Systems & Control: Foundations & Applications

Series EditorTamer Basar, University of Illinois at Urbana-Champaign

Editorial BoardKarl Johan Astrom, Lund University of Technology, Lund, SwedenHan-Fu Chen, Academia Sinica, BeijingWilliam Helton, University of California, San DiegoAlberto Isidori, University of Rome (Italy) and

Washington University, St. LouisPetar V. Kokotovic, University of California, Santa BarbaraAlexander Kurzhanski, Russian Academy of Sciences, Moscow

and University of California, BerkeleyH. Vincent Poor, Princeton UniversityMete Soner, Koc University, Istanbul

Chang-Hee WonCheryl B. SchraderAnthony N. Michel

Advances in StatisticalControl, Algebraic SystemsTheory, and DynamicSystems CharacteristicsA Tribute to Michael K. Sain

BirkhauserBoston • Basel • Berlin

Chang-Hee WonTemple UniversityDepartment of Electrical &Computer EngineeringPhiladelphia, PA 19122USAcwon@temple.edu

Cheryl B. SchraderBoise State UniversityCollege of EngineeringBoise, ID 83725-2100USAschrader@boisestate.edu

Anthony N. MichelUniversity of Notre DameDepartment of Electrical EngineeringNotre Dame, IN 46556-5637USAamichel@nd.edu

Series EditorTamer BasarCoordinated Science LaboratoryUniversity of Illinois at Urbana-Champaign1308 W. Main St.Urbana, IL 61801-2307USA

ISBN: 978-0-8176-4794-0 e-ISBN: 978-0-8176-4795-7DOI: 10.1007/978-0-8176-4795-7

Library of Congress Control Number: 2008923475

c©2008 Birkhauser Boston, a part of Springer Science+Business Media, LLCAll rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Birkhauser Boston, c/o Springer Science+Business Media, LLC, 233 SpringStreet, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly anal-ysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks and similar terms, even if they arenot identified as such, is not to be taken as an expression of opinion as to whether or not they are subjectto proprietary rights.

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Mathematics Subject Classification: 93-02, 93-06

Dedicated to our mentor and colleague Michael K. Sain on theoccasion of his seventieth birthday

A group picture during the workshop, “Advances in Statistical Control, SystemTheory, and Engineering Education: A Workshop in Honor of Dr. Michael K. Sain,”on Saturday, 27 October 2007.

Bottom row, from the left: Joe Cruz, Mary Sain, Elizabeth Sain, John Sain, FrancesSain, Mike Sain, Patrick Sain, Barbara Sain, Peter Hoppner, and Shirley Dyke.Middle row, from the left: Giuseppe Conte, Anna Maria Perdon, Jody O’Sullivan,Anthony Michel, Edward Davison, Frank Lewis, Peter Dorato, Bostwick Wyman,Khanh Pham, Cheryl Schrader, Ron Cubalchini, Erik Johnson, Matthew Zyskowski,Ying Shang, and Qingmin Liu.Top row, from the left: Ken Dudek, Panos Antsaklis, Steve Yurkovich, Mike Schafer,B. F. Spencer, Gang Jin, Eric Kuehner, Chang-Hee Won, Leo McWilliams, Stan Lib-erty, and Ronald Diersing.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Part I Statistical Control

Introduction and Literature Survey of Statistical Control: Going Beyondthe MeanChang-Hee Won, Ronald W. Diersing, and Stephen Yurkovich . . . . . . . . . . . . . . 3

Cumulant Control Systems: The Cost-Variance, Discrete-Time CaseLuis Cosenza, Michael K. Sain, Ronald W. Diersing, and Chang-Hee Won . . . . 29

Statistical Control of Stochastic Systems Incorporating IntegralFeedback: Performance Robustness AnalysisKhanh D. Pham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Multi-Cumulant and Pareto Solutions for Tactics Change Predictionand Performance Analysis in Stochastic Multi-Team NoncooperativeGamesKhanh D. Pham, Stanley R. Liberty, and Gang Jin . . . . . . . . . . . . . . . . . . . . . . . 65

A Multiobjective Cumulant Control ProblemRonald W. Diersing, Michael K. Sain, and Chang-Hee Won . . . . . . . . . . . . . . . . 99

Part II Algebraic Systems Theory

Systems over a Semiring: Realization and DecouplingYing Shang and Michael K. Sain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

viii Contents

Modules of Zeros for Linear Multivariable SystemsCheryl B. Schrader and Bostwick F. Wyman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Zeros in Linear Time-Delay SystemsGiuseppe Conte and Anna Maria Perdon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Part III Dynamic Systems Characteristics

On the Status of the Stability Theory of Discontinuous DynamicalSystemsAnthony N. Michel and Ling Hou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Direct Adaptive Optimal Control: Biologically Inspired FeedbackControlDraguna Vrabie and Frank Lewis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Characterization and Calculation of Approximate Decentralized FixedModes (ADFMs)Edward J. Davison and Amir G. Aghdam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Some New Nonlinear and Symbol Manipulation Techniques to MitigateAdverse Effects of High PAPR in OFDM Wireless CommunicationsByung Moo Lee and Rui J.P. de Figueiredo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Homogeneous Domination and the Decentralized Control Problemfor Nonlinear System StabilizationJason Polendo, Chunjiang Qian, and Cheryl B. Schrader . . . . . . . . . . . . . . . . . . 257

Volterra Control SynthesisPatrick M. Sain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Part IV Engineering Education

The First Professional Degree: Master of Engineering?Peter Dorato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

Theology and Engineering: A Conversation in Two LanguagesBarbara K. Sain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

Publications of Michael K. Sain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

Preface

Life has many surprises. One of the best surprises is meeting a caring mentor, anencouraging collaborator, or an enthusiastic friend. This volume is a tribute to Pro-fessor Michael K. Sain, who is such a teacher, colleague, and friend. On the beautifulfall day of October 27, 2007, friends, families, colleagues, and former students gath-ered at a workshop held in Notre Dame, Indiana. This workshop brought togethermany people whose lives have been touched by Mike to celebrate his milestone 70thbirthday, and to congratulate him on his contributions in the fields of systems, cir-cuits, and control.

