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Introduction to Control of Oscillations and Chaos
 9810230699, 9789810230692

Table of contents :
Contents......Page 12
Preface......Page 6
Notations and definitions......Page 16
1.1 What is control?......Page 20
1.1.1 Models of the controlled plants......Page 21
1.1.2 Control goals......Page 23
1.1.3 "Naive" control......Page 25
1.1.4 Feedback......Page 27
1.1.5 Uncertainty......Page 28
1.1.6 Nonlinearity......Page 30
1.2 What is chaos?......Page 33
1.3.1 Mechanics and mechanical engineering......Page 36
1.3.2 Electrical engineering and telecommunications......Page 37
1.3.3 Chemistry and chemical engineering......Page 39
1.3.4 Biology, biochemistry and medicine......Page 41
1.3.5 Economics and finance......Page 42
2.1 Mathematical models of controlled systems......Page 44
2.2.1 State-space stability......Page 52
2.2.2 Stability theorems and Lyapunov functions......Page 58
2.2.3 Absolute stability......Page 68
2.3 Feedback linearization and normal forms......Page 76
2.4.1 Introductory comments......Page 82
2.4.2 Passivity and dissipativity......Page 84
2.4.3 Passification as a control design problem......Page 93
2.4.4 Input-to-state stability......Page 98
2.5.1 Goal-oriented formulation of the control problem......Page 102
2.5.2 Design of Speed Gradient Algorithms......Page 105
2.5.3 Properties of the speed gradient algorithms......Page 109
2.6 Robustness of speed gradient algorithms with respect to disturbances......Page 127
2.7 Gradient control of discrete-time systems......Page 131
3.1.1 General concepts......Page 136
3.1.2 Oscillations in dynamical systems......Page 140
3.2.1 Convergence and synchronization......Page 147
3.2.2 Lyapunov stability, Lyapunov exponents, Bol exponents......Page 154
3.2.3 Computation of the Bol exponents......Page 159
3.2.4 Orbital stability......Page 160
3.3.1 Definition and properties of the Poincare map......Page 164
3.3.2 Controlled Poincare maps......Page 166
3.3.3 Controlled closing lemma......Page 171
3.4 What is chaos? (continued)......Page 173
4.1 Adaptive control problem statement......Page 178
4.2 Direct and identification approaches to adaptive control design......Page 184
4.3.1 Problem statement......Page 188
4.3.2 State feedback......Page 189
4.3.3 Output feedback......Page 197
4.4 Controlled synchronization of dynamical systems......Page 201
4.5 Decomposition based synchronization......Page 204
4.6.1 Semipassivity and L- dissipativity......Page 208
4.6.2 Synchronization of two linearly coupled systems......Page 211
4.6.3 Synchronization of several systems with multiple interconnections......Page 218
4.6.4 Adaptive synchronization......Page 222
4.6.5 Adaptive synchronization of uncertain semipassive systems......Page 225
4.6.6 Adaptive synchronization of hyper-minimum -phase systems......Page 230
4.7 Adaptive suppression of forced oscillations......Page 233
4.8.1 Integrator backstepping......Page 247
4.8.2 Adaptive control of unmatched systems......Page 253
4.9.1 Control of energy......Page 262
4.9.2 The swinging (small control) property......Page 267
4.9.3 Control of first integrals......Page 268
4.9.4 Control of generalized Hamiltonian systems......Page 272
4.10.1 Background and motivation......Page 275
4.10.2 Linearization of the controlled Poincare map......Page 276
4.11 Control of bifurcations......Page 284
5.1.1 Swinging a simple pendulum......Page 288
5.1.2 Pendulum with a controlled suspension point......Page 295
5.2 Stabilization of the equilibrium point of the thermal convection loop model......Page 301
5.3 Adaptive synchronization of two forced Duffing's systems......Page 310
5.4 Adaptive synchronization of Chua's circuits......Page 319
5.5 Gradient control of the Henon system......Page 324
5.5.1 Stabilizing the unstable equilibrium of the Henon system......Page 325
5.5.2 Synchronizing two identical Henon systems......Page 327
5.6 Control of periodic and chaotic oscillations in the brussellator model......Page 329
6.1 How to tow a car out of a ditch......Page 338
6.2 Synchronization of generators based on tunnel diodes......Page 342
6.3 Stabilization of swings in power systems......Page 348
6.4 Adaptive control of the thin film growth from a multicomponent gas......Page 357
6.5 Control of oscillatory behavior of populations......Page 361
6.6 Control of a nonlinear business-cycle model......Page 366
Chapter 7 Conclusions: What is the message of the book?......Page 374
Exercises......Page 378
Bibliography......Page 382
Index......Page 404

Citation preview

* i WORLD SCIENTIFIC SERIES ON

^

NONLINEAR SCIENCE

Alexander L. Fradkov

Series Editor: Leon O. Chua

INTRODUCTION TO CONTROL OF OSCILLATIONS AND CHAOS Alexander L. Fradkov & Alexander Yu.

World Scientific

Pogromsky

INTRODUCTION TO CONTROL OF OSCILLATIONS RNA CHAOS

WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE

Editor: Leon O. Chua University of California, Berkeley Series A. MONOGRAPHS AND TREATISES Published Titles Volume 15: One-Dimensional Cellular Automata B. Voorhees Volume 16: Turbulence, Strange Attractors and Chaos D. Ruelle Volume 17: The Analysis of Complex Nonlinear Mechanical Systems: A Computer Algebra Assisted Approach M. Lesser Volume 19: Continuum Mechanics via Problems and Exercises Edited by M. E. Eglit and D. H. Hodges Volume 20: Chaotic Dynamics C. Mira, L. Gardini, A. Barugola and J.-C. Cathala Volume 21: Hopf Bifurcation Analysis: A Frequency Domain Approach G. Chen and J. L. Moiola Volume 22: Chaos and Complexity in Nonlinear Electronic Circuits M. J. Ogorzalek Nonlinear Dynamics in Particle Accelerators R. Dila"o and R. Alves-Pires Volume 25: Chaotic Dynamics in Hamiltonian Systems H. Dankowicz

Volume 23:

Volume 30: Quasi-Conservative Systems: Cycles, Resonances and Chaos A. D. Morozov Volume 31: CNN: A Paradigm for Complexity L. O. Chua Forthcoming Titles Volume 4: Methods of Qualitative Theory in Nonlinear Dynamics (Part I) L. Shilnikov, A. Shilnikov, D. Turaev and L. O. Chua Volume 18: Wave Propagation in Hydrodynamic Flows A. L. Fabrikant and Y. A. Stepanyants Volume 27: Thermomechanics of Nonlinear Irreversible Behaviours G. A. Maugin Volume 32: From Order to Chaos II L. P. Kadanoff

a | WORLD SCIENTIFIC SERIES ON rm

NONLINEAR SCIENCE

Series A

Vol. 35

Series Editor: Leon 0. Chua

INTRODUCTION TO CONTROL OF OSCILLOTIONS ONO CHAOS A. L. Fradkov A. Yu. Pogromsky Russian Academy of Sciences

X YP World Scientific

Singapore •NewJersey•London •HongKong

Published by World Scientific Publishing Co. Pte. Ltd. P 0 Box 128, Farrer Road, Singapore 912805 USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Fradkov, A. L. (Aleksandr L'vovich) Introduction to control of oscillations and chaos / Alexander L. Fradkov, Alexander Yu. Pogromsky. p. cm. -- (World Scientific series on nonlinear science, Series A ; vol. 35) Includes bibliographical references and index. ISBN 9810230699 1. Nonlinear control theory. 2. Chaotic behavior in systems. I. Pogromsky, Alexander Yu. II. Series: World Scientific series on nonlinear science. Series A, Monographs and treatises ; v. 35. QA402.35.F73 1998 629.8'36--dc2l 98-40055 CIP

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Copyright m 1998 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Preface

During the last quarter of the century, numerous studies in different countries confirmed that a complex and particularly chaotic behavior is observed in systems of various types in physics , mechanics , engineering, chemistry and chemical technology, biology, economics , etc., see [223 , 329, 214, 217, 87, 66] . Chaos may serve as an indicator of the complexity of the system as well as an instrument of its further evolution . A dramatic growth of interest in the problem of controlling chaotic systems has been recently observed . An indication of this is the increase in the number of publications. For instance , the bibliography on control and synchronization of chaos [59] contains almost 800 titles , with about 700 published in 1993-1996. One of the reasons for such an interest is that control promises both a better understanding of chaotic behavior and the means of influencing and modifying it. Various applications were reported , such as eliminating chaotic regimes in lasers, increasing the reaction rate in chemical technologies by means of chaotic stirring , providing secure communications by using chaotic carrier signals and treating cardiac arrhythmia. Another reason is the interdisciplinary nature of the problem which attracts the attention of different scientific communities and makes it attractive to even wider audience. Evidence of this is given by the surprisingly large number of publications in the scientific mass media . The headings speak for themselves: • • • •

"Putting gentle reins on unruly systems" [11], "Mastering chaos" [79], "Do chaos-control techniques offer hope for epilepsy ?" [127], "Keeping chaos at bay " [150], V

Preface

vi

• "Chaos in harness" [181], • "Chaos under control" [216], • "How to get order out of stirring things up" [242]. It seems it is expected that "Coping with chaos" would cause a kind of revolution in science and technology! However, analysis and control of oscillatory and chaotic systems is extremely difficult due to their intricate nonlinear dynamics. There are some interesting questions that still require unbiased scientific investigation:

• Is it possible to control chaotic systems? • What are the most efficient methods of controlling oscillatory systems?

• What are the possibilities and limitations of controlling oscillatory and chaotic systems? It is important to answer these questions in view of the numerous potential applications in laser and plasma technologies, communications, biology and medicine, economics, ecology, etc. The first question has a positive answer. A few approaches to controlling chaos were proposed; linearization of the Poincare map [232]; periodic forcing [174],[149], linear and optimal control [306],[142] and others. However, the most promising methods for controlling nonlinear dynamics are those of nonlinear control theory [152], [227]. Also, the uncertainty which is always present in real problems demands special control methods which can in fact be provided by the existing theory of nonlinear and adaptive control (e.g. [89, 94, 168] ). However, many of the published papers on the control of chaos do not make full use of the existing nonlinear control theory; many results are obtained by means of computer simulations. On the other hand, most control theorists and engineers are not familiar with the potential applications of control of chaotic systems. That is why the authors' primary goal was to write a book which would give a reasonably rigorous exposition of modern nonlinear control theory as applied to various oscillatory and chaotic systems. The authors' approach is based on the consideration of chaotic systems in the broader context of oscillatory systems, keeping in mind that oscillations may be either periodic or nonperiodic, e.g. chaotic. We demonstrate that the classical concepts of Lyapunov function, Poincare map and G.

Preface

vu

Birkhoff 's recurrence are suitable both for the analysis and for the design of systems with oscillatory behavior. Such a view may simplify the analysis and facilitate the usage of modern control theory. On the other hand, we do not use the concept of probability and the probabilistic theory of random processes staying within the deterministic framework. The book begins with an introduction where the idea of control and the motivating examples from the different fields of science and technology are discussed. Chapter 2 presents the main concepts and results of nonlinear and adaptive control theory which serve as a solid mathematical background for the control of oscillatory systems. The necessary mathematical concepts and tools for the analysis of oscillatory and chaotic systems are introduced in Chapter 3. Chapter 4 contains nonlinear and adaptive control algorithms and their applicability conditions with respect to different assumptions about the structure, parameters and measured outputs of the controlled oscillatory systems. The control design methods are based on the concepts of Lyapunov functions, passivity, Poincare maps, speed gradient, and gradient algorithms. The control objectives include the excitation or suppression of oscillations to the desired energy level, transformation of the oscillation mode from chaotic to periodic, synchronization, etc. The chapter contains a number of theorems which establish the system stability, performance and robustness under disturbances. The described methods and algorithms are illustrated in Chapter 5 by a number of model examples, including classical models of oscillatory and chaotic systems: pendulums, the Lorenz, Duffing, Henon, Chua systems. The performance of the proposed algorithms is derived theoretically from the results of the previous chapters and also evaluated by computer simulation. Chapter 6 contains application problems from different fields of science and technology. These problems include: • • • • •

Synchronization of chaotic generators based on tunnel diodes; Stabilization of swings in power systems; Control of thin films growth process; Control of oscillatory behavior of populations in ecology; Synchronization of business cycles.

