.1: Illustration of convex and nonconvex sets.

Definition B.2 f : Rn → R is a convex function, if for all x ,y ∈ R

n the following holds:

f (λx+ (1−λ)y) ≤ λf (x)+ (1−λ)f (y) , ∀ λ ∈ [0 1] .

A sketch of a convex function is presented in Figure B.2.

Lemma B.1 Let f : Rn → R be a convex function. Then for arbitrary constant δ, the set

�δ � {θ ∈ Rn|f (θ ) ≤ δ} is convex. The set �δ is called a sublevel set.


Lemma B.2 Let f : Rn → R be a continuously differentiable convex function. Choose

a constant δ and consider the convex set �δ � {θ ∈ Rn|f (θ ) ≤ δ}. Let θ ,θ∗ ∈ �δ and

291

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292 Appendix B. Projection Operator for Adaptation Laws

t

f (t)

x yz

f (z)

Figure B.2: Illustration of convex function.

f (θ∗)< δ and f (θ ) = δ (i.e., θ∗ is not on the boundary of �δ , while θ is on the boundaryof �δ). Then, the following inequality takes place:

(θ∗ − θ )�∇f (θ ) ≤ 0,

where ∇f (θ ) is the gradient vector of f (·) evaluated at θ .

The proof of this lemma immediately follows from [145, Theorem 25.1]. Next, weintroduce the definition of the projection operator.

Definition B.3 (see [144]) Consider a convex compact set with a smooth boundary given by

�c � {θ ∈ Rn | f (θ ) ≤ c} , 0 ≤ c ≤ 1,

where f : Rn → R is the following smooth convex function:

f (θ ) � (εθ +1)θ�θ − θ2max

εθ θ2max

,

with θmax being the norm bound imposed on the vector θ , and εθ > 0 is the projectiontolerance bound of our choice. The projection operator is defined as

Proj(θ ,y) �

y if f (θ )< 0,y if f (θ ) ≥ 0 and ∇f�y ≤ 0,

y− ∇f‖∇f ‖

⟨ ∇f‖∇‖f , y

⟩f (θ ) if f (θ ) ≥ 0 and ∇f�y > 0.

Property B.1 (see [144]) The projection operator Proj(θ ,y) does not alter y if θ belongsto the set �0 � {θ ∈ R

n | f (θ ) ≤ 0}. In the set {θ ∈ Rn | 0 ≤ f (θ ) ≤ 1}, if ∇f�y > 0, the

Proj(θ ,y) operator subtracts a vector normal to the boundary �f (θ ) = {θ ∈ Rn | f (θ ) =

f (θ )}, so that we get a smooth transformation from the original vector field y to an inwardor tangent vector field for �1.

Property B.2 (see [144]) Given the vectors y ∈ Rn, θ∗ ∈�0 ⊂�1 ⊂ R

n, and θ ∈�1, wehave

(θ − θ∗)�(Proj(θ ,y)−y) ≤ 0. (B.1)

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Appendix B. Projection Operator for Adaptation Laws 293

yProj(θ ,y)

∇f (θ )

projection

scaled by (1−f (θ ))

0

θ

f (θ ) = 0

f (θ ) = 1

y

Proj(θ ,y) ∇f (θ )

projection

scaled by 0

0

θ

f (θ ) = 0

f (θ ) = 1

Figure B.3: Illustration of the projection operator.

Indeed,

(θ∗ − θ )�(y−Proj(θ ,y))

=

0 if f (θ )< 0,0 if f (θ ) ≥ 0 and ∇f T y ≤ 0,(θ∗ − θ )�∇f︸︷︷︸

≤0

∇f T y︸︷︷︸≥0

f (θ )︸︷︷︸≥0

‖∇f ‖2 if f (θ ) ≥ 0 and ∇f�y > 0.

Changing the signs on the left side, one gets (B.1). An illustration of the projection operatoris shown in Figure B.3.

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294 Appendix B. Projection Operator for Adaptation Laws

Example B.1 In order to illustrate the use of the projection operator in the adaptation laws,we consider the system in Section 1.2.1 with the same direct model reference adaptivecontroller. The only difference is that we replace the adaptive law in (1.7) by the followingprojection-based adaptation law:

kx(t) = �Proj(kx(t), −x(t)e�(t)Pb), kx(0) = kx0 . (B.2)

Since the structure of the control law and the definition of the reference model do not change,the tracking error dynamics can still be written as

e(t) = Ame(t)+bk�x (t)x(t) , e(0) = 0.