Mike was born on March 22, 1937, in St. Louis, Missouri. After obtaining hisB.S.E.E. and M.S.E.E. at St. Louis University, he went on to study at the Universityof Illinois at Urbana-Champaign for his doctoral degree. With his Ph.D. degree com-plete, he came to the University of Notre Dame in 1965 as an assistant professor. Hebecame an associate professor in 1968, a full professor in 1972, and the Frank M.Freimann Chair in Electrical Engineering in 1982. He has remained at and loved theUniversity of Notre Dame for over 40 years. Mike also held a number of consult-ing jobs throughout his career. Most notably, he consulted with the Energy ControlsDivision of Allied-Bendix Aerospace from 1976 to 1988 and the North AmericanOperations branch of the Research and Development Laboratory of General MotorsCorporation for a decade, 1984–1994.

Mike’s research interests have been wide and varied. He worked on statisticalcontrol and game theory with a focus on the use of cumulants, system theory onsemirings, generalized pole and zero techniques, nonlinear multivariable feedbackcontrol with tensors, structural control for buildings and bridges subject to highwinds and earthquakes, jet engine gas turbine control, algebraic systems theory, andgeneralization of H∞ control.

Mike is a pioneer in statistical control theory, which generalizes traditional linear-quadratic-Gaussian control by optimizing with respect to any of the cost cumulantsinstead of just the mean. For over 30 years, Mike and his students have contributedto the development of minimal cost variance control, kth cumulant control, and sta-tistical game theory. In statistical game theory, the statistical paradigm generalizedmixed H2/H∞ control and stochastic H∞ control concepts. Although there is more

x Preface

work to be done in this area, Mike has pioneered a promising new stochastic optimalcontrol method.

Another major contribution of Mike’s research is in the field of algebraic sys-tems theory, expanding the algebraic system-theoretic concepts of poles and zerosof a linear system. Mike and his collaborators also researched a module-theoreticapproach to zeros of a linear system and the application of these ideas to inversesystems. Mike’s 1981 monograph Introduction to Algebraic Systems Theory bridgedthe gap between systems theory and algebraic theory and is considered a definitiveintroduction to algebraic systems theory.

More recently, Mike has applied concepts from feedback control theory to modelCatholic moral teachings and decision making, showing that analogous structuresexist in the two fields, and that one can construct a framework to support selectionof “good” outcomes and rejection of what is “not good.”

Mike has also been a valuable resource to the Institute of Electrical and Elec-tronics Engineers (IEEE). In particular, he was the founding editor-in-chief of theflagship Circuits and Systems Magazine. Mike, with the support of then IEEE Cir-cuits and Systems Society president Rui De Figueiredo, changed the IEEE Circuitsand Systems Society Newsletter into the Circuits and Systems Magazine, a highlyregarded magazine within the IEEE. Mike was also the editor-in-chief of the jour-nal of record in the field of control systems, the IEEE Transactions on AutomaticControl. He also served on numerous award committees, including the IEEE AwardBoard, where he chaired the Baker Prize Committee which annually determines thebest publication from among all those in the transactions and journals of the IEEE.During his 42 years of service, he has received numerous awards and honors includ-ing the IEEE Centennial Medal, IEEE Circuits and Systems Society Golden JubileeMedal, IEEE Fellow, and University of Notre Dame President’s Award.

Perhaps more importantly, Mike is widely recognized by his peers and students asan outstanding educator, and he has received several teaching awards for his excellentpedagogy. He has directed over 47 theses and dissertations, 19 of which are doctoraldissertations, and his students have become leaders in academic research, teaching,and administration, and in industry and government.

This Festschrift volume is divided into four parts: statistical control theory, alge-braic systems theory, dynamic systems characteristics, and engineering education.The statistical control theory part begins with a survey. Statistical control is a gener-alization of Kalman’s linear-quadratic-Gaussian regulator. Here, we view the optimalcost function as a random variable and optimize the cost cumulants. The current stateof research is discussed in the first chapter. In the second chapter, Cumulant ControlSystems: The Cost-Variance, Discrete-Time Case, the authors address the secondcumulant optimization for a discrete-time system. In this digital world, this is animportant addition to statistical control theory. The third chapter, by Pham, discussesstatistical control for a system with integral feedback, and extends the statistical con-trol idea to both regulation and tracking problems. The fourth chapter uses a sta-tistical control paradigm for decision making, using multi-player game theory. Thefinal chapter of Part I deals with multi-objective cumulant control. Here the cumulantidea is applied to mixed H2/H∞ control. Instead of optimizing the mean in the H2

Preface xi

cost function, the authors optimize the variance while constraining the system’s H∞norm. Interestingly, this idea generalizes stochastic H∞ control.

The second part of the book is dedicated to algebraic systems theory. Its firstchapter describes a new system theory for linear time-invariant systems with coef-ficients in a semiring motivated by applications in communication networks, manu-facturing systems, and queueing systems. In addition to revealing realization issuesof systems over semirings, this theory connects geometric control with the frequencydomain and provides methods to compute invariant sets associated with decou-pling. The second chapter, by Schrader and Wyman, discusses the module-theoreticapproach to zeros and poles of a linear multivariable system. By examining the intu-ition that the zeros of a linear system should become the poles of its inverse system,this chapter emphasizes Mike’s contributions to this body of knowledge. The mainresult provides a complete understanding of the connection between all poles andzeros of a transfer function matrix, including those at infinity and those resultingfrom singularities. The final chapter of this section, by Conte and Perdon, presentsthe notion of zeros for linear time-delay systems by generalizing the algebraic notionof a zero module. Additional control problems such as inversion and tracking are alsoaddressed using this framework.