Most of the results from Chapter 6 are based on the joint research of the

viii

Preface

authors and experts in the relative fields. To make the book more suitable for the teaching process a number of exercises are placed at the end of the book. In fact , the fields of nonlinear control and nonlinear oscillations were developing suprisingly independently. The present book is perhaps the first one to bring together these two important branches of nonlinear science. The original methods presented in the book were first published in [96, 110], and summarized in [99, 111]. The additional feature of the book is that it incorporates some results in nonlinear control , adaptive control and nonlinear oscillations obtained by the researchers of the former Soviet Union who had good scientific schools in those fields . These results were published in Russian and were not well known in the West. The authors hope that the book may be interesting and useful to researchers , to theoretically oriented engineers , teachers and students from the fields of electrical and mechanical engineering , physics , chemistry, biology, economics . Control engineers in various fields of technology dealing with complex oscillatory systems may also apply the methods given in the book. Finally, mathematicians might find there some new results , unsolved problems and motivating examples.

The prospective reader should have some degree of familiarity with standard university courses of calculus , linear algebra and ordinary differential equations . Knowledge of deterministic chaos and linear control theory will also help in the understanding some parts of the book. The authors would like to acknowledge the valuable help of their colleagues associated with the Laboratory "Control of Complex Systems" of the Institute for Problems of Mechanical Engineering of Russian Academy of Sciences: B.R. Andrievsky, M.V. Druzhinina, P.Yu. Guzenko, I.A. Makarov, A.Yu. Markov, V.O. Nikiforov, E.V. Panteley, I.G. Polushin, A.A. Stotsky, A.S. Shiriaev. We are pleased to thank our colleagues and friends from other universities who contributed a lot in shaping our view of the subject: I.I. Blekhman, F.L. Chernousko, K. Furuta, D.J. Hill, H. Nijmeijer, R. Ortega, A.A. Pervozvansky, V.Ya. Yakubovich. We are also thankful to our colleagues from the different fields: E.G. Dymkin, S.A. Kukushkin, A.V. Lyamin, A.V. Osipov, G.S. Simin and V.P. Shiegin, for their contributions. Our research was significantly supported by the Institute for Problems of Mechanical Engineering . The results of researches supported by the

Preface

IX

Russian Foundation of Basic Research (grants 93-013-16322, 96-01-01151), INTAS (project 94-0965), ISF (J58100-1995) and by the St. Petersburg Scientific and Educational Center for the Problems of Machine Building, Mechanics and Control Processes (project 2.1-589 of the Russian Federal Program "Integration") are also reflected in the book. Important contributions stem from contacts with students and from the lecture course "Control of Oscillations and Chaos" delivered by one of the authors at the Baltic State Technical University and the St. Petersburg University in 1996-1997. Finally it is our pleasure to acknowledge valuable comments made during our invited visits and seminars given in more than 50 universities in 15 countries.

Alexander Fradkov, Alexander Pogromsky St. Petersburg, 1997.

This page is intentionally left blank

Contents

Preface

v

Notations and definitions

1

Chapter 1 Introduction 1.1 What is control? . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Models of the controlled plants . . . . . . . . . . 1.1.2 Control goals . . . . . . . . . . . . . . . . . . . . 1.1.3 "Naive" control . . . . . . . . . . . . . . . . . . . 1.1.4 Feedback . .. . . . .. . . . . . . . . . . . . . . 1.1.5 Uncertainty . . . . . . . . . . . . . . . . . . . . . 1.1.6 Nonlinearity . . . . . . . . . . . . . .. . . . . . 1.2 What is chaos? . . . . . . . . . . . . . . . . . . . . . . . 1.3 What use is it? . . .. . . . .. . . . . . . . . . . . . .. 1.3.1 Mechanics and mechanical engineering . . . . . . 1.3.2 Electrical engineering and telecommunications . 1.3.3 Chemistry and chemical engineering . . . . . .. 1.3.4 Biology, biochemistry and medicine . . . . . . . 1.3.5 Economics and finance . . . . . . . . . . . . . . .

5 .. .. .. .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. .. .. .. .. .. .. .. .. .. ..

.. .. .. .. ..

.. .. .. .. ..

Chapter 2 The mathematics of nonlinear control 2.1 Mathematical models of controlled systems . . . . . 2.2 Stability and boundedness . . . . . .. . . . . . . . . 2.2.1 State-space stability . . . . . . . . . . . . . . 2.2.2 Stability theorems and Lyapunov functions . 2.2.3 Absolute stability . . . . . . . . . . . . . . . xi

5 6 8 10 12 13 15 18 21 21 22 24 26 27

29 .. .. .. .. ..

29 37 37 43 53

Contents

xii

2.3 Feedback linearization and normal forms .... .. . .. . . .. 61 2.4 Feedback stabilization and passivity . . . . . . . . . . . . . . . 67 2.4.1 Introductory comments . . . . . . . . . . . . . . . . . . 67 2.4.2 Passivity and dissipativity . . . . . . . . . . . . . . . . . 69 2.4.3 Passification as a control design problem . . . . . . . . . 78 2.4.4 Input-to-state stability . . . . . . . . . . . . . . . . . . . 83 2.5 Speed gradient algorithms . . . .. . . . . . . . . . . . . . . . . 87 2.5.1 Goal-oriented formulation of the control problem .. . . 87 2.5.2 Design of Speed Gradient Algorithms . . . . . . . . . . 90 2.5.3 Properties of the speed gradient algorithms . . . . . . . 94 2.5.4 Identifying properties of SG algorithms . . . . . . . . . 110 2.6 Robustness of speed gradient algorithms with respect to disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2.7 Gradient control of discrete-time systems . . . . . . . . . . . . 116 Chapter 3 The mathematics of oscillations and chaos 121 3.1 What is oscillation? . . . . . . . . . . . . . .. . . . . . . . . . 121 3.1.1 General concepts . . . . . . . . . . . . . . . . . . . . . . 121 3.1.2 Oscillations in dynamical systems . . . . . . . . . . . . 125 3.2 Stability of oscillations . . . .. ... . ... ...... ... .. 132 3.2.1 Convergence and synchronization . . . . . . . .. . . . . 132 3.2.2 Lyapunov stability, Lyapunov exponents, Bol exponents 139 3.2.3 Computation of the Bol exponents . . . . . . . . . . . . 144 3.2.4 Orbital stability . . . . . . . . . . . . . . . . . . . . . . 145 3.3 Poincare maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3.3.1 Definition and properties of the Poincare map . . . . . . 149 3.3.2 Controlled Poincare maps . . . . . . . . . . . . . . . . . 151 3.3.3 Controlled closing lemma ... ... ........ ... 156 3.4 What is chaos? (continued) . . . . . . . .. . . . . . . . . . . . 158

Chapter 4 Methods of nonlinear and adaptive control of oscillations 163 4.1 Adaptive control problem statement . . . . . . . . . . . . . . . 163 4.2 Direct and identification approaches to adaptive control design 169 4.3 Adaptive systems with reference models . . . . . . . . . . . . . 173 4.3.1 Problem statement . . ... ....... .. . .. .... 173 4.3.2. State feedback . . . . . . . . . . . . . . . . . . . . . . . 174 4.3.3 Output feedback . . .. .... .. ... . .. .. .. 182

4

Contents

xiii

4.4 Controlled synchronization of dynamical systems .. .. ....

186

4.5 Decomposition based synchronization . . . . . . . . . . . . . . 4.6 Passivity based synchronization . . . . . .. . . . . . . . . . . . 4.6.1 Semipassivity and G-dissipativity . . . . . . . . . . . . . 4.6.2 Synchronization of two linearly coupled systems . . . . 4.6.3 Synchronization of several systems with multiple interconnections . . . . . . . . . . . . . . . . . . . . .. 4.6.4 Adaptive synchronization . . . . . . . . . . . . . . . . . 4.6.5 Adaptive synchronization of uncertain semipassive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.6 Adaptive synchronization of hyper-minimum-phase systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Adaptive suppression of forced oscillations . . . . . . . . . . . . 4.8 Control of cascaded systems. Relaxation of the matching condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Integrator backstepping . . . . . . . . . . . . . . . . . . 4.8.2 Adaptive control of unmatched systems . . . . . . . . . 4.9 Speed Gradient control of Hamiltonian systems . . . . . . . . . 4.9.1 Control of energy . . . . . . . . . . . . . . . . . . . . . . 4.9.2 The swinging (small control) property . . . . . . . . . . 4.9.3 Control of first integrals . . . . . . . . . . . . . . . . . . 4.9.4 Control of generalized Hamiltonian systems . . . . . . . 4.10 Discrete adaptive control via linearization of Poincare map . . 4.10.1 Background and motivation . . . . . . . . . . . . . . . . 4.10.2 Linearization of the controlled Poincare map . . . . . . 4.11 Control of bifurcations . . . . . . . . . . . . . . . . . . . . . . .

189 193 193 196 203 207 210 215 218 232 232 238 247 247 252 253 257 260 260 261 269

Chapter 5 Control of oscillatory and chaotic systems 273 5.1 Control of pendulums . . . . . . . . . . . . . . . . . . . . . . . 273 5.1.1 Swinging a simple pendulum . . . . . . . . . . . . . . . 273 5.1.2 Pendulum with a controlled suspension point . . . . . . 280 5.2 Stabilization of the equilibrium.point of the thermal convection loop model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

5.3 Adaptive synchronization of two forced Duffing's systems . . .

295

5.4 Adaptive synchronization of Chua's circuits . . . . . . . . . . . 5.5 Gradient control of the Henon system . . . . . . . . . . . . . . 5.5.1 Stabilizing the unstable equilibrium of the Henon system . . . . . . . . . . .. . . . . . . . . . . . . . . . .