If we consider the same Lyapunov function candidate as in (1.8), the adaptation law in (B.2)leads to

V (t) = −e�(t)Qe(t)+2k�x (t)(

Proj(kx(t), −x(t)e�(t)Pb)+x(t)e�(t)Pb)

.

Then, Property B.2 implies that

k�x (t)(

Proj(kx(t), −x(t)e�(t)Pb)+x(t)e�(t)Pb)

≤ 0,

which yields

V (t) ≤ −e�(t)Qe(t) ≤ 0.

From Barbalat’s lemma one concludes that e(t) → 0 as t → ∞. The advantage of us-ing projection-type adaptation is that one ensures boundedness of the adaptive parame-ters by definition. This property is exploited in the analysis of the L1 adaptive controlarchitectures.

Remark B.1 Since the MRAC architecture and the predictor-based MRAC architecturelead to the same error dynamics from the same initial conditions, one can also employ theprojection-based adaptation law in predictor-based MRAC and refer to Barbalat’s lemmato conclude asymptotic stability.

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Appendix C

Basic Facts on Linear MatrixInequalities

C.1 Linear Matrix Inequalities and Convex OptimizationIn this section, we give some properties of LMIs. An LMI is an inequality of the form

F (x) = F0 +m∑i=1

xiFi < 0, (C.1)

where x ∈ Rm is the decision variable andFi ∈ R

n×n, i = 0,1, . . . ,m, are constant symmetricmatrices. By F (x)< 0 we mean that F (x) is negative definite.

From the definition of F (x), it follows that it is an affine function of the elementsof x. An important property of LMIs is that the set {x|F (x)< 0} is convex, that is, the LMIin (C.1) forms a convex constraint on x.

Lemma C.1.1 (see [23]) A matrix A is positive (or negative) definite if and only if T �ATis positive (or negative) definite for arbitrary nonsingular matrix T .

Lemma C.1.2 (Schur complement lemma [23]) The nonlinear inequalities

R(x)< 0, Q(x)−S(x)R(x)−1S�(x)< 0,

where Q(x) =Q�(x), R(x) = R�(x) and S(x) depend affinely on x, are equivalent to theLMI [

Q(x) S(x)S�(x) R(x)

]< 0.

We consider the following two problems [23]:

LMI Problem Given an LMI in (C.1), the corresponding LMI problem is to find xf thatverifies F (xf )< 0, or to prove that the LMI is infeasible. Further, by solving the LMIF (x)< 0, we mean solving the corresponding LMI problem.

Eigenvalue Problem The eigenvalue problem is the minimization of the maximum eigen-value of a matrix that depends affinely on a variable, subject to an LMI constraint (orproof that the constraint is infeasible), i.e., minimize λ subject to A(x, λ)< 0, whereA(x, λ) is affine in (x, λ).

295

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296 Appendix C. Basic Facts on LMIs

C.2 LMIs for Computation of L1-Norm of LTI SystemsConsider the following LTI system:

x(t) = Ax(t)+bu(t) , x(0) = x0 ,

y(t) = Cx(t) ,(C.2)

where A ∈ Rn×n, C ∈ R

l×n are given matrices; b ∈ Rn is a given vector; x(t) ∈ R

n is thestate; y(t) ∈ R

l is the system output; and u(t) ∈ R is the bounded exogenous input. Let g(t)be the impulse response for this system. The following theorem provides a conservativeupper bound for the L1-norm of the system.

Theorem C.2.1 (An upper bound on the L1-norm [2, 133]) If there exists a symmetricpositive definite matrix Pα ∈ R

n×n solving the LMI

APα+PαA� +αPα+ 1

αbb� ≤ 0 (C.3)

for some α > 0, then

‖g‖L1 ≤ ‖CPαC�‖2 .

Remark C.2.1 From the Schur complement lemma, it follows that the inequality in (C.3)can be written as [

APα+PαA� +αPα b

b� −α]

≤ 0. (C.4)

Remark C.2.2 (Bound on feasible solution for α) The inequality in (C.4) implies that aconservative bound on the feasible solution for α is given by

α ∈ (0,−2�(λmax(A))] ,

where �(λmax(A)) is the maximum real part of the eigenvalues of the matrix A.

Based on the result of Theorem C.2.1 we define the least upper bound on the L1-normof the system.