The third part starts with the overview of stability results for discontinuous hybriddynamical systems. Michel and Hou show that if the hypotheses of a classical Lya-punov stability and boundedness result are satisfied for a given Lyapunov function,then the hypotheses of the corresponding stability and boundedness result for discon-tinuous dynamical systems are also satisfied for the same Lyapunov function. Theyalso show that the converse is not true in general. The second chapter solves complexsystems using a neural network structure. In particular, it discusses two algorithms,based on a biologically inspired structure, in solving for an optimal state feedbackcontroller. The third chapter tackles the characterization and calculation of approx-imate decentralized fixed modes. The fourth chapter is concerned with a communi-cations system, wherein Lee and de Figueiredo discuss two approaches to mitigateadverse effects due to the high peak-to-average power ratio in orthogonal frequencydivision multiplexing systems. Then Polendo et al. discuss constructive techniquesfor stabilization of nonlinear systems with uncertainties and limited information. Thefinal chapter of this part presents a systematic method for deriving and realizing non-linear controllers and nonlinear closed-loop systems using Volterra control synthesis.

Mike has been a lifelong mentor and teacher to many students. So, appropriately,we have chosen two important subjects in education for this volume. One impor-tant topic is the issue of the first professional degree in engineering. In this context,Dorato argues that the first professional degree in engineering should be the Mas-ter of Engineering degree rather than the bachelor’s degree. In order to maintainAmerica’s competitiveness, advances in engineering education are prerequisite. Thischapter should generate some insight into the question of what constitutes a trueengineering education. A relatively new interest of Mike has been the research of therelationship between theology and engineering. In this research he has been collabo-rating with his daughter at St. Thomas University. Thus, it is appropriate to end thisvolume with a chapter about theology and engineering, authored by Barbara Sain.

xii Preface

There she answers the question: What does the discipline of engineering have to dowith the life of faith? It is interesting and insightful to see models of the will in blockdiagrams!

Religion is an important part of Mike’s life. He is a devoted Catholic with a greatlove and devotion for the Virgin Mary. He attends daily Mass and has visited Medju-gorje in Bosnia-Herzegovina four times. Perhaps this is why his view on life is largerthan just research or teaching. We would like to end this preface with a prayer—thesame prayer that begins all Mike’s classes—because this is another commencementfor Mike.

Our Father, Who art in heavenHallowed be Thy Name;

Thy kingdom come,Thy will be done,

on earth as it is in heaven.Give us this day our daily bread,

and forgive us our trespasses,as we forgive those who trespass against us;

and lead us not into temptation,but deliver us from evil. Amen.

Notre Dame, IN P. AntsaklisEvansville, IN R. DiersingNotre Dame, IN E. KuehnerLos Angeles, CA P. SainBoise, ID C. SchraderPhiladelphia, PA C. WonColumbus, OH S. Yurkovich

October 2007 Workshop Organizing Committee

Acknowledgments

The editors are most grateful to the Notre Dame Electrical Engineering Departmentand the authors of the papers for the support of the workshop and this volume, espe-cially Dr. Thomas Fuja. We thank all the participants of the workshop. There wasmuch enthusiastic support from the planning stage from many people, including Drs.Bostwick Wyman, Edward Davison, Frank Lewis, and Tony Michel. For the logisticssupport, we acknowledge the aid from Ms. Lisa Vervynckt and Ms. Fanny Wheelerof the University of Notre Dame. We also acknowledge valuable advice from our col-leagues and friends, especially Drs. Derong Liu, Panos Antsaklis, Rui deFigueiredo,and Peter Dorato. The first editor would like to acknowledge the support from Dr.Saroj Biswas and Dr. Keya Sadeghipour. For the LaTeX typesetting help, we thankJong-Ha Lee, Zexi Liu, and Bei Kang of Temple University. Jong-Ha was responsi-ble for the wonderful workshop web site. The editors thank Mr. Tom Grasso and Ms.Regina Gorenshteyn of Birkhauser Boston for their professional support and advice.

List of Contributors

Amir G. AghdamDept. of Elect. and Comp Eng.Concordia UniversityMontreal, Quebec H3G 1M8, Canadaaghdam@ece.concordia.ca

Panos AntsaklisDept. of Elect. Eng.University of Notre DameNotre Dame, IN 46556, USAantsaklis.1@nd.edu

Giuseppe ConteDip. di Ingegneria InformaticaGestionale e dell’AutomazioneUniversita Politecnica delle MarcheVia Brecce Bianche60131 Ancona - Italygconte@univpm.it

Luis Cosenza4412 Residencial PinaresEl Hatillo, TegucigalpaHondurasluis cosenza@yahoo.com

Edward J. DavisonDept. of Elect. and Comp. Eng.University of TorontoToronto, Ontario M5S 1A4, Canadated@control.utoronto.ca

Rui J.P. de FigueiredoCalifornia Institute for Telecommunica-tions and Information TechnologyUniversity of CaliforniaIrvine, CA 92697-2800, USArui@uci.edu

Ronald W. DiersingDepartment of EngineeringUniversity of Southern IndianaEvansville, IN 47712, USArwdiersing@usi.edu

Peter DoratoDepartment of Electrical and ComputerEngineeringMSC01 1100University of New MexicoAlbuquerque, NM 87131-0001, USApdorato@ece.unm.edu

Ling HouDept. of Electrical and ComputerEngineeringSt. Cloud State UniversitySt. Cloud, MN, USAlhou@stcloudstate.edu

xvi List of Contributors

Gang JinElectronics & Electrical EngineeringFord Motor CompanyDearborn, MI 48124, USAgjin1@ford.com

Eric KuehnerDept. of Electrical EngineeringUniversity of Notre DameNotre Dame, IN 46556, USAekuehner@nd.edu

Byung Moo LeeInfra Laboratory, KT17 Woomyeon-dong, Seocho-guSeoul, 137-792, South Koreablee@kt.com

Frank LewisAutomation and Robotics ResearchInstituteThe University of Texas at Arlington7300 Jack Newell Blvd. S.Ft. Worth, TX 76118, USAlewis@uta.edu

Stanley R. LibertyOffice of PresidentKettering UniversityFlint, MI 48504, USAsliberty@kettering.edu

Anthony N. MichelDept. of Electrical EngineeringUniversity of Notre DameNotre Dame, IN, 46556 USAamichel@nd.edu

Anna Maria PerdonDip. di Ingegneria InformaticaGestionale e dell’AutomazioneUniversita Politecnica delle MarcheVia Brecce Bianche60131 Ancona - Italyperdon@univpm.it