304 309 310

xiv

Contents

5.5.2 Synchronizing two identical Henon systems . . . . . . . 312 5.5.3 Adaptive model reference control of the Henon system . 313 5.6 Control of periodic and chaotic oscillations in the brussellator model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Chapter 6 Applications 323 6.1 How to tow a car out of a ditch .. .. ..... .. ... .. .. 323 6.2 Synchronization of generators based on tunnel diodes ...... 327 6.3 Stabilization of swings in power systems . . . . . . . . . . . . . 333 6.4 Adaptive control of the thin film growth from a multicomponent gas .................................. 342 6.5 Control of oscillatory behavior of populations . . . . . . . . . . 346 6.6 Control of a nonlinear business-cycle model . . . . . . . . . . . 351 Chapter 7 Conclusions: What is the message of the book? 359 Exercises

363

Bibliography Index

367

389

Notations and definitions

Throughout the book we shall use the following notations and definitions. The vector space of real numbers is denoted as R and R' stands for the vector space which is the direct product of the n copies of R: R' = R x R x • • • x R. An element of R' will be denoted as x = col(x1i x2, .... xn) where col stands for the column vector composed of the elements x1i x2, ... , X. This notation will also be used in the case where the components x; are vectors again. The set of positive real numbers and zero is denoted as R+ or [0, oo). The Euclidean norm in R' is denoted simply as i.e. Ixl = x2+x2+...+xn

if x = col(xl, ... , xn), xi E 1R1. The scalar product of two vectors x1, x2 E R' is denoted as xi x2 i where "T" stands for the transpose operation, 1x12 = xT x. Let us consider the quadratic form xT Px for any symmetric square n x n matrix P, PT = P. If this form is positive for any x 0 0 then the matrix P is called positive definite and denoted as P > 0. Matrices satisfying the nonstrict inequality xT Px > 0, Vx E R' are called positive semidefinite or nonnegative. A nonnegative scalar function v : ]Rn -+ R+ is called positive definite if v(0) = 0 and v(x) > 0, Vx ,- 0. A scalar function v : R' x 1R+ -+ R is called radially unbounded if lxi -+ oo implies inft>o v(x, t) -+ oo. For example, the functions v(x) = Jx12, v(x) = xT Px, P > 0 are radially unbounded while the function v(x, t) _ Jx12/(t + 1) is not radially unbounded. A function y : R+ --* R+ is called a 1C-function if it is continuous, 1

2

Notations and definitions

strictly increasing and -y(O) = 0. it is referred to as a K".-function if it is a K-function and radially unbounded; A function ,3 : R+ x R -> 1R is called a KG-function if for each fixed t the function (3(., t) is a K-function and for each fixed s > 0 /3(s, t) -+ 0 as t->oo. Given a system of differential equations (N.1)

i = F(x, t),

where x(t) E R' , F : ][8' x [to, oo) -+ R , and i = dx/dt, we will interpret its solution x as a function of time: x : [to, oo ) -+ R'. The solution of (N.1) with initial conditions x(to) = xo calculated at the instant t is denoted as x(t, to , xo ). For autonomous systems (N.2)

i = F(x),

we can take an arbitrary value of to as the initial time (in view of the semigroup property of dynamical systems) and therefore we will assume that to = 0. The solution of the autonomous system (N.2) calculated at the instant t with initial condition x(0) = xo will be denoted simply as x(t, xo). In both cases, for the sake of simplicity we will also use notation x(t) whenever it causes no confusion. The set of all points ry E Rn such that 3t > to : ry = x(t, to, xo) will be referred to as the trajectory and will be denoted as r(xo, to) (r(xo) for autonomous system (N.2)) or simply {x(t)}. In many works on differential equations this object is often called semi-trajectory but we will not consider the behavior of the system for t < to and therefore there is no abuse in this definition. A closed trajectory which corresponds to a periodic solution sometimes will be referred to as the orbit.

Consider the linear time-varying system of differential equations: (N.3)

i = A(t)x,

where A(t) is the time-dependent n x n matrix. The matrix -P(t, z) which satisfies

a'k (t, -0

at

=

A(t)4)(t,-r),

4(r

,rr)

=

In

is called the fundamental matrix of the system (N.3). Here In stands for n x n identity matrix : In = diag{1 ,1, ... ,1}.

Notations and definitions 3

Given a differentiable scalar function f : Il8n -+ R, denote V f (x) as the column vector of its first derivatives calculated at the point x: Vf(x)

= (L(x))T T.

For example, if f (x) = aT x, a E Rn then V f (x) = a. If f (x) = xT Px, P = PT, then V f (x) = 2Px. If the function f is a function of two vector variables xl and x2 then Vxl f (X1, x2) stands for the vector composed of the partial derivatives of f with respect to xl calculated at the point col(xl, x2). Differentiation of the matrix functions is understood componentwise, that is if f : Rn -> R' then a f (x)/ax is an m x n matrix composed of the partial derivatives a fi(x)/ax„ i = 1,... m; j = 1, ... , n. For a given differentiable scalar function f : R' x R -+ 1R and given system (N.1) the time derivative of f with respect to the system (N.1) is as follows:

AX, t) = at + (V.f (x, t))T F(x, t). For autonomous system (N.2) and time-independent scalar function f we have

j (X) = (Vx f (x))T F(x), which coincides with the Lie derivative LF f (x) of the function f with respect to the vector field F:

LF.f (x) = (Oxf (x))T F(x). Let y be a function defined on the set of positive numbers, i.e. y : R+ -+ Rn. The LP norm, 1 < p < oo is introduced as follows: Y

HAP =

(10 00

I y(t) I Pdt)

For example, Ilylli

= f

y(t)Idt,

4

Notations and definitions

II Y II2 = Uo "o I y(t)I2dt) One can also define the L

norm as follows: Iy(t)I IIyIIo = sup t

If the norm IIyIIn is finite we write y E Gp.

Chapter 1

Introduction

1.1 What is control?

It would be naive to think that you understand what "control of oscillations and chaos" is before you have understood what "control" is. So what is control? The idea of control should be familiar to you if you have tried to turn the steering wheel of your car, or to turn the tap of your shower or to ask your spouse to bring you your coffee. The result may be good or bad but in any case the concept of control should contain the following four components: A. The controlled system (or controlled plant, as control engineers say) - which is what we want to control. B. The control goal (or the desired behavior of the controlled system) - which is what we want to achieve. C. The set of measurable variables (or outputs) - the things we can measure. D. The set of controlling variables (or inputs) - the things we can change in order to act on the controlled system.

Exercise 1.1 Find all the four components in the above-mentioned examples. A Another important component is the controller (or regulator) that produces the control inputs to achieve the desired goal. However this fifth component is not present where the control problem is at the stage of formulation; it only appears after the problem has already been solved. By 5

6

Introduction

controlled plant

output

controller I

Fig. 1.1 Control system

the solution of the control problem (or the control design) we mean just finding the control law (or the control algorithm). Once the control law is found, it can be used by the controller to evaluate the control inputs based on the measured outputs of the controlled plant. The system, composed of the plant and the controller, is called the control system or the closed loop system (Fig. 1.1). Even simple rules accumulated by human experience may solve some simple control problems, for example "If the water is too hot, turn the tap to the left", (1.1) "If the water is too cold, turn the tap to the right", However, such an approach does not work in more complicated cases. What can really help is a good mathematical model of the controlled system.

1.1.1

Models of the controlled plants

It was probably R. Descartes who first pursued the idea that a single real world system can be represented by a variety of mathematical models. We may illustrate this idea by the following simple example. Example 1.1 "Tap" model . The model of the plant used for the control purposes is usually as simple as possible. For example let us build a model of the system where the input is the angular position of a hot water tap and the output is the temperature of running water. We will call this system the "tap". Let u(t) be the angle of the tap at the time t and y(t) be the temperature of the water. When the displacement of the input angle u(t)

What is

control?

7

from the base value u is small, we may assume that the corresponding displacement of y(t) from the value y is proportional to u(t) - u: y(t) - y =NUM - il),

(1.2)

where /3 is constant. What we get is quite a simple model of the controlled plant "Tap". It may be also rewritten as follows: y(t) = a + 13(u(t) - v,),

(1.3)

where a = y - 0u. The models (1.2) or (1.3) belong to to the class of linear static (or memoryless) models: indeed, the value of its output y(t) depends on the value of the input u(t) measured at the same time t. However, the real life systems must always have some inertia or memory, i.e. one cannot instantaneously change their states. It means that more accurate models should include the predetermined values of the input and output and/or their derivatives. Such more complicated models are called dynamical models. Thus, when taking into account the fact that the temperature of the mixed flow cannot be changed instantaneously after the change of the flow rate of the component, we come to the first order differential equation Tdy(t) + y(t)

= a + f3u(t), (1.4)

where T > 0 is the time constant. Further, if there is a long piece of pipe between the tap and the place of mixing, we must take into account the time delay:

T ddtt) + y(t) = a +,(3u(t -,r),

(1.5)

where r > 0 is the delay lag. Finally, when we need a valid model not only in the region of small deviations, we have to take into account its nonlinearity. The nonlinear static model of the tap according to the heat transfer law is as follows: y(t) = f(u) =

a+Of(u(t)), kluyl + k2 u2y2 k1 u + k2u2 '

(1.6) ( 1 . 7)

where Y1, Y2 are the temperatures of hot and cold water, k1u,k2u2 are the flow rates of hot and cold water respectively and k1, k2 are the scaling

8

Introduction

coefficients.

The main classes of mathematical models used in the control theory will be surveyed in Sec. 2.1. In many cases we need to design the discrete-time control algorithm for the computerized control of a continuous-time real life system. It can be performed by transforming both a controlled plant model and a controller model either into the continuous-time or into discrete-time form. Therefore, the designer is free to choose between the continuous-time and discrete-time approaches. This book is mainly focused on continuoustime systems, although the discrete-time gradient control method is also considered here.

1.1.2

Control goals

After examining the models of the controlled systems we pass to another component of the control problem, the control goal. According to tradition two kinds of the control goals are considered here: regulation and tracking. Goal 1: Regulation. Regulation (or stabilization, or positioning) is understood as driving the vector of the state variables x(t) of the control system to some equilibrium x, (the fixed point). In some cases it is only the regulation of the output variable y(t) that is of significance. Formally, we can write Goal 1 as the limit relation lim x(t ) = X. t00

(1.8)

lim y(t) = y,.

(1.9)

or t-.00

Sometimes more realistic approximate relations are used , such as tlyr l x(t ) - X.1 < 0 00

(1.10)

) - y.I < 0 t1iiy(t 00

(1.11)

or

where 0 is the parameter of accuracy.

What is control? 9

Goal 2: Tracking. The term "tracking" means driving the variable x(t) to the desired trajectory x.(t), i.e. slim [x(t) - X. (t)] = 0 00

(1.12)

or driving the output y(t) to the desired function y.(t), i.e. slim [y(t) - y.(t)] = 0. 00

(1.13)

The desired output y.(t) is often interpreted as the reference or command signal. Sometimes, the reference signal is defined explicitly as a function of time. In other cases it may be defined as a solution of another auxillary system called the reference model, or the goal model.- In the latter case, the problem of finding the controller to ensure the goal (1.12), or (1.13) is called the model reference control problem. Turning to the control of the oscillatory behavior, we can see at least two more kinds of the control goals: Goal 3: Synchronization. Synchronization is understood as a coincidence of the variables of two or several systems; or as a concurring change of some quantitative characteristics of the systems. The synchronization problem differs from that of the model reference control problem because it allows the coincidence of different variables, taken at different time instants, i.e. with time shifts, which may be either constant, or tending to constants (the asymptotic phases). In addition, in the typical synchronization problems the bidirectional (mutual) connection between the systems is allowed. It means that the desired limit (synchronous) solution cannot be predetermined. On the other hand, synchronization with unidirectional connections (master-slave or drive-response synchronization) may be of interest as well. The synchronization phenomenon which occurs in systems without control (self-synchronization) is well known and quite well studied [32, 33, 187, 38, 19]. However, far less attention was paid to synchronization as a control goal. Investigations of the controlled synchronization have started only recently [213, 237, 230, 97]. The general representation of synchronization as the control problem is presented in [35]. A few approaches to the controlled synchronization problems will be explored in this book (see Chapter 4 and examples in Chapter 5).

Goal 4: Modification of asymptotic behavior of the system. This class of goals includes various special cases, such as

Introduction

10

- the change of the type of equilibrium (e.g. turning an unstable fixed point into a stable one and vice versa); - the change of the type of limit set (e.g. turning a limit cycle into a chaotic attractor and vice versa); - the change of the bifurcation point and/or the type of bifurcation point in the parameter space;

- the creation of oscillations with desired properties. An important special case of the latter subclass is represented by the socalled swinging/upwinding which implies the excitation of the oscillations up to the desired energy, (or frequency, or some other parameters of intensity). A related problem is the modification of the fractal dimension of the invariant compact set (attractor). Some general approaches to the swinging problem are described in our book (Sec. 4.9). Examples of its application to the pendulum-like systems and to the control of the oscillatory behavior of populations can be found in Chapters 5 and 6. An important distinctive feature of controlling the oscillations and chaos is that the low level (or low power, or cheap) control is needed. The reason lies in the fact that the life time of an oscillatory system covers a large number of cycles and the average power of the controlling action per cycle should be rather small in order to retain the total energy of the control to be within a certain specified value. Fortunately, in many cases the "small control" requirement can be achieved in the way, shown below in Chapter 4.

1.1.3

"Naive" control

Now, assume that the control problem has been already well posed. What we need to do is to solve it. At the first glance, it seems that the simple controlled plant model should yield a simple solution of the control problem. Occasionally this is actually true, but in many cases "underwater reefs" may appear.