Definition C.2.1 The least upper bound of the L1-norm over all possible α is called∗∗∗-norm [2],

‖g‖∗ � infα

‖CPαC�‖2 . (C.5)

Remark C.2.3 An important question is, how tight is the ∗-norm for approximation of theL1-norm? In [168], a family of systems was constructed that illustrates that this approxi-mation can be very poor. It is, nonetheless, quite possible for the ∗-norm upper bound tobe useful. It is known, for example, that the simplex algorithm for solving linear program-ming problems has very poor worst-case computational complexity, but this algorithm hasproved to be effective on real-world problems. This example suggests that the engineeringexperience, and not the theoretical worst-case behavior, will be the definitive test of the∗-norm approach. In [2], the authors suggest that for many systems the ∗-norm is a quitetight upper bound on the L1-norm.

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C.3. LMIs for Stability Analysis of Systems with Time Delay 297

C.3 LMIs for Stability Analysis of Systems with Time DelayConsider the following LTI system in the presence of time delay:

x(t) = Ax(t)+Adx(t− τ ) , t > 0,

x(t) = φ(t) , t ∈ [−τ ,0] ,(C.6)

where A and Ad ∈ Rn×n are the matrices of appropriate dimension, φ(t) is the given initial

condition, and τ > 0 denotes the time delay. The next theorem gives a sufficient conditionfor stability of this system dependent upon the delay.

Theorem C.3.1 (see [44]) The system (C.6) is asymptotically stable for τ ∈ [0, τ ] for someτ > 0 if there exist P > 0,P1 > 0, and P2 > 0 of appropriate dimensions, satisfying

E τPA� τPA�d

τAP −τP1 0τAdP 0 −τP2

< 0, (C.7)

where E � P (A+Ad )� + (A+Ad )P + τAd (P1 +P2)A�d .

Letting P = P1 = P2, we obtain the following (conservative) result.

Lemma C.3.1 If the LMIP (A+Ad )� + (A+Ad )P PA� PA�

d AdP

AP − 1ηP 0 0

AdP 0 − 1ηP 0

PA�d 0 0 − 1

2η P

≤ 0 (C.8)

has a positive definite solution for P , then the system (C.6) is stable for arbitrary τ ∈ [0, η].

The proof follows from the Schur complement (Lemma C.1.2), applied to the inequalitiesin (C.7).

C.4 LMIs in the Presence of Uncertain ParametersWe finally recall two well-known results from [23], which help to obtain a finite number ofLMIs when the uncertain parameter in the original LMI lies in a convex polytope.

Lemma C.4.1 (Vertexization of uncertain LMIs) Let � be a convex hull and let �0 bethe set of its vertices with finite number of elements. Then, the set{

x ∈ Rm : F (x,θ ) = F0(θ )+

m∑i=1

xiFi(θ )< 0 ∀ θ ∈�}

is nonempty if and only if the set{x ∈ R

m : F (x,θ ) = F0(θ )+m∑i=1

xiFi(θ )< 0 ∀ θ ∈�0

}

is nonempty, provided that Fi(θ ) affinely depend on θ ∈� for each i = 0, . . . ,m.

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298 Appendix C. Basic Facts on LMIs

Proof. For the only-if direction, the proof immediately follows from the fact that �0 ⊂�.For the if direction, it follows from the definition of a convex hull and the convexity ofF (x,θ ) in θ , given by

F (x,θ ) = F(x,∑i

ρiθi

)<∑i

ρiF (x,θi)< 0,

which holds for every fixed x ∈ Rm and completes the proof. �

Corollary C.4.1 (Polytopic uncertain system) Assume that the system matricesA andAdin (C.6) are not precisely known but belong to a polytope, such that they can be representedas a convex combination of the vertices of the polytope:

[A Ad

]=nv∑j=1

ρj

[A(j ) A

(j )d

], (C.9)

whereA(j ), A(j )d are the vertices of the polytope, ρj ∈ [0,1] for each index j ,

∑nvj=1ρj = 1,

with nv being the number of vertices. Then, for arbitraryA andAd from the polytope (C.9),there exists a feasible solution P > 0 for the LMI in (C.8) if and only if the LMI

E(j ) P (A(j ))� P (A(j )

d )� A(j )d P

A(j )P − 1τP 0 0

A(j )d P 0 − 1

τP 0

P (A(j )d )� 0 0 − 1

2τ P

< 0

has a feasible solution P > 0 for all j = 1, . . . ,nv , where E(j ) � P (A(j ) +A(j )d )� + (A(j ) +

A(j )d )P .