Khanh D. PhamSpace Vehicles DirectorateAir Force Research LaboratoryKirtland AFB, NM 87117, USA

Jason PolendoSouthwest Research Institute6220 Culebra Rd.San Antonio, TX78229, USAjason.polendo@swri.org

Chunjiang QianDept. of Electrical & ComputerEngineeringUniversity of Texas at San AntonioOne UTSA CircleSan Antonio, Texas78249, USAChunjiang.Qian@utsa.edu

Barbara K. SainUniversity of St. Thomas2115 Summit Ave.Saint Paul, MN 55105, USAbksain@stthomas.edu

Michael K. SainDepartment of Electrical EngineeringUniversity of Notre DameNotre Dame, IN 46556, USAavemaria@nd.edu

Patrick M. SainRaytheon CompanyP.O. Box 902El Segundo, CA 90245, USApsain@ieee.org

Cheryl B. SchraderCollege of EngineeringBoise State University1910 University DriveBoise, ID 83725-2100, USAschrader@boisestate.edu

List of Contributors xvii

Ying ShangDepartment of Electrical and ComputerEngineeringSouthern Illinois UniversityEdwardsville, IL 62026, USAyshang@siue.edu

D. VrabieAutomation and Robotics ResearchInstituteThe University of Texas at Arlington7300 Jack Newell Blvd. SFt. Worth, TX 76118, USAdvrabie@uta.edu

Chang-Hee WonDepartment of Electrical and ComputerEngineeringTemple UniversityPhiladelphia, PA 19122, USAcwon@temple.edu

Bostwick F. WymanDepartment of MathematicsThe Ohio State University231 West 18th AvenueColumbus, OH 43210, USAwyman.1@osu.edu

Stephen YurkovichDepartment of Electrical EngineeringThe Ohio State UniversityColumbus, OH 43210, USAyurkovich@ece.osu.edu

Part I

Statistical Control

Introduction and Literature Survey of StatisticalControl: Going Beyond the Mean

Chang-Hee Won,1 Ronald W. Diersing,2 and Stephen Yurkovich3

1 Department of Electrical and Computer Engineering, Temple University, Philadelphia, PA19122, USA. cwon@temple.edu

2 Department of Engineering, University of Southern Indiana, Evansville, IN 47712, USA.rwdiersing@usi.edu

3 Department of Electrical Engineering, The Ohio State University, Columbus, OH 43210,USA. yurkovich@ece.osu.edu

Summary. In traditional optimal control, the system is modeled as a stochastic differentialequation and an optimal controller is determined such that the expected value of a cost func-tion is minimized. An example is the well-known linear-quadratic-Gaussian problem that hasbeen studied extensively since the 1960s. The mean or the first cumulant is a useful perfor-mance metric, however, the mean is only one of the cumulants that describe the distribution ofa random variable. It is possible for the operator to optimize the whole distribution of the costfunction instead of just the mean. In fact, a denumerable sum of all the cost cumulants hasbeen optimized in risk-sensitive control. The key idea behind statistical control is to optimizeother statistical quantities such as the variance, skewness, and kurtosis of the cost function.This leads to the optimal performance shaping concept. Furthermore, we use this statisticalconcept to generalize H∞ and multiple player game theory. In both traditional H∞ theory andgame theory, the mean of the cost function was the object of optimization, and we can extendthis to the optimization of any cumulants if we utilize the statistical control concept. In thischapter, we formulate and provide a literature survey of statistical control. We also review min-imal cost variance (second cumulant) control, kth cost cumulant control, and multiobjectivecumulant games. Furthermore, risk-sensitive control is presented as a special case of statisticalcontrol. When we view the cost function as a random variable, and optimize any of the costcumulants, linear-quadratic-Gaussian, minimal cost variance, risk-sensitive, game theoretic,and H∞ control all fall under the umbrella of statistical control. Finally, we interpret the costfunctions via utility functions.

1 A Brief Introduction to Statistical Control

In statistical control, the cost function is viewed as a random variable and theperformance is shaped through the cost cumulants. In other words, the densityfunction is shaped through the mean (first cumulant), variance (second cumu-lant), skewness (third cumulant), kurtosis (fourth cumulant), and other higher order

C.-H. Won et al. (eds.), Advances in Statistical Control, Algebraic Systems Theory,and Dynamic Systems Characteristics, DOI: 10.1007/978-0-8176-4795-7 1,© Birkhauser Boston, a part of Springer Science+Business Media, LLC 2008

4 Chang-Hee Won, Ronald W. Diersing, and Stephen Yurkovich

cumulants. The density function shaping shown in the cover of this book is acartoon representation of this density shaping concept. In this respect, statisticalcontrol generalizes classical linear-quadratic-Gaussian optimal control, where onlythe mean of the cost function is minimized. Moreover, when applied to stochasticgame theory, the statistical control approach also generalizes H-infinity control.This chapter is a brief survey of historical and recent developments in statisticalcontrol.

Mathematically, the statistical control problem is formulated as follows. Considerthe Ito-sense stochastic differential equation with control,

dx(t) = f (t,x,k)dt + E(t)dw(t),

y(t)dt = g(t,x,k)dt + dv(t),

and the cost function,

J(t,x(t),k) =∫ tF

tL(t,x,k)dt.

Here, x(t) is a state vector, k(t,x) is an input vector, w(t) is a disturbance vectorof Brownian motion, y(t) is an output vector, and v(t) is an output noise vector ofBrownian motions.

To define the cumulants, we need the following first and second characteristicequations:

φ(t) = E{exp(−sJ)}

and

ψ(s) = logφ(s)∞

∑i=1

(−1)i

iβis

i,

where {βi} are known as the cumulants of the cost function, J. Functions of thesecumulants are maximized or minimized to shape the cost distribution. The statisticalcontrol problem finds the optimal controller that optimizes the distribution of the costfunction through cost cumulants.

In the linear-quadratic-Gaussian (LQG) optimal control problem, the system islinear, f (t,x,k) = A(t)x(t)+B(t)k(t,x), and the cost function is quadratic, L(t,x,k) =xT Qx + kT Rk, where the superscript T denotes the transpose. Then one finds thecontollers, k, such that the expected value of the quadratic cost is minimum. Thisproblem was popularized by R. E. Kalman [Kal60]. This is the first cumulant (mean)problem and it is a special case of statistical control. This is extensively studied inthe literature with a number of textbooks about the subject, for example see [Ast70,KS72].