As an example, we consider the controlled plant described by the first order differential equation (1.4). It is convenient to rewrite the plant equation in the form resolved for the derivative (the so-called Cauchy form) y=ay+bu +ao ,

(1.14)

What is control?

11

where the constant coefficients a, b, ao depend on the parameters of (1.4): a = -11T, b = i3/T, ao = a/T. Note that in (1.14) we do not need to suppose that T > 0, i.e. (1.14) may also describe the systems with unstable (a > 0) or neutrally stable (a=0) internal dynamics. Let the control goal be the regulation of the output y(t) to the desired value y.. An apparent solution is to calculate the equilibrium y = -ao/abu/a of the system (1.4) with the constant input u(t) - u and then to choose the constant input u which provides y = y., i.e. to choose u(t) = u., where u. = -(ay. + ao)/b,

(1.15)

Then the goal y(t) - y. will be fulfilled for all t > 0 if it holds at the initial time instant t = 0, i.e. if y(O) = y.. In addition, the goal will be achieved asymptotically (when t -+ oo) for any initial conditions y(O) when the plant (1.14) is asymptotically stable (a < 0). The solution of the tracking problem with the desired output function y.(t) can be performed in a similar way, i.e. just by substituting y.(t) into (1.14) and solving (1.14) for u. (t):

u.(t) = ((it) dt - ay.(t) - ao I /b. (1.16) For example, to achieve the desired oscillatory behavior y. (t) = A+B sin wt, the following control u. (t) = (Bw cos wt - aA - aB sin wt - ao) /b can be applied to system (1.14). However , this simple solution has a few drawbacks: a) If the controlled plant is unstable, then the constant input (1.15) will leave it unstable, i.e. arbitrary small mismatch in the initial conditions will allow the output to grow unlimitedly. b) Suppose the plant (1.4) is stable, but its parameters are known inaccurately (e.g. the available values of a, b used in the controller differ from the true ones). Then the error of evaluation of u. may be significant and the desired goal will not be achieved.

c) Sudden changes in the plant parameters will imply the irreversible goal breakdown.

12

1.1.4

Introduction

Feedback

Experience in engineering has shown that much more efficient solutions can be obtained by means of the feedback principle, i.e. by means of the control algorithms which use the available measurements of y(s), 0 < s < t to compute u(t). The arsenal of feedback laws is quite wide. One of the simplest and most frequently used control laws is the proportional feedback law: u(t) = K(y(t) - y*).

(1.17)

The relay (or sign) feedback law is also commonly used: u(t) = Ksign(y(t) - y*).

(1.18)

It can be easily seen that the negative feedback in (1.17) or (1.18) (with K < 0 for b > 0) is able to turn the unstable plant (1.14) (with a > 0) into a stable one. However, a steady-state error e,,,, = limt.,,. [y(t)-y*] = y00 -y* may appear. One can see that the equilibrium of the closed loop system (1.14), (1.17) is stable if a+bK < 0 and it gives a steady state solution y'," = y*-(ay*+ao )/(a+bK) with a steady-state error e,,, = -(ay*+ao)/(a+bK), which becomes smaller when K takes a larger value. To eliminate the steady-state error completely, we may add the bias: u(t) = -(ay* + ao)/b + K(y(t) - y*)

(1.19)

or introduce the integral feedback: u(t) = ki (y(t) - y*) + K2 J t (y(s) - y *) ds. 0

(1.20)

The proportional-integral (PI) controllers (1.20) are widely used, since they require less knowledge of the plant parameters. Note that when a = ao = 0, the system has a zero steady-state error for all y* and all K, such that bK < 0. Therefore, in this case any desired output y* can be achieved by feedback (1.17) with an arbitrary small K, i.e. by means of the arbitrarily small controlling force. The simple example given above demonstrates that the feedback is able to improve the stability and accuracy of the closed loop system whereas the uncertainty of the plant parameters may prevent the introduction of any improvements to the system performance. Among other obstacles, the most important are:

What is

control?

13

- nonlinearity; - incompleteness of measurements; - incompleteness of control.

The incompleteness of measurements means that either not all the plant state variables are available for measurement or that the measurement noise is too high. The incompleteness of controlling means that it is impossible to change the controlled plant model in any desired way by means of admissible changes of the controlling variables. We try to show in this book that the modern theory of nonlinear and adaptive control is able to overcome the above-mentioned difficulties. Some general methods of nonlinear control are briefly outlined in Chapter 2. While it is by no means a substitute for a textbook, Chapter 2 may acquaint scientists, engineers or university teachers with the most fruitful concepts and ideas of modern nonlinear control theory. It can also help the reader to better understand the methods and results of the other chapters of the book.

1.1.5

Uncertainty

It is worthwhile to illustrate the main ideas of the nonlinear and adaptive control with simple examples . Let us start with an adaptive control, which helps one to cope with the plant parameter uncertainty. The problem of coping with the uncertainty is very important indeed. The use of an imprecise plant model for control purposes may actually lead to poor performance of the control system. If this is not the case, the control system is called robust. However, it is difficult to design a robust controller if the plant parameters change over a wide range. In this case, we may attempt to design an adaptive controller. Suppose again that the plant is described by the linear first order differential equation (1.14) and the control goal is specified as the regulation lim y(t) = y*.

t-. 00

(1.21)

As mentioned above, the control law (1.19) is able to ensure the goal (1.21) when the precise values of the parameters a, b, ao are known. To obtain a controller working well for the arbitrary values of the plant parameters, we may try to adjust the parameters of the controller (1.19) based

14

Introduction

on on-line measurements. To this end we present the control law (1.19) in the form u(t) = S1y* + ^2 + 6(y - y*),

(1.22)

where S1 = -a/b, ^2 = -ao/b, 63 = K. After that we replace the constant coefficients 6i, i = 1, 2, 3 by adjustable ones*:

u(t) = 91 (t)y* +02 (t) +03(t )(y(t) - y*),

(1.23)

where Bi, i = 1, 2, 3 are the adjustable parameters. The question is: how to update 9i(t) in order to ensure the goal (1.21) for arbitrary values of a, b? To solve the problem the so-called Speed-Gradient (SG) method may be employed, which is described in detail in Sec. 2.5. In order to design the SG algorithm, we reformulate the goal (1.21) in the following way:

tlim Q(y(t)) = 0,

(1.24)

where Q(y) = (y - y*)2/2 is the so-called objective function. Then we calculate its time derivative Q which is the speed of change of Q(y(t)) along the solutions of the system (1.13), (1.23) with the fixed Bi, i = 1, 2, 3. Apparently, Q = (y - y*)y = (y - y* )( ay + b91y* + b02 + b03(y - y*) + ao)• Finally, we calculate the partial derivatives aQ/490i, form the speed gradient vector V9Q = colOQ/8Bi where 0 = col(01, 02i03), and write down the SG algorithm -'Y(y - y *) y *, 92

--Y(y - y*),

93

-'Y(y - y*)2,

(1.25)

where 7 > 0 is the gain factor. Algorithm (1.25) suggests a change of the vector 0(t) in the direction of decreasing Q, which eventually implies (after achieving the inequality Q < 0 for all y # 0) a decrease in the initial functional Q(y(t)). An experienced reader may notice that in this example we use the overparametrization: since the command signal is constant , it is sufficient to introduce two adjustable parameters . However , in this overparametrized form the structure of the adaptive controller becomes more apparent.

What is control?

15

It follows from the results of Sec. 2.5 that the goal (1.24) is achieved for all initial conditions in the closed loop system, if b > 0. The same is valid for b < 0 when the gain factor is chosen to be negative. Thus, to solve the problem by means of the controller (1.23), (1.24) it is sufficient to know the sign of the coefficient b. The idea of replacing the unknown coefficients by their estimates is sometimes called the certainty equivalence principle. It is widely used in the adaptive control theory and will be constantly employed in Chapter 4.

1.1.6

Nonlinearity

In some cases the control algorithms for the nonlinear systems can be designed in a way quite similar to those used for the linear systems. For example, let the plant model (1.13) be augmented by nonlinear term sin y as follows: y = ay + ao + d sin y + bu,

(1.26)

where a, ao, d, b are the unknown constant parameters. It can be easily noticed that the overall system closed by the linear feedback can possess multiple equilibria, some of which may be stable and may lie far from those desired. To overcome the increase in the plant complexity, we may introduce a corresponding sine term into the control law: UM = 01(t )y* + 92(t ) + 03 sin y + 04(t)(y(t) - y*).

(1.27)

Then, the parameter update algorithm may be designed similarly to (1.25) as follows

O1 =

--Y(y - y.)y.,

e2 =

-'Y(y - y.),

03 =

-y(y - y.) sin y,

03 =

-'Y(y - y.)2.

(1.28)

In many cases the nonlinearity increases the complexity of the plant behavior but does not increase the complexity of the control design. On the other hand, nonlinearity may give rise to severe problems, when we take into account additional constraints, e.g. the "small control" requirement.

Introduction

16

m Fig. 1.2 A simple pendulum

We illustrate this idea by the simple example of a swinging pendulum (Fig. 1 . 2) which is one of the classical examples for an oscillatory system. Consider the controlled pendulum described by the following second order equation: M12 2 cp + mgl sin V = u,

(1.29)

where cp(t) is the angular deflection of the pendulum from the vertical position ; u(t) is the controlling torque applied to the axis of rotation; m is the mass; l is the length and g is the gravity acceleration. Let the control goal be the excitation of the pendulum oscillations up to the given level of the energy E,,:

E(t)-+E„

as t-+oo, (1.30)

where 2 E = m(2) + mgl(1 - cos V)

(1.31)

is the total energy of the pendulum. The goal (1.30) differs slightly from the conventional regulation or tracking goals (e.g. positioning a robot arm with drill or welding electrode). It looks rather like the goal of a monkey, who wants to move from one tree to another swinging on a liana . Similar problems arise in the case of putting into operation some vibratory equipment , pendulum clocks , etc. Common sense suggests that swinging requires much less power than holding the arm in some fixed position . So, how can a monkey solve the problem?

What is control?

17

The conventional solution based on the linearization of the plant model does not work in this case. Indeed, consider the plant model linearized near the equilibrium y = 0 + w0cp = bu where wo = signal,

(1.32)

2g/d, b = 2/(ml'). Using the "resonant " harmonic input

u(t) = rysinwot, (1.33) one can make the pendulum energy arbitrarily large, even for the arbitrarily small input amplitude ry. However, it is not true when the open loop control (1.33) acts on the system (1.29), because the frequency of its free oscillations depends on the amplitude! Now, we turn to the feedback laws. Again, applying the speed gradient method for the objective function Q = (E - E. )2 /2 and taking into account the Hamiltonian nature of the controlled system, we can find very simple solutions, e.g. u(t) _ -ry(E(t) - E*)cp

(1.34)

u(t) = -'ysign ((E(t) - E„)cp).

(1.35)

or

Using the general results of Sec. 4.9, one can show that the goal (1.30) is achieved for any E. > 0 and any -y > 0 and for almost all initial conditions W(O), 0(0) (see Sec. 5.1). The control laws (1.34), (1.35) are also applicable to systems with dissipation. However, in this case the gain ry cannot be chosen arbitrarily small. More difficulties can arise due to the incompleteness of measurements (e.g. the velocity 0 the solution ( 2.26) is unique . However , if xo 0 then Eq. (2.25) has two different solutions starting from xo = 0 at t = 0: xl(t) = t2 ,

x2(t) 0.

The reason for nonuniqueness is that the Lipschitz condition is violated in the neighborhood of x = 0 (i.e. locally) because the derivative of the right hand side d(2 f)/dx = 1// is unbounded near x = 0. Note that the right hand side itself is still continuous for all x E R+. A Motivated by the examples considered we give the following standard definition. Definition 2.1 A system (2.21) is called forward complete if the corresponding solution x(t) exists for all initial conditions x(to) = xo E Rn on the entire interval [to, oo). It is well known that if each solution of the system (2.21) is bounded, then the system (2.21) is forward complete. In what follows we will give more interesting conditions of extending of the solutions to the infinite time interval.