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Index

L1 design system, 25L1 filter design, 26, 111, 189L1 adaptive augmentation, 243, 259

Adaptation law modificationsσ -modification, 2, 248e-modification, 2, 248projection operator, 248

Adaptation sampling time, 162AirSTAR

GTM, 159, 254Mobile Operations Station, xixpiloted evaluations, 254T1 research aircraft, xixT2 research aircraft, xix

ApplicationsAirSTAR, 159, 254crew launch vehicle, 192, 259flexible aircraft, 259nuclear power plant, 260rotorcraft, 259SIG Rascal, 243smart materials, 260vision-based control, 259well drilling, 260X-29, 259X-48B, 159, 259

Backstepping, xii, 2, 121Barbalat’s lemma, 275Base sampling time, 252, 253Bursting, 77

Certainty equivalence, 2Control reconfiguration, 242Control signal saturation, 261Controllability

strong, 286uniform, 286

Convexfunction, 291set, 291

Crossover frequencygain, 9phase, 76

Decentralized control, 261Disturbance rejection, 33

Event triggering, 261

Fault detection and isolation, 246Feedforward prefilter, 145, 161Final Value Theorem, 23Flight control systems, 241Flight envelope, 241, 254Function

KL class, 276K class, 276K∞ class, 276Dirac-delta, 267Lyapunov, 273positive definite, 272truncated, 266

Gain scheduling, 3, 4, 211, 242, 260Gap metric, 260Guidance system, 242

Handling qualities, 254Cooper–Harper rating, 254,

255High-fidelity simulator, 255High-gain feedback, 2, 3, 26, 260

Impulse response, 268matrix, 268

Input quantization, 261

315

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316 Index

Invariant setpositively, 82, 97

Limit cycle, 91Linear matrix inequality, 112, 295Locked-in-place failure, 245Loss of control, 242, 259Lyapunov equation, 265

Margingain, 9, 59phase, 9time-delay, 11, 51

Matching conditions, 140, 159Matrix

Hurwitz, 265positive definite, 265state transition, 268transfer, 268

MIL-Standard requirements, 254Model Reference Adaptive Control

direct, 1, 4indirect, 1neural-network based, 89state-predictor based, 6

Model-following control, 8

Naval Postgraduate Schoolflight tests, xviii, 243hardware-in-the-loop, 243SIG Rascal 110, xviii

Noise sensitivity, 33Norm, 263

∗-norm, 114, 296L1-norm for LTI systems, 277L1-norm for LTV systems, 280function, 266induced matrix, 264truncated, 266vector, 263

Nyquist criterion, 8

Optimization problemconstrained, 112convex, 114generalized eigenvalue, 115, 295nonconvex, 112

Parameter drift, 2, 76, 77Path-following control, 243Performance optimization, 112, 114Persistency of excitation, 1, 3, 242,

260Piecewise-constant adaptation law, 163,

192, 198Pilot-induced oscillations, 255Pitch break, 257Polytopic uncertain system, 297, 298Projection operator, 18, 292

Recursive design methods, 121, 140Reference system

LTI, 17LTV, 211non-SPR, 192strictly positive real, 179

Robust Multiple-ModelAdaptive Control,207

Rohrs’ example, 1, 76in flight, 247

Scaled response, 260Self-oscillating adaptive controller, xiSelf-tuning regulator, xiSemi-linear system, 84, 99, 125, 147Space

L-space, 266extended L-space, 266

Stabilityasymptotic, 272, 273BIBO, 276

uniform, 280BIBS, 277exponential, 273Lyapunov, 272, 273uniform, 273

State predictor, 6modification, 22, 172, 207

Stochastic optimization, 113adaptive random search, 113meta-control methodologies, 114particle swarm optimization, 113randomized algorithms, 113

Strict-feedback system, 3, 121Supervisory control, 2

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Index 317

Task execution time, 252Trajectory initialization error, 44Transmission zeros, 142, 160Two-cart benchmark problem, 207

Uncertaintiesactuator, 68, 94, 223internal dynamics, 94matched, 17system input gain, 35, 121, 142, 160unmatched, 121

Uniform performance, 14, 24scaled response, 19, 24, 29transient specification, 25

Verification and Validation, 241, 260

Wing rock, 90

X-15, 1

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State Predictor

Fast Adaptation

Control Law with Low-pass Filter

Adaptive Controller

Reference Command

Predictable Response

Uncertain System

L1

L1 Adaptive Control

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