Minimal cost variance (MCV) control, where the idea is to minimize the vari-ance (i.e., the second cumulant) of the cost function instead of the mean, has beenpioneered by Sain [Sai65, Sai66]. Sain and Souza examined the MCV concept forproblems of estimation in [SS68]. In 1971, an open loop result on the MCV opti-mal control problem was solved in [SL71]. They observed that all cost cumulants ofany finite-horizon integral quadratic form in the system state were quadratic affine

Introduction and Literature Survey of Statistical Control 5

in the inital state mean for a linear system with Gaussian input noise. Some yearslater, Liberty and Hartwig published the results of generating cost cumulants in thetime domain [LH76]. Recently, Sain, Won, and Spencer showed the relationshipbetween risk-sensitive control and MCV control in [SWS92,Won95]. The full-state-feedback MCV control results are given in [Won95]. There, the problem formulationis for a nonlinear system with a nonquadratic cost function. Further development wasmade by Pham et al. for a linear system and quadratic cost function [Pha04]. This isdescribed in Section 3. Won developed the necessary condition of MCV control fora nonlinear system and a non quadratic cost function in [Won05]. Finally, the statis-tical control idea was generalized to game theory by Diersing [Die06], see Section 4.Surprisingly, this concept leads to the generalization of the H∞ control theory. Thedevelopment of statistical control is summarized in the Table 1.

Here, we note that minimal variance control and cost moment control are cov-ered in the literature, but not statistical (cost cumulant) control. Minimal variancecontrol, which minimizes the variance of the output, has not been very successful.This phrase, minimal variance control, was coined by Karl Astrom and his students.It is not cost variance, but rather output variance. From an engineering point ofview, Astrom’s variance control is very similar to LQG, E{xT x} = traceE{xxT}.As such, we may lump it together with the work of Kalman. Therefore, it is inessence the same idea as cost average control. Minimal cost moment control alsoexists, however, it is different from cost cumulant control, and in fact Sain was oneof the first researchers to study cost moment control in the 1960s [Sai67]. How-ever, this research was not very successful. Even though from a mathematician’spoint of view, cost moment may be similar to cost cumulant, in control engineeringapplications a cumulant gives very different results from a moment. For example,in cost moment minimization, a nonlinear controller may result from a linear sys-tem, quadratic cost case. However cost cumulant control gives a linear controller.Cost moment controllers are much too complicated, giving nonlinear controllers forthe LQ case. Moreover, the control of higher moments is problematic. If we controljust the first few of them, the neglected higher moments may have more effect thanthe ones that we have chosen to control. This makes moment control very sensitiveto model errors. This is not desirable in theory, in computation, in approximation,or in application. With cumulants, controlling the first few is an excellent approx-imation to controlling them all, as the neglected ones tend to produce increasinglysmaller effects. This is true at least in the multiple large applications that we havestudied.

Basar and Bernhard noted the relationship between deterministic dynamic gamesand H∞ optimal control in their book [BB91]. Various researchers have pointed outthat the time domain characterization of H∞ controllers contains a “generalized”Riccati-type equation that is also found in linear-quadratic zero-sum differentialgames or risk-sensitive linear-exponential-quadratic stochastic control [GD88], sug-gesting a possible relation between different approaches to robust control. Recently,in the area of robust control, we are noticing just such a synthesis of various differentareas such as H∞, deterministic differential game theory, and risk-sensitive control.

6 Chang-Hee Won, Ronald W. Diersing, and Stephen Yurkovich

Table 1. The development of statistical control.

Year Authors Title Remarks1965 Sain On Minimal-Variance

Control of Linear Systemswith Quadratic Loss

Started minimal cost variancecontrol research

1966 Sain Control of Linear SystemsAccording to the MinimalVariance Criterion

Open loop minimal costvariance control

1968 Sain andSouza

A Theory for LinearEstimators Minimizing theVariance of the Error Squared

Cost variance estimators

1969 Cosenza On the Minimum VarianceControl of Discrete-TimeSystems

Cost variance control indiscrete time

1971 Sain andLiberty

Performance MeasureDensities for a Class of LQGControl Systems

Performance measuredensities

1971 Liberty Characteristic Functions ofLQG Control

Characteristic function of thequadratic cost

1976 LibertyHartwig

On the Essential QuadraticNature of LQGControl-PerformanceMeasure Cumulants

Published in Information andControl journal

1978 LibertyHartwig

Design-Performance-Measure Statistics forStochastic Linear ControlSystems

Performance measuredensities for nth cumulantcase, linear case

1992 Sain,WonSpencer

Cumulant Minimization andRobust Control

Relations between costvariance control andrisk-sensitive control

1995 Won Cost Cumulants inRisk-Sensitive and MinimalCost Variance Control

Full state feedback costvariance control

2000 Sain,Won,Spencer,Liberty

Cumulants andRisk-Sensitive Control: ACost Mean and VarianceTheory with Application toSeismic Protection ofStructures

Published in Annals of theInternational Society ofDynamic Games

2004 Pham Statistical Control Paradigmfor Structural VibrationSuppression

kth cumulant control, linearsystem

2006 Diersing H∞, Cumulants and Games kth cumulant, nonlinearproblem formulation, gametheory

Introduction and Literature Survey of Statistical Control 7

Fig. 1. Relations between various robust controls.

See Figure 1 for an overview of the connection between various different areas ofrobust control. For more detailed descriptions of these relationships, see [Won04].

In Section 2 we review the MCV control, where we minimize the variance ofthe cost function while keeping the mean at a prespecified level. In Section 3, wereview kth cumulant control results for a linear system and quadratic cost function.In Section 4, we survey the cumulant game results and show that this statistical con-trol concept generalizes mixed H2/H∞ as well as H∞ control. In Section 5, we surveyrisk-sensitive control and relate to the statistical control by showing that the exponen-tial cost function is a denumerable sum of all the cumulants. Then we provide someinterpretation of the cost functions through the utility function concept in Section 6.Our conclusions and future work are provided Section 7.