As one can see that the system considered in Example 2.4 is not complete. Similarly one can establish that the system described by the following model x=-x+x2

The mathematics of nonlinear control

40

is also not forward complete, although solutions starting from the neighborhood of the origin tend to zero and therefore exist on the infinite time interval. Nevertheless solutions which correspond to large initial conditions may exist only on a finite time interval. More detailed investigation of this system is left to the reader as an exercise. Now it is high time to introduce some stability concepts. Definition 2.2 Let the solution x(t) of the system (2.21) be well defined for all t > to. The solution t(t) is called Lyapunov stable if for any e > 0, tl > to there exists S > 0 such that any solution x(t) with the initial condition x(ti) satisfying jx(tl) - x(tl)I < S is well defined for all t > to and the inequality jx(t) - x(t)j < e holds for all t > t1. If, additionally, the value of S does not depend on t1, then x(t) is called uniformly Lyapunov stable. Note that Lyapunov stability is a local property of the system. Loosely speaking, Lyapunov stability means that small deviations of any solution from the stable one remain small for all t > to. It may be interpreted as continuous dependence of the solutions on the initial conditions over an infinite time interval with respect to the uniform norm (C-norm) llx(•)II _ suet>t, Ix(t)I• Many applications also need stronger forms of stability. The most important among them is asymptotic stability. The well-defined solution Wi(t), t > to of the system (2.21) Definition 2.3 is called asymptotically stable if it is Lyapunov stable and Ix(t) - x(t)I -+ 0 as t -+ oo for sufficiently small jx(to)-x(to)l. If, additionally the Lyapunov stability is uniform, then the solution x(t) is called uniformly asymptotically stable. It is well known [327] that for autonomous systems

x = F(x)

(2.27)

the concept of (asymptotic) stability coincides with the concept of uniform (asymptotic ) stability and in the sequel we will talk about uniform stability only if we deal with nonautonomous systems of differential equations. It is important for applications not only to know that some solution is stable but also to know the set of initial conditions for which the solutions converge. It motivates the following definition.

41

Stability and boundedness

Definition 2.4 The well-defined solution x(t), t > t0 is called asymptotically stable with the domain of attraction X0 C R n if it is Lyapunov stable and \x(t) - x(t)\ -* 0 as t -» oo whenever x(to) € Xo- If X0 — R n , then x{t) is called globally asymptotically stable (GAS). The global asymptotic stability is an important property which means that x(t) is the limit mode of the system. In practice it is important to know when the limit mode is unique. However in general the limit mode is not unique: if the system has a GAS solution then any of its solutions is GAS. A reasonable answer can be found for systems defined on the twosided time interval. The system (2.21) with the right-hand side F(x,t) defined for x £ R n and for - o o < t < oo is called convergent if it has a unique globally attracting solution x(t) bounded on - c o < t < oo. The convergent systems will be studied in more detail in Sec. 3.2. Obviously no system can possess more than one GAS equilibrium point. Therefore with a little modification of terminology we will say that the system (2.21) is GAS if it has a GAS equilibrium. The GAS system cannot have more than one equilibrium. Now we introduce one more stability concept which is more restrictive than global asymptotic stability. Definition 2.5 The solution x(t) of (2.21) defined for all t>t0 is called exponentially stable if there exist C > 0, a > 0 which do not depend on to such that the inequality \x{t) - x(t)| < Ce-a{t-to)\x(t0) is valid for sufficiently small \x(to) -

- x(t0)\

(2.28)

x(t0)\.

The validity of (2.28) means that the convergence rate of the solutions is exponential, i.e. as for the solutions of the asymptotically stable linear timeinvariant systems. If, in Definition 2.5, the initial mismatch \x(to) -x{to)\ is arbitrary, then the solution x(t) will be referred to as globally exponentially stable. The above definitions express different forms of continuous dependence of the solutions on the initial conditions. The next one expresses continuous dependence of the solutions on the variations of the right hand sides of (2.21), which is also important for applications. Consider together with (2.21) the perturbed system

x = F(x,t) + F(x,t),

(2.29)

42

The mathematics of nonlinear control

where F satisfies the same assumptions as F in (2.21) to ensure the existence of solutions of (2.29) for all t > to (at least for some initial conditions). The solution x(t) defined for all t > to is called stable Definition 2.6 under persistent disturbances (SPD), if for any e > 0 there exist 6l > 0, 62 > 0 such that for any solution x(t) of (2.29) with initial conditions x(to) satisfying Ix(to) - x(to)I < 6l and function P satisfying IF(x, t) I < 62 in the cylinder t > to, ix - ^EI < e the inequality Ix(t) - x(t)I < e holds for allt>to. The meaning of the SPD property is very natural: small (in C-norm) perturbations of the right hand sides lead to small (again in C-norm) perturbations of the solutions. Note that it differs from the commonly used concept of structural stability (or roughness) meaning that small in Cl-norm (i.e. small together with derivatives) perturbations of the right hand sides lead to a small change of the solutions (see [14, 16, 136]). Nowdays the concept of robustness has become more popular and it encompasses the above properties. Some scalar characteristics or index S(p) is called robust with respect to p near po, where p is an abstract parameter varying in some metric space P, if IS(p) - S(po)I < e(b) as long as dist(p,po) < S. The function e(b) (which does not necessarily satisfy e(0) = 0) is called the robustness measure. Next we introduce a useful definition expressing ultimate boundedness of the solutions of (2.21). This property corresponds to a typical behavior of the uniformly GAS system under bounded state-independent disturbance. A system possessing such a property was called dissipative by N. Levinson [182, 64]. Since other versions of the dissipativity concept are also in use both in the nonlinear control theory and in this book, we will refer to the property in question as the dissipativity in the sense of Levinson, Levinson dissipativity or L-dissipativity. Definition 2.7 such that

System (2.21) is called L-dissipative if there exists R > 0

slim jx(t)I < R 00 for all initial conditions x(to) = xo E R.

In other words there exists a ball of radius R such that for any solution x(t) there exists a time instant tl = to + T(xo, to) > to such that for all t > tl we have Ix(t)I < R. Obviously the ball can be replaced by any

Stability and boundedness

43

compact set in R. If T(xo, to) can be chosen independently on to then the system is called uniformly G-dissipative. As one can see all solutions of the G-dissipative system are bounded and tend to some set which depends neither on the initial instance to nor on the initial conditions x(to). A milder concept is that of Lagrange stability. Definition 2.8 The system (2.21) is called Lagrange stable if each solution of (2.21) is bounded. Any L-dissipative system is Lagrange stable while the converse statement is not true. We have introduced some useful stability properties which are important for system analysis. In the sequel, we will find the conditions by which to check that the given system possesses these properties.

2.2.2 Stability theorems and Lyapunov functions The key contribution of A. M. Lyapunov into the nonlinear analysis was not only the formulation of the stability properties but also the establishment of different stability criteria. He developed two methods of analysing stability, both of which are in common use today. In this section we will briefly describe the so-called second (or direct) Lyapunov's method which is based on some scalar functions whose time-derivatives along the solutions of the system satisfy some inequalities. Such functions are commonly referred to as Lyapunov functions. The value of Lyapunov functions is now far beyond the stability theory. It turns out that employment of Lyapunov functions allows one to give the criteria for other forms of dynamical behavior like oscillatory behavior, boundedness of solutions, etc. Lyapunov functions allow one to evaluate the system performance indices, to establish its optimality, etc. Finally (and it is extremely important for this book) Lyapunov functions can be used as a control design tool providing the closed loop system with a prespecified behavior. In this section however we will formulate only the basic stability theorems leaving more advanced applications of the Lyapunov function concepts for the subsequent sections and chapters. First we will give a sufficient condition for the completeness of the system (2.21). Theorem 2 .1

Assume that there exists a smooth radially unbounded

44

The mathematics of nonlinear control

function V : Rn -+ 1181 with the property

V(x, t) < KV(x) for some real K where V (x, t) = (VV (x))T F(x, t) is the time derivative of V along the solutions of the system (2.21). Then the system (2.21) is forward complete. More generally it can be shown [173] that if V : R"` x [to, oo) --' R1 is a radially unbounded scalar function such that

V(x, t) < k(t)L(V(x, t)),

(2.30)

where L is a continuous positive scalar function satisfying rr L(r) --oo as r -> oo (2.31) Jo and k is a nonnegative continuous function, then the system ( 2.21) is for-

ward complete. Now we present some theorems which establish Lyapunov stability conditions . It is convenient to reduce the stability analysis of the solution x(t) to that of the trivial solution x(t) = 0 by the time-dependent coordinate transformation and replacing x(t) - x(t) by x(t ). After such a coordinate change the identity F(0, t) 0 in (2.21) is valid for all t > to. Let V : 1l8' x [to , oo) -; R1 be a smooth scalar function . Its speed of change along the trajectories of the system ( 2.21) (time derivative with respect to the system ( 2.21)) will be denoted as V(x,t):

T/ (x, t) = att (x, t) + (VV ( x, t))T F(x, t). Theorem 2.2 Suppose that there exists a continuously differentiable scalar function V : X x [to, oo) -> R+ such that V(0, t) = 0, V(x, t) > ao(x) and V(x, t) < 0, where ao(x) > 0 for x # 0. Then the trivial solution x(t) - 0 of the system (2.21) is Lyapunov stable. If, additionally, V satisfies the inequality a1(JxJ) > V (x, t) for some continuous strictly increasing function a1 i a1(0) = 0, then x(t) is uniformly Lyapunov stable. Theorem 2.2 is usually referred to as the "first Lyapunov theorem". The addition about uniform stability belongs to K. Persidsky (1933) [239]. The following result is related to what is called the "second Lyapunov theorem".

Stability and boundedness 45

Recall that the scalar function a : R+ R+ is a class IC function if it is continuous , strictly increasing and a (O) = 0, see [139].

Theorem 2 . 3 Suppose that there exists a continuously differentiable function V : Xo x [to, oo) -+ R+, Xo C X, such that ao(Ixl) oo. (2.41)

Condition ( 2.41) guarantees the global asymptotic stability of the trivial solution and it turns out that violation of this condition may cause instability of the solution starting sufficiently far from zero [166]. In other words, in general the posed question has negative answer. 0 Essentially, Theorem 2.4 provides conditions for stability of the set M. The concepts of stability of sets and partial stability play an important role for nonlinear and adaptive control. It was first introduced by A. M. Lyapunov in 1893 [197] and by V. V. Rumyantsev [265] and extensively studied later (see [266, 262, 310]). The asymptotic stability of the set M means that M is Lyapunov stable and dist(x(t), M) -> 0 when t -> oo. However, it is more restrictive than the property W(x(t)) -' 0, M = {x : W(x) = 0}

(2.42)

for a smooth nonnegative function W as it can be demonstrated by the following example. Example 2.8 Consider the following second order system { xl = x1 2x x2= 1

(2.43)

For the initial conditions x1(0) = 1, x2 (0) = a it has the following solution xi(t) = et, x2 ( t) = all 2e-2t ) (2.44)

Taking a Lyapunov-like function as follows: x2 V(xl, x2) = 2 2 1+x1

50

The mathematics of nonlinear control

we get 4x2 x2_2 _ r xi - -4V(xi, x2) V(xl, x2) = -1 +x2 + (1 +x2)2 1 - 4(1+x21 ) ) Therefore -4V < V < -3V < 0 and, clearly, V(x1(t)x2(t)) -* 0 and V(xl (t), x2( t)) --> 0, as t -^ oo. However, no solution to (2.43) with x2(0) = a 0 tends to the set S = {(x1, x2):V(x1, x2) = 0} = {(x1, x2 ): x2 = 0} as seen from (2.44). IL

Since properties like (2.42) are frequently used in the control of oscillations we give here some general result for time-varying systems which will be used below. Theorem 2. 6 (Partial stability theorem ) [250] Let the system (2.21) possess a smooth nonnegative function V : R' x [to, oo) -+ R+ such that

W (x, t) < 0, Vt > to, Vx E R" where W (x, t) is the derivative of V(x, t) with respect to the system (2.21). Assume further that boundedness of V(x, t) implies boundedness of F(x, t), uniform continuity of W (x, t) in x and boundedness of 8W/at. Then V(x(t), t) is bounded on [to , oo) and

slim W(x(t), t) = 0 . 00

(2.45)

Remark 2 .2 To verify uniform continuity of the function W in x and boundedness of aW/8t one can prove that the time derivative W is bounded when V is bounded.