2 Minimal Cost Variance Control: Second Cumulant Case

In statistical control, certain linear combinations of the cost cumulants are con-strained or minimized. Thus, the classical minimal mean cost problem can be seenas a special case of statistical control, in which the first cumulant is minimized. Thisidea of minimizing the expected value of the cost function was developed in the1960s, and it is well known in the literature. For example, see [Ath71,Dav77,Sag68].The fundamental idea behind minimal cost variance (MCV) control is to minimizethe variance of the cost function while keeping the mean at a prespecified level. MCV

8 Chang-Hee Won, Ronald W. Diersing, and Stephen Yurkovich

control is a special case of statistical control where the second cumulant (variance)is optimized. MCV control was first developed in a dissertation in 1965 [Sai65], andappeared in a journal in 1966 [Sai66]. In [SS68], Sain and Souza examined the MCVconcept for problems of estimation. The discrete time version of MCV control wasinvestigated by Cosenza [Cos69]. In 1971, Sain and Liberty published an open loopresult on minimizing the performance variance while keeping the performance meanat or below a prespecified value [SL71]. Liberty continued to study characteristicfunctions of integral quadratic forms, further developing the open loop MCV con-trol idea in a Hilbert space setting [Lib71]. Some years later, Liberty and Hartwigpublished the results of generating cost cumulants in the time domain [LH76]. Sain,Won, and Spencer showed that MCV control is related to risk-sensitive control undersome appropriate assumptions [SWS92]. The MCV formulation is for a nonlinearsystem and a non-quadratic cost framework, however, the controller is solved for alinear system and quadratic cost function [Won95, SWS95, SWS00]. This result issummarized in this section.

Both risk-sensitive and MCV control are a special case of statistical control.In [SWS00], we also show that MCV control is an approximation of risk-sensitivecontrol, where first two cumulants are optimized. A time-line comparison of MCVcontrol and risk-sensitive control is shown in Figure 2.

Minimal Cost Variance Control RS Control[Sain] 1965[Sain] 1966

[Sain and Souza] 1968[Liberty], [Sain and Liberty] 1971

1973 [Jacobson]1974 [Speyer, Deyst, Jacobson]

[Liberty and Hartwig] 1976 1976 [Speyer][Liberty and Hartwig] 1978

1981 [Kumar and van Schuppen],[Whittle]

1985 [Bensoussan, van Schuppen]

1990 [Whittle]1991 [Whittle]

[Sain, Won and Spencer] 19921994 [Won, Sain, and Spencer]

[James, Baras, Elliott],[Runolfsson]

[Won] 1995

Fig. 2. A time-line comparison of MCV and RS control.

Introduction and Literature Survey of Statistical Control 9

2.1 Open Loop Minimal Cost Variance Control

In this section, we present the solution of the open loop MCV control problem. Thisis a summary of the results presented in [Sai66]. Consider a linear system

x(t) = A(t)x(t)+ B(t)u(t)+ E(t)w(t) (1)

and the performance measure

J =∫ tF

0[xT (t)Qx(t)+ uT (t)Ru(t)] dt + xT (tF )Px(tF), (2)

where w(t) is zero mean with white characteristics relative to the system, tF is thefixed final time, x(t) ∈ Rn is the state of the system, and u(t) ∈ Rm is the controlaction. The weighting matrices P and Q are symmetric and positive semidefinite,and R is a symmetric and positive definite matrix. Note that

E{w(t)wT (σ)} = Sδ (t −σ), (3)

where δ denotes the Dirac delta function and the superscript T denotes the transpo-sition.

The fundamental idea behind minimal cost variance control [Sai65, Sai66] is tominimize the variance of the cost function J

JMV = VARk{J} (4)

while satisfying a constraintEk{J} = M, (5)

where J is the cost function and the subscript k on E denotes the expectation basedupon a control law k generating the control action u(t) from the state x(t) or from ameasurement history arising from that state. By means of a Lagrange multiplier μ ,corresponding to the constraint (5), one can form the function

JMV = μ(Ek{J}−M)+VARkJ, (6)

which is equivalent to minimizing

JMV = μEk{J}+VARk{J}. (7)

In [SL71], a Riccati solution to JMV minimization is developed for the open loopcase

u(t) = k(t, x(0)). (8)

The solution is based upon the differential equations

z(t) = A(t)z(t)− 12

B(t)R−1BT (t)ρ(t) (9)

10 Chang-Hee Won, Ronald W. Diersing, and Stephen Yurkovich

˙ρ(t) = −AT (t)ρ(t)−2Qz(t)−8μQv(t) (10)

v(t) = A(t)v(t)+ E(t)SET(t)y(t) (11)

y(t) = −AT (t)y(t)−Qz(t) (12)

with boundary conditions

z(0) = x(0) (13)

ρ(tF) = 2Pz(tF)+ 8μPv(tF) (14)

v(0) = 0 (15)

y(tF) = Pz(tF) (16)

and the control action relationship

u(t) = −12

R−1BT (t)ρ(t). (17)

2.2 Full-State Feedback Minimal Cost Variance Control

This section deals with feedback MCV control in the completely observed case as aspecial case of statistical control. The admissible controller is defined, then the costvariance is minimized within that admissible controller. If T = {t : t0 ≤ t ≤ tF} is aset of real time instants, then the system which is to be controlled has the stochasticdifferential equation

dx(t) = f (t,x(t),u(t))dt + E(t)dw(t), t ∈ T, (18)

where x(t) ∈Rn is an n-tuple state at time t, u(t) ∈Rm is an m-tuple control actionat time t, and f is a continuous mapping from T ×Rn ×Rm to Rn. We assumethat f and the gradient of f with respect to x are bounded. The stochastic nature ofthe problem arises from the Wiener process (equivalently Brownian motion) w(t).Equation (18) must be viewed as a formal representation of the way in which manystochastic models arise: by means of a classical derivation of a differential equationand the addition of a noise process to describe, and to compensate for, uncertaintiesand approximations. These uncertainties may include assumptions in obtaining f(including neglected higher order terms) and physical disturbances such as thermalnoise or interference.

The interpretation of (18) is given in terms of Ito’s integral equation

x(t)− x(t0) =∫ t

t0

f (s,x(s),u(s))ds+∫ t

t0

E(s) dw(s), (19)

in which the second integral is a Wiener integral.In order to assess the performance of (18), it is convenient to associate with each

realization of w(t) a penalty

Introduction and Literature Survey of Statistical Control 11

J(t0,x0;u) =∫ tF

t0

L(s,x(s),u(s))ds+ψ(x(tF )), (20)

where L is a continuous mapping from T ×Rn×Rm to the nonnegative real line R+,

u = {u(s) : s ∈ T} (21)

is a control action segment, and x0 = x(t0). We assume that L and the gradient ofL with respect to x are bounded. With regard to observing (18), it is assumed thata noise-free measurement of x(t) can be made over T . The partially observed casewill be treated in the next section. In order to control the performance of (18), amemoryless feedback control law is introduced in the manner

u(t) = k(t,x(t)), t ∈ T, (22)

where k is a nonrandom function with random arguments. The Markovian natureof the problem suggests that it is sufficient to consider the process of equation (22)[FS92, p. 136]. Accordingly, the notation of (20) is replaced by J(t0,x0;k).

The class of admissible control laws, and comparison of control laws within theclass, is defined in terms of the first and second moments of (20). Let E{·} denotethe mathematical expectation. Define

V1(t0,x0;k) = E{J(t0,x0;k)|x(t0) = x0;k} (23)

V2(t0,x0;k) = E{J2(t0,x0;k)|x(t0) = x0;k}. (24)

Definition 1. A function M(t,x), from T ×Rn to R+, is an admissible mean costfunction if it has continuous second partial derivatives with response to x and acontinuous partial derivative with respect to t, and if there exists a continuous controllaw k such that

V1(t,x;k) = M(t,x) (25)

for t ∈ T and x ∈Rn.A minimal mean cost (MMC) control law k∗M satisfies

V1(t,x;k∗M) = V ∗1 (t,x) ≤V1(t,x;k), (26)

for t ∈ T, x ∈Rn, whenever k �= k∗M. Clearly, M(t,x) ≥V ∗1 (t,x).

Definition 2. Every admissible M(t,x) defines a class KM of admissible control lawsk corresponding to M in the manner that k ∈ KM if and only if k satisfies (25). It isnow possible to define an MCV control law k∗V .

Definition 3. Let M(t,x) be an admissible mean cost function, and let KM be itsinduced class of admissible control laws. An MCV control law k∗V then satisfies

V2(t,x;k∗V ) = V ∗2 (t,x) ≤V2(t,x;k), (27)

12 Chang-Hee Won, Ronald W. Diersing, and Stephen Yurkovich

for t ∈ T , x ∈ Rn, whenever k ∈ KM is such that k �= k∗V . The corresponding costvariance is given by

V ∗(t,x) = V ∗2 (t,x)−M2(t,x) (28)

for t ∈ T, x ∈Rn.

An MCV control problem, therefore, is quite general in its scope. It presupposesthat an average cost M(t,x) has been specified (arbitrarily, within the bounds set byDefinition 1), and it seeks the control law which minimizes the variance of (20) aboutthe average (25).

The main result of MCV optimal control is summarized in the following theorem.This is a necessary condition for optimality.

Theorem 1. Let M(t,x), t ∈ T , x ∈Rn, be an admissible mean cost function, and letM induce a nonempty class KM of admissible control laws. Then the MCV functionV ∗(t,x) satisfies a Hamilton–Jacobi–Bellman equation

mink∈KM

O(k)[V ∗(t,x)]+∥∥∥∥

∂M(t,x)∂x

∥∥∥∥

2

E(t)W (t)ET (t)= 0, (29)

for t ∈ T, x ∈Rn, together with the terminal condition

V ∗(tF ,x) = 0. (30)

In equation (29), ‖a‖2A = 〈a,Aa〉. The proof of Theorem 1 is given in [SWS00].

The sufficient condition—verification theorem—is also given in [SWS00].The solution of the general nonlinear system, nonquadratic cost function MCV

problem is still an ongoing research area, however, for a linear system, f (t,x,k) =A(t)x(t)+ B(t)k(t,x), and the quadratic cost function, L(t,x,k) = xT Qx + kT Rk, thesolution is as follows [SWS00, Won95].

Theorem 2. Out of all the admissible controllers that satisfy (25), the optimal linearcontroller that minimizes the following:

V ∗(t,x) = mink∈KM

[

E{J2(t,x,k)}−M2(t,x)]

is given byk∗V (t) = −R−1BT [M + γV ]x(t),

where M and V are the solutions of the coupled Riccati equations

M + AT M +M A + Q−M BR−1BT M + γ2V BR−1BT V = 0 (31)

and

V + 4M EWET M + AT V +V A−M BR−1BT V

−V BR−1BT M −2γV BR−1BT V = 0 (32)

with boundary conditions M (tF) = QF and V (tF) = 0.

Introduction and Literature Survey of Statistical Control 13

3 kth Cost Cumulant Control: kth Cumulant Case

Pham developed finite horizon kth cumulant state feedback optimal control [PSL02].Won and Pham also investigated the infinite horizon version of the MCV and kthcumulant cases, respectively [WSL03, PSL04]. The output feedback statistical con-trol for the kth cumulant case is considered in [PSL02].

Here we will introduce the kth cost cumulant kcc control problem for a linear sys-tem with a quadratic cost function. The cost function is a linear combination of thefirst kth cumulants of a finite horizon integral quadratic form cost. In kCC control, weminimize this cost function. The following results were given by Pham in [PSL04].