The proof of this Theorem is not difficult and we present it here. Proof: Since V < 0 along solutions of (2.21), we have 0 < V(x(t), t) < V(x(to), to) = Vo Therefore F(x(t), t) is bounded and all solutions x(t) of (2.21) are well defined for any t > to. Since V(x(t), t) is bounded and nonincreasing , lim V(x(t), t) = V. exists. The identity V(x(t), t) - Vo = t-,o

t

f W(x(s), s) ds implies that W(x(t), t) is integrable on [to, oo). Rewriting to t

(2.21) in integral form x (t) - x(s) = f F( x(T),

r)

dT and taking into ac-

count boundedness of F(x(T), r) we get that x(t) is uniformly continuous.

Stability and boundedness

51

Hence the function W(x(t), t) is uniformly continuous in t and integrable on [to, oo) and the statement follows immediately from the Barbalat lemma which is an important tool in the stability analysis: Lemma 2.1 (Barbalat) [254] Consider the function i> : R + —» R+. / / tp is uniformly continuous and lim fn°° rf>(s)ds exists and is finite then

Theorems concerning stability of sets (or partial stabilty) are useful for the studying stability of oscillatory motions in the case when the limit set is some manifold or even a more complex set, perhaps unbounded. The next theorem provides necessary and sufficient conditions for the exponential stability. Theorem 2.7 (N. Krasovskii) [167] The trivial solution x{t) = 0 of the system (2.21) with a continuously differentiable right-hand side is globally exponentially stable if and only if there exist a function V : R n x [to, oo) —> R+ and positive numbers cto,ct\, a?, 03 satisfying the following inequalities: ao|x| 2 < V(x,t) < ai\x\2,

\VxV(x,t)\

< a3\x\, V(x,t) < - a 2 | x | 2 (2.46)

To complete this section we will formulate the boundedness conditions for the solutions of the system (2.21). We will present conditions for stability under persistent disturbances (which may be interpreted as local boundedness) and also for £-dissipativity and Lagrange stability (interpreted as global boundedness). Theorem 2.8 (Malkin) [198] Suppose that (2.21) has a continuously differentiable right hand side and uniformly asymptotically stable solution x(t) = 0. Then it is stable under persistent disturbances (SPD). Theorem 2.8 expresses the SPD property of the disturbed system in terms of undisturbed system behavior. According to Theorem 2.3 another equivalent condition of SPD is the existence of the Lyapunov function satisfying (2.32). One more (much overlooked) equivalent condition in terms of general (Lyapunov) exponents (see Chapter 3) was given by Bol in [40] who in fact did not use the term "stability under persistent disturbances". Next let us formulate the conditions of the £-dissipativity.

The mathematics of nonlinear control

52

Theorem 2 . 9 (Yoshizawa) [326] Assume that there exist R > 0 and a function V : {x E Rn : IxI > R} x [to, oo) -+ R+ which satisfies ao(IxI) < V(x, t) < al(Ixj); V(x, t) < -a2(IxI) as long as IxI > R for continuous nondecreasing radially unbounded function ao, continuous increasing function al, and positive continuous function a2. Then the system (2.21) is uniformly G-dissipative. Corollary 2.1

Consider the system (2.47)

x = F(x, t) + f (t)

where F is smooth, the disturbance vector function is bounded : I f (t) I _< Af for all t > to and the undisturbed system x = F(x, t) is uniformly globally exponentially stable. Then the system (2.47) is L-dissipative and the following inequality is valid Ix(t)I < as Af + I I x(to)l a2 - a3 0f,

exp (__( t _to)

)

(2.48)

where ao, al, a2, a3 are positive numbers from (2.46) and the function [R]+ is defined as follows [R]+ = R if R > 0, [R]+ = 0 if R < 0. To prove this result just evaluate W, where W(t) = V(x(t)) along the trajectories of (2.47) using (2.46): W(t) < -2a1W(t)+ 2a, Af. Vfa-_1

(2.49)

Integrating linear differential inequality (2.49) over [to, oo) and again taking into account (2.46) we obtain inequality (2.48). Finally we formulate conditions of the Lagrange stability. As before we will assume that the right hand side of (2.21) satisfies conditions guaranteeing existence of the solutions to (2.21) at least on some time interval. Theorem 2.10 [325, 183] The system (2.21) is Lagrange stable if and only if there exists a nonnegative scalar radially unbounded function V : Ili" x [to, oo) -4 R [ such that the function V(x(t), t) is a nonincreasing function of time.

Remark 2 .3 It is easy to notice that the sufficient condition for a differentiable function V to be a nonincreasing function of time is V < 0.

Stability and boundedness 53

Some of the results discussed above can be summarized in Table 2.1.

2.2.3

Absolute stability

A very fruitful approach to nonlinear system analysis relies on the concept of absolute stability [193, 7, 254, 323]. This approach originated from the practical problems of automatic control in the presence of uncertainty. Start with a motivating example. Consider again the system (2.37) from Example 2.7 but in this case rewrite it in the following form: y=bx+u f i=cy+dx

(2.50)

where u plays the role of the system input. Assume that the system (2.50) is closed by some output feedback u = f (y) with the purpose of stabilizing the system at the origin for all possible initial conditions. Here y is the system output which is assumed as available for measurement. Using methods of linear analysis it is not difficult to design such a feedback in the form f (y) = ky where the gain factor must be chosen to satisfy the standard Routh-Hurwitz condition. Next suppose that because of the nonlinearity in the actuator, the linear feedback cannot be realized in practice. The question is: can we rely on the linear analysis dealing with a nonlinear system? A similar question arises in the case when the actuator model possesses an uncertainty, i.e. the dependence of the control action on the measured output is not known exactly. In both cases the system designer needs conditions guaranteeing the global asymptotic stability of the closed loop system.

All above observations can be extended to the case of nonlinear systems with a vector input u and vector output y for example for affine systems x = f(x) + 9(x)u, y = h(x). The feedback in this case may be described by some static (memoryless) relation: u = U(y),

(2.51)

54

The mathematics of nonlinear control

x = F(x, t) V : X0 x R1 — R + a0(\x\) R} ai is increasing, a2(r) > 0, Vr > R ao is nondecreasing rad. unbounded X0 = {x 6 Rn : |x| < R} ao € JC oa(r) > 0, X0 = {x G Rn : |x| < i*}

Result System is forward complete

System is Lagrange stable

System is uniformly £-dissipative

x(t) = 0 is Lyapunov stable

ao. 1) with respect to the output y in the class of inputs satisfying the integral constraint (2.58) if for any initial conditions x(0) = xo and for any admissible input satisfying (2.58) the corresponding solution x(t) to (2.59) is unique and well defined on the infinite interval [0, oo) and there exist 1C-functions a„Q such that the following inequality holds:

fo

00

jy(t)1Pdt < a (Ix(0)I) +Q('y).

(2.60)

For the uniform C-norm the concept of absolute stability can be defined as follows Definition 2.11 The system (2.59) is called C-absolutely stable with respect to the output y in the class of inputs satisfying the integral constraint (2.58) if for any initial conditions x(0) = xo and for any admissible input satisfying (2.58) the corresponding solution x(t) to (2.59) is unique and well defined on the infinite interval [0, oo) and there exist 1C-functions a„Q such that the following inequality holds: sup jy ( t)j < a(lx(0 )1) +)3(-y)t>o

(2.61)

Definition 2.12 The system (2.59) is called asymptotically GP ( or C)absolutely stable with respect to the output y in the class of inputs satisfying the integral constraint (2.58) if it is GP (or C)-absolutely stable and lim x(t) = 0. t-00 Remark 2.4 It is worth mentioning that even in the case when p 54 oo and y - x Definition 2.10 is not equivalent to Definition 2.12. Indeed in the definition of absolute stability we do not require that y E G00 and therefore in general y E £, does not imply that y(t) --; 0. The above definitions of absolute stability can be further extended to the case of several integral constraints. Definition 2.13 The system (2.59) is called Cp-absolutely stable with respect to the output y in the class of inputs satisfying the constraints

Jo T w,(u(t), y(t))dt + y, > 0, s = 1, ... , r,

(2.62 )

58

The mathematics of nonlinear control

where w9 : II8' x R' - R', s = 1, ... , r are some functions, if for any initial conditions x(O) = xo and for any admissible input satisfying (2.62) the corresponding solution x(t) to (2.59) is unique and well defined on the infinite interval [0, oo) and there exist IC-functions a, 01 , ... , 0r such that the following inequality holds:

fo 00

Iy(t)IPdt < a (Ix(0)I) + E Q('Y)• s_i

(2.63)

The definition of asymptotic GP-absolute stability and/or C-absolute stability for several integral constraints can be composed similar to Definitions 2 . 11 and 2.12. In the study of absolute stability for complex nonlinear systems it is convenient to interpret the initial system (2.59) as the base or nominal system. Then the constraints (e.g. (2.56 ), ( 2.58), (2.62)) can be understood as descriptions of additional nonlinearities and uncertainties. The most complete results are known for the case when the nominal system is described by a linear state space model and the constraints are homogeneous quadratic forms (e.g. (2.57 ) with a4 = a5 = a6 = 0). In this case the criterion of absolute stability can be formulated in terms of frequency- domain inequalities which are convenient for many applications. We formulate here a seminal result due to V. A. Yakubovich [323]. To give precise formulation we need to define the following minimal stability property. Definition 2.14

Consider the linear system dx dt= Ax + Bu , y = Cx + Du , (2.64)

where x(t) E 118", u(t) E 118', y (t) E R' with the matrices A, B, C, D of appropriate dimensions. The system (2.64) is said to be minimally stable in class of inputs satisfying the constraints (2.58) if for any xo E II8", xo $ 0 there exists a pair {x- (•,xo),u(•,xo )} satisfying (2.58) with ry = ry(xo) such that t(O,xo) = xo, (Tj, xo ) --> 0 when T3 -+ oo and

inf y(A o) = 0.

Stability and boundedness

59

Theorem 2 .11 (Yakubovich) [323] Let the pair (A, B) be controllable and the system (2.64) be minimally stable in the class of inputs satisfying the constraint (2.58) where w is a quadratic form of u, y. Then the system (2.5) is G2-asymptotically absolutely stable with respect to the output y with quadratic a and linear /3 in (2.60) in the class of inputs satisfying (2.58) if and only if the following inequality holds for some b > 0, all w E R1 such that det(iwln - A) 0 0 and all complex-valued vectors u E C': w(u, X(zw)u) < -b IX(iw)uI2

(2.65)

where i2 = -1, w is the Hermitian extension * of the form w to the complex space and X(X) = D + C(AII - A)-1B is the transfer function of the system (2.64).