Theorem 3. Consider the stochastic linear-quadratic control problem defined on[t0,tF ]

dx(t) = (A(t)x(t)+ B(t)u(t))dt + E(t)dw(t), x(t0) = x0, (33)

and the performance measure

J(t0,x0;u) =∫ tF

0[xT (τ)Qx(τ)+ uT (τ)Ru(τ)] dτ+ xT (tF)Q f x(tF), (34)

where coefficients A ∈ C ([t0,tF ];Rn×n);B ∈ C ([t0,tF ];Rn×m); E ∈ C ([t0,tF ];Rn×p);Q∈C ([t0,tF ];Sn) positive semidefinite; R∈C ([t0,tF ];Sm) positive definite; and W ∈S

p). Suppose further that both k ∈Z+ and the sequence μ = {μi ≥ 0}k

i=1 with μ1 > 0are fixed. Then the optimal state-feedback kCC control is achieved by the gain

K∗(α) = −R−1(α)BT (α)k

∑r=1

μrH∗(α,r), α ∈ [t0,tF ], (35)

where the real constants μr = μi/μ1 represent parametric control design freedomand {H∗(α,r) ≥ 0}k

r=1 are symmetric solutions of the backward differential equa-tions

ddα

H∗(α,1) = −[A(α)+ B(α)K∗(α)]T H∗(α,1)−H∗(α,1)[A(α)+ B(α)K∗(α)]

−K∗T (α)R(α)K∗(α)−Q(α),d

dαH∗(α,r) = −[A(α)+ B(α)K∗(α)]T H∗(α,r)−H∗(α,r)[A(α)+ B(α)K∗(α)]

−r−1

∑s=1

2r!s!(r− s)!

H∗(α,s)E(α)W ET (α)H∗(α,r− s),

with the terminal conditions H∗(tF ,1) = Q f , and H∗(tF ,r) = 0 when 2 ≤ r ≤ k,whenever these solutions exist.

The solution is obtained using modified dynamic programming, which is an iter-ative algorithm to find approximate solutions. See [PSL04] for the complete descrip-tion of the solution procedure.

14 Chang-Hee Won, Ronald W. Diersing, and Stephen Yurkovich

4 Cumulant Games: Multiobjective Case

A natural extension from cumulant control is into game theory. Stochastic differen-tial game theory has a great deal of work using the mean of cost functions as theperformance indices for the players. It makes sense therefore to use further cumu-lants as performance indices. Furthermore, application of game theory is in the areaof H2/H∞ and H∞ control. With the use of cumulants, we can find a cumulant gener-alization of these two well-known control techniques.

The game is given through a stochastic differential equation

dx(t) = f (t,x(t),u(t),w(t))dt +σ(t,x(t))dξ (t), (36)

where u is the control, w is the second player (the disturbance), and ξ is Brownianmotion. The costs for the players are given as

J1(t,x,u,w) =∫ tF

t0

L1(t,x(t),u(t),w(t))dt +ψ1(tF ,x(tF ))

J2(t,x,u,w) =∫ tF

t0

L2(t,x(t),u(t),w(t))dt +ψ2(tF ,x(tF ))(37)

with J1 being the control’s cost and J2, the disturbance’s cost.This result is found in [Die06].

Theorem 4. Let M be an admissible mean cost function, M ∈ C1,2p (Q)∩C(Q), with

an associated UM. Also consider the function V ∈C1,2p (Q)∩C(Q) that is a solution to

minμ∈UM

{

∂V∂ t

(t,x)+ f T (t,x,μ ,ν∗)∂V∂x

(t,x)

+12

tr

(

σ(t,x)W (t)σT (t,x)∂ 2V∂x2 (t,x)

)

+∣∣∣∣

∂M∂x

(t,x)∣∣∣∣

2

σ(t,x)W (t)σT (t,x)

}

= 0

(38)

with V (tF ,x f ) = 0 and the function P ∈C1,2p (Q)∩C(Q) that satisfies

minν∈WF

{∂P∂ t

(t,x)+ f T (t,x,μ∗,ν)∂P∂x

(t,x)

+12

tr

(

σ(t,x)W (t)σT (t,x)∂ 2P∂x2 (t,x)

)

+ L2(t,x,μ∗,ν)}

= 0

(39)

with P(tF ,x f ) = ψ2(x f ). If the strategies μ∗ and ν∗ are the minimizing argumentsof (38) and (39), then the pair (μ∗,ν∗) constitutes a Nash equilibrium solution.

Introduction and Literature Survey of Statistical Control 15

Now consider a linear system,

f (t,x(t),u(t),w(t)) = dx(t) = (Ax(t)+ Bu(t)+ Dw(t))dt + Edη(t),

and the quadratic costs

L1(t,x(t),u(t),w(t)) = zT1 (t)z1(t)

L2(t,x(t),u(t),w(t)) = δ 2wT (t)w(t)− zT2 (t)z2(t),

where

z1(t) = G1(t)x(t)+ H1(t)u(t)z2(t) = G2(t)x(t)+ H2(t)u(t),

where HT1 H1 = R1 > 0, HT

2 H2 = R2 > 0, GT1 G1 = Q1 ≥ 0, and GT

2 G2 = Q2 ≥ 0. Intraditional mixed H2/H∞ control such as the one in [LAH94], one finds a controller,u, such that the mean of the first cost function, J1, is minimized while the secondplayer, w, maximizes the mean of the second cost function, J2. Maximizing the meanof J2 is equivalent to

supw

‖z2‖2,[t0,tF ]

‖w‖2,[t0,tF ]≤ δ ,

which implies that δ is a constraint on the H∞ norm of the system. The solution tothis mixed H2/H∞ control problem is given in [LAH94].

One can generalize this problem to determine a controller, u, such that the vari-ance of the first cost function, J1, is minimized while the mean is kept at a prespec-ified level. This can be called mixed MCV/H∞ control. The equilibrium solution ofmixed MCV/H∞ control is determined as

u∗(t) = μ∗(t,x(t)) = −R−11 BT (t)(M (t)+ γV (t))x(t)

w∗(t) = ν∗(t,x(t)) = − 1δ 2 DT (t)P(t)x(t)

(40)

with Riccati equations

M + AT M +M A + Q1−M BR−11 BT M

− 1δ 2 PDDT M − 1

δ 2 M DDT P

+ γ2V BR−11 BT V = 0,

(41)

where M (tF) = Q1f ,

V + AT V +V A− γM BR−11 BT V − γV BR−1

1 BT M

− 1δ 2 PDDT V − 1

δ 2 V DDT P −2γV BR−11 BT V

+ 4M EWET M = 0,

(42)