Similar results hold true for C-absolute stability and for several quadratic constraints ([113], [211] ). Recently some of the results were extended to the case of nonlinear nominal system [270]. Let us demonstrate how to apply Theorem 2.11 by the following example Example 2 .9 Consider again the system from Example 2.7 x = cy + dx y=bx+u

(2.66)

where all quantities are scalars . Suppose the system (2.66) is closed by the nonlinear negative feedback (2.51) where U(y) meets the sectorial constraints (2.55) or, in other words it satisfies the inequality 0 < -f(y)/y < A. The constraints (2.55) can be considered as a local quadratic constraint (2.56) where w(u, y) = u(-y - µ-1u). Therefore the feedback satisfying (2.56) will also satisfy the integral constraint (2.58) for any T; > 0, j = 1, 2 ... and for ry = 0, that is the system ( 2.66) is minimally stable and Theorem 2.11 is applicable. Any real quadratic form Q(x) = xTGx, x E R" with a real symmetric n x n matrix G = GT can be extended to a Hermitian form Q defined for every complex valued vector z = x + iv, x, v E R" by Q(z) = Q(x) + Q(v). In coordinate representation x = col(xl ,..., x,) it means that any term of the form xjxk in Q (x) is replaced by Reza zk, where the asterisk stands for complex conjugation . Obviously, Q(z) - Q(z) for all real z E R".

60

The mathematics of nonlinear control

To check the conditions of Theorem 2.11 we first evaluate the transfer function X(p) of the linear part of (2.66). To this end employ the mnemonic rule: replace the time derivative by the symbol p and solve the arising linear algebraic equations for y with fixed u. Then the transfer function X(p) can be found through inspection of the relation y = X(p)u. For the system (2.66) we have px=cy+dx, py=bx+u.

Solving the first equation for x and substituting the result into the second equation it is easy to obtain y

_ _ p-d p2 - dp - bcu.

Therefore the transfer function of the system (2.66) is given by X(p)= 2 p-d

P - dp - bc*

Assume that d < 0, be < 0. In this case the system matrix is Hurwitz because its characteristic polynomial has positive coefficients. Calculate the frequency response OW) - iw - d - (iw-d)(idw-be-w2) -idw - be - w2 (bc + w2)2 + d2w2 bcd - iw(d2 + be + w2)

(bc + w2) 2 + d2w2 . The Hermitian extension of the quadratic form w(u, y) looks as follows tv = -Re{u*y} - µ-lu*u,

where "*" stands for complex conjugation. Hence the frequency domain inequality becomes

[-ReX(iw) - µ -1 + bI X(iw)12] Iu12 -bcd + 6(d2 + w2) - µ-1 (w2d2 + (bc + w2)2) w2d2 + (bc + w2)2

It is clear that the frequency domain condition (2.65) can be fulfilled for all d < 0, be < 0, 0 < µ < oo, if 6 is chosen sufficiently small. Therefore the system (2.66) is G2-absolutely stable for all nonlinearities with graphs within a finite sector on the plane. Theorem 2.11 says nothing

Feedback linearization and normal forms

61

about absolute stability within the infinite sector (µ = oo). In this case the absolute stability can still be established by means of a degenerated version of the Kalman-Yakubovich lemma (see Sec. 2.4). Note that Theorem 2.11 and its extensions provide absolute stability conditions not only for memoryless nonlinearities but also dynamical feedback arising in complex interconnected systems. 0 The proofs of Theorem 2.11 and related results are based on the algebraic result , the so-called frequency theorem or Kalman- Yakubovich lemma, which allows one to formulate frequency- domain solvability conditions for some matrix inequalities arising in the design of Lyapunov functions. The standard formulation of the Kalman-Yakubovich lemma will be given later in Sec. 2.4. A new approach in the field of absolute stability is based on the study of averaged integral constraints of the form 1 T, w(u(t), y(t ))dt + ry > 0 lim T, -a o Tj o

J

(2.67)

As shown in [252] validity of the constraints ( 2.67) along the solutions of (2.59) implies L-dissipativity of the system , i.e. boundedness instead of stability. This result was extended to the case of several constraints (see [112]). The averaged constraints seem to be a promising tool for analysis and design of oscillatory systems. Some extensions and examples can be found also in [109].

2.3 Feedback linearization and normal forms After giving a brief exposition of the main concepts and ideas of nonlinear systems analysis we move on to surveying some approaches to the main problem: nonlinear systems design. The essence of the design problem is to find a feedback providing a closed loop system with the desired behavior. There is no good solution of this problem in general case. One of the most natural approaches to this problem is as follows: first to simplify the controlled system description in some way and then to solve the initial problem for the simplified system. As is well known the most simple and well-studied systems are linear ones. Therefore the design methods based

62 The mathematics of nonlinear control

on the reduction of the system description to a linear one (linearization) are of special interest. The standard way of system linearization is approximate linearization near some base point or base solution. It changes only the system equations, while the state, input and output vectors remain unchanged. Another approach is to change the state vector in such a way that in the new coordinates the system equations are simpler. Finally we may allow for changing both the state cordinates and inputs (feedback transformations). If the transformations are invertible, then the transformed system is equivalent to the initial one. Choosing the state coordinates and feedback transformation properly we may pose the problem of exactly reducing the system description to a linear one. If it is possible, the initial system is called feedback linearizable. The procedure of finding such state and feedback transformations is called feedback linearization. Below we give more precise treatment of this problem for the case of the systems affine in control: x = f(x) + g(x)u.

(2.68)

Definition 2.15 The system (2.68) is called feedback linearizable in the open domain fl E Rn if there exist the smooth coordinate change z = 4P(x), x E fl and the feedback transformation u = a(x) +,Q(x)v

(2.69)

with smooth functions a, 8 such that 4D and ,6 are smoothly invertible in Il and the closed loop system (2.68), (2.69) is linear, i.e. there exist constant matrices A E Rn R+ (called storage function, here we assume that the dimension of input equals to the dimension of output). The inequalities like (2.81) or (2.82) are valid for passive electric circuits with the stored energy as a storage function. It motivates introducing the term passivity in more general cases.

It is clear that (2.81) for the positive definite function V yields Lyapunov stability of the unforced (u(t) - 0) system. Moreover the feedback of the form u = --yy,

(2.83)

where ry > 0, yields asymptotic stability under some additional observabilitylike conditions. Instead of (2.83) a more general feedback law can be used: u = -z/i(y),

(2.84)

68

The mathematics of nonlinear control

where the vector V) forms an acute angle with the vector y: O(y)Ty > 0 when y # 0. Note that if the affine system x = f(x) + g(x) u

(2.85)

posesses an asymptotically stable trivial solution for u - 0 then by virtue of the converse Lyapunov theorem there exists a positive definite function V : Iltn -, ]I8+ such that (VV (x))T f (x) < 0 for x 0. It immediately yields dissipation inequality (2.81) with the output y = (VV(x))Tg(x) since the derivative of the function V with respect to (2.85 ) is equal to V = (VV (x))T f (x) + (VV (x))T g(x)u. If the output function h(x, t) is not fixed and the storage function is positive definite, then the passivity of the system is equivalent to its stability. The problem of finding an output function h(x, t) and a feedback law rendering the system ( 2.85) passive is called the passification problem. It can be considered as an intermediate step on the way to stabilization. It is also of interest in its own right because in many applications (e.g. in the energybased control of oscillations , see Chapter 4) the storage function is neither proper (radially unbounded ) nor positive definite and the desired behavior of the closed loop system differs from its stability. In the light of the above the following questions are of interest: (1) When the system (2.85) is passive; (2) When there exists the state or output feedback making the closed loop system passive; (3) When there exist the output function h(x, t) and the state or output feedback which passifies the system ( 2.85);

(4) How to passify a complex system by means of step by step passification of its subsystems. The answers to the above questions constitute the passification theory which has been developed during the last two decades. It has achieved some degree of maturity especially for the case of linear systems. We will expose briefly some important results in this section. Also in Sec. 2.5 the Speed-Gradient Method will be described which provides a simple and efficient design of passifying algorithms.

Feedback stabilization and passivity

2.4.2

69

Passivity and dissipativity

When we are concerned with input-output properties of the controlled systems, particularly, when solving the problem of output feedback design, we need some toolkit similar to the Lyapunov's stability theorems. Such a general and convenient toolkit was developed recently based on the concepts of passivity and dissipativity. These concepts were born in physics where the energy dissipation is a key property of the systems. Similar concepts are also well known in the electrical circuit theory: recall that a circuit is called passive if it does not contain any energy sources. In the theory of oscillations the concept of dissipativity was introduced by A. Andronov and coworkers [16], in the stability theory it was studied by N. Levinson [182] but in a different sense. In the control theory the definition of dissipativity was proposed by J. C. Willems [315] and now this definition has become widely used. In this section we will show connections between Lyapunov stability, passivity and dissipativity properties.

We will deal with the systems of the following form i = F(x, u, t), y = h(x, t),

(2.86)

where x(t) E R" is the state, u(t) E R' is the input, and y(t) E R1 is the output. To avoid the problems of choosing the initial time we will consider the systems (2.86) on the time interval [0, oo) with the initial condition x(0) = xo. The case when the initial conditions are specified for t = to can be reduced to the previous one by the time shift t' = t - to. It is assumed that the set of admissible inputs functions U consists of all piecewise continuous, locally bounded functions u : ][8+ -. R'' and F : R' x R' x Il8+ -+ R' is a locally Lipschitz in x, u uniformly in t function, F(0, 0, t) - 0, Vt > 0, the function h is assumed to satisfy h(0, t) _- 0, Vt > 0. Additionally when studying the behavior of systems with inputs we will assume that the control is applied at the instant t = 0. Therefore for any initial conditions x(0) and input u E U the solution to the system (2.86) is well defined at least on some time interval [0, Tu,zp). If Tu,.u = oo, then the system (2.86) is referred to as forward complete. In Sec. 2.2.2, Theorem 2.1, we gave sufficient conditions of completeness for time-varying systems without input. Considering input as a (bounded) function of time it is possible to apply them to the case already discussed .

Associated with the system (2.86) consider a real-valued defined on ][8'"' x R' piecewise continuous function w called the supply rate.

70

The mathematics of nonlinear control

Definition 2.18 A system (2.86) with the supply rate w is said to be dissipative in the sense of Willems or W-dissipative if there exists a continuous nonnegative function V : Rn X Il8+ -' R+, such that the following dissipation inequality holds: V(x(t), t) - V(x(to.), to) < f w( u(s), y ( s))ds tot

(2.87)

for all it E U , x(to) E Rn' 0 < to < t < Tu,.o, where Tu,yo is the upper time limit for which the solution corresponding to the input it and initial conditions x(0) = xo exists. The function V in the definition is called the storage function. Remark 2 .5 If the function V is differentiable in x and t, then the dissipation inequality can be rewritten in the infinitesimal form

at

(x, t) + (V.V (x, t))T F(x, u, t) < w(u, h(x, t))

(2.88)

which is equivalent to (2.87) Indeed, to obtain (2.88) just integrate (2.87) over [to, t]. Conversely, if the function V is differentiable in x and t, then dividing both sides of (2.87) by t - to and approaching t to to under the condition x(to) = x we readily obtain (2.88) from the mean value theorem.

The dissipativity property is closely connected with the concepts of stability and absolute stability. Consider for simplicity autonomous systems . In this case we may consider only time-invariant storage functions V = V (x). Assume that the system (2.86) is W-dissipative with a positive definite storage function with respect to the supply rate w satisfying w(0, y) < 0 for all y E R1. Then substituting the zero input u(t) 0 into (2.87) (or (2.88)) yields Lyapunov stability of the trivial solution x(t) _- 0 of the free (u = 0) system (2.86). Next assuming that V is radially unbounded we obtain that all the solutions of the uncontrolled system are bounded, i.e. the system (2.86) with u(t) = 0 is Lagrange stable. Finally, if V E 1C, and the input u satisfies the following integral constraint

J0 c w(u(s), y(s))ds < ry

(2.89)

for all t _> 0 some ry > 0, then the system (2.86) is C- absolutely stable with respect to the output y in the class of inputs it satisfying (2.89) (Note that

I I ..•. 4 -4.. b,- . .. ^,

Feedback stabilization and passivity 71

in the definition of absolute stability the integral constraint corresponds to (2.89) when the function w is replaced by -w). Throughout the book we are interested in a particular case when the supply rate is linear in control that is w(u, y) = q(y)Tu for some function q : II8! -i IR In this case mapping q o h defines a new output of the same dimension as the input u. Motivated mostly by this observation we will consider the case of square systems: 1 = m. An important special case of the dissipativity in the sense of Willems is the concept of passivity. Definition 2.19 A system (2.86) with m = 1 is said to be passive if it is dissipative with the supply rate w = yTu and the storage function V satisfies V(O,t)=0, for alit ^ 0. In other words the system (2.86) is passive if the following dissipation inequality is valid for all solutions of (2.86). V(x(t), t ) - V(x(to ), to) < ft y( s)Tu(s )ds. to

(2.90)

It is seen from ( 2.90) that the storage function V(x(t), t) does not increase along any solution consistent with the constraint y(t) - 0. Since such solutions define the zero dynamics (see Definition 2.17) one can deduce that passive systems having a positive definite storage function V have a zero dynamics with the Lyapunov stable trivial solution. Definition 2.20 A system (2.86) is called weakly minimum phase (resp. minimum phase, exponentially minimum phase ) if its zero dynamics is Lyapunov stable (resp. asymptotically stable, exponentially stable). It is convenient to introduce some modifications of the passivity property: Definition 2.21 A passive system (2.86) is said to be state strictly passive (SSP), if the following relation V(x(t), t) - V(x(to), to) =

Jtot (y(s)Tu (s) - S(x(s)))ds

(2.91)

holds for all 0 < to < t < Tu,,, for some positive definite function S : R n -+ R+ Motivated by Krasovskii's theorem (Theorem 2.7) of exponential stability the following definition can be introduced.

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The mathematics of nonlinear control

Definition 2.22 A state strictly passive system is called exponentially passive if there exist positive numbers 01,02,03 such that the following inequalities hold: a1\x\20.

(2.97)

Feedback stabilization and passivity 73

The following result is a version of the Kalman -Yakubovich (or KalmanYakubovich-Popov) lemma (see [320 , 156, 179], for its origins see [24]) Lemma 2 . 2 (Kalman-Yakubovich lemma) Let the pair (A, B) be controllable . Then the following statements are equivalent: 1. There exists a matrix P = PT which satisfies (2.96). 2. The following frequency- domain inequality w(u, X(awu) > 0 (2.98) holds for all real w such that det(iwI - A) # 0 and for all complexvalued u E C". Here iv is Hermitian extension of the form (2.95). 3. There exist matrices P, v, i which satisfy the following identity 2xTP(Ax + Bu) = w(u, y) - IvTx - r u12. (2.99) Note that Lemma 2.2 says nothing about the condition P > 0 or P > 0 which is important for studying passivity and dissipativity. The necessary and sufficient conditions which ensure P > 0 or P > 0 are still unknown. However, the case of passive systems has been studied completely. We formulate another version of the Kalman-Yakubovich lemma for the special case D = 0, m = 1, w(u, y) = yTu which will be useful in the following chapters. Lemma 2 .3 (Yakubovich), [320, 179] Let rankB = m. Then the following two statements are equivalent:

1. There exists a positive definite matrix P = PT > 0 such that PA+ATP < 0, PB = CT.

(2.100)

2. The polynomial det(AIn-A) is Hurwitz and the following frequencydomain inequalities are satisfied Re uT X(iw)u > 0,

Jim w2Re uT X(iw)u > 0

(2.101)

W -100

for allwER anduERm, u#0. Remark 2.6 It can be shown that inequalities (2.101) are equivalent to the following condition: Re X(A) > 0 if Re A > 0. (2.102)

74 The mathematics of nonlinear control

This condition is referred to as the strict positive real condition and it corresponds to the case of strictly passive systems , while passive linear systems satisfy a weaker condition referred to as the positive real condition: ReX(A ) > 0 if ReA > 0 .

( 2.103)

Remark 2.7 The condition rankB = m looks natural because it means that some columns of the matrix B are linearly independent and therefore if this is not the case, then some input variables can be discarded. Lemma 2.3 gives efficient frequency-domain conditions of strict passivity of linear systems. Namely, strict passivity is equivalent to the strict positive realness of the transfer matrix of the system. This result has broad applications in the circuit theory. It is also useful for analyzing the stability of control systems [177, 180].

However the control design problems require different tools. For example assume that we closed the system (2.94) by the linear feedback u = Ky,

(2.1-04)

where m = l and K is an m x m matrix. Then the conditions (2.100) read as follows

P(A + BKC) + (A + BKC)T P < 0, PB = CT . And the question is: whether it is possible to find a positive definite matrix P and matrix K which satisfy these relations. The solution to a slightly more general problem (for the non-square matrix K) is given by the following statement which can be called the feedback Kalman- Yakubovich lemma.

Lemma 2 .4 [91] Let rankB = m. Then for the existence of the matrix P= PT > 0 and m x l matrix K satisfying P(A + BKC) + (A + BKC)TP < 0, PB = CTG,

(2.105)

where G is a given 1 x m matrix, it is necessary and sufficient that the polynomial det(AIn - A) det GT X(A) is Hurwitz and the matrix

r = lim AGTX(A) 1 XI-00 is symmetric and positive definite.

Feedback stabilization and passivity 75

Corollary 2.3 Let m = 1 and GT X(A) = ,3(A)/a(A) where /3(X) = ,Qn_1a"-1 + • • • + 00 and a(A) = an\n + • • • + ao are the numerator and denominator of the transfer function GTX(.X ). Then the problem (2.105) is solvable if and only if the polynomial O(A) is Hurwitz and /3n_1 > 0. The Kalman-Yakubovich lemma is one of the most important results of the nonlinear and optimal control theory and stability theory (see survey [24]). Therefore it is interesting to obtain analogous results for nonlinear systems and nonquadratic supply rates . The known proofs of this lemma use a number of results from algebra , complex analysis and optimization theory and cannot be directly extended to the fully nonlinear case. Nevertheless some results which are close in spirit to the Kalman -Yakubovich lemma are known for nonlinear systems and often referred to as the nonlinear Kalman-Yakubovich lemma . We present here the extension of the standard result [218, 148] to the time-varying systems . It will be used in the sequel. Let us consider again nonlinear affine in control systems: i = f(x,t) + g( x, t)u, y = h(x, t),

(2.106)

where x(t) E R', u(t) E lRt, y(t) E lRm, and f, g, h are smooth enough to ensure existence of solutions to (2.106) at least on some time interval.

As one can see the condition PB = CT

(2.107)

appearing in (2.100) is equivalent to the identity xT PB = (Cx)T for all x E R' which can be interpreted as (VV (x))T B = y',

(2.108)

where V (x) = xT Pxl2, y = Cx. An extension to the nonlinear affine systems i = f (x) + g(x)u, y = h(x) turns (2.107) into (VV (x))T g(x) = h(x)T .

(2.109)

It motivates the following definition. Definition 2.24 A system (2.106) has the KYP (Kalman-YakubovichPopov) property if there exists a differentiable nonnegative function V :

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The mathematics of nonlinear control

Rn x R+ -, R+, V (O, t) = 0, for all t, such that aV (x, t) + (V=V (x, t))' f (x, t) < 0,

-

(2.110)

(V.V (x, t))T g(x, t) = h(x, t)T T.

(2.111)

at

The two relations (2.110), (2.111) can be interpreted as an infinitesimal

version of the dissipation inequality for a passive system. The following result may be called a nonlinear version of the Kalman-Yakubovich lemma. Lemma 2.5 A system (2.106) is passive with a differentiable storage function if and only if it has the KYP property. For time-invariant case the lemma was formulated in [218] (for proof see [148] ) Proof: If the system (2.106) has the KYP property, then along any of its trajectories which exist on the time interval [to, Tu,=o )

1 (x(t), t) = aV( (t), ut, t) + (V V(x(t), t))T f(x(t), t) +(V V (x(t), t))T g(x (t), t)u(t) < y(t)T u(t)

(2.112)

and integration from to to t, where t < Tu,.o, yields the dissipation inequality. Therefore the system ( 2.106 ) is passive with the storage function V. Conversely, if the system (2.106) is passive, then the dissipation inequality (2.90) is satisfied for all to < Tu,.o. Rewrite the dissipation inequality in the following form: V (x(to + 6), to + 6) - V (x(to ), to ) < 1 jt0+S y 6

6

o

(s ) T u(s)ds

(2.113)

with some positive 6 such that to + 6 < Tu,,o. Taking the limit for 6 -+ 0 yields (2.112). Since (2.112) is valid for any admissible input and the set of admissible inputs contains u(t) - 0, we immediately obtain that V satisfies (2.110) at t = to and the following relation must hold for all admissible inputs at t = to:

((V.V (x(t), t))T g(x(t), t) - y(t))T u(t) < 0.

Feedback stabilization and passivity 77

Taking into account that the left multiplier in this inequality does not depend on u and since to may be arbitrary from [0, Tu,x0) we readily see that the relation (2.111) holds for all t E [0, Tu,,O ), that is, the system has ■ the KYP property with the storage function V.

Remark 2.8 Notice that a passive system can have solutions defined only on a finite time interval as one can see from the following example.

Example 2.11 Consider the following system x = -x + x2u, y = x3, x(t) E R. This system has the KYP property. Indeed, consider the following storage function candidate: V(x) = x2/2. Simple calculations give VV(x)(-x) = -x2 < 0,

VV(x)x2 = x3 = y,

i.e. the system has the KYP property. On the other hand for any bounded piecewise continuous u there exists a nonempty set of initial conditions x(0) such that the corresponding solution exists only on a finite time interval (see

Example

2.4).

A

Let us briefly discuss the connection between the concepts of passivity and stability. It is not difficult to notice that imposing some additional assumptions on the storage function V we may obtain different stability properties of the uncontrolled (u(t) - 0) system. Besides, the Lyapunov function which proves those stability properties is just the storage function. For example the following result (which is a slightly modified result of [148] ) can be easily established ([168], Lemma D.3): Lemma 2.6 Suppose the system (2.106) is passive (state strictly passive). If V is positive definite, radially unbounded and decrescent, that is, there exist class 1C,,^ functions yl and y2 such that 'yi(JxJ) < V(x,t) < ry2(JxI) for all x E R", t E R+, then, for u(t) - 0, the equilibrium x(t) - 0 is globally uniformly stable (asymptotically stable).

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The mathematics of nonlinear control

Proof of this fact is based on the following relationship which holds for u(t) - 0:

V(x(t), t) = -S(x(t)), where the function S is from the definition of state strictly passive systems.

2.4.3

Passification as a control design problem

As we have seen , the passivity property is closely connected to the property of.stability. It motivates the following control design problem: to find a feedback which makes the system passive. We will need the. following definition for a local form of passivity similar to the one given in [255]. Definition 2.25 A system (2.86) is called state strictly passive in the region SZ C R , if there exist a nonnegative function V : SZ x R+ -' R+ and a positive definite smooth function S : Si -+ R+ such that the relation (2.91) holds for all 0 < to < t < Tu,xo and all u E U which ensure that x(7-) E Si for any r E [Ol t] I t < Tu,xo .

Remark 2 .9 If the set Si is compact then we restrict the set of admissible inputs to the set which renders all the corresponding solutions bounded and therefore Tu,,, = oo for so defined inputs. Definition 2.26 A system (2.86) is said to be globally state-feedback passive, or state passifiable if there exists a smooth control law, u = a(x, t) +,8(x, t)v

(2.114)

where a(0, t) - 0 for all t > 0, and v(t) E ll8' is a new input, such that any solution of the closed loop system satisfies the dissipation inequality (2.90) for all admissible inputs v.

Similarly to Definition 2.22 the concept of a state-feedback exponentially passive, or exponentially passifiable system can be introduced. We are also interested in formulating an output feedback version of Definition 2.25. Definition 2.27 A system (2.86) is called semiglobally output feedback exponentially passive if for any compact set Si C 1Rn there exists a smooth

Feedback stabilization and passivity 79

feedback u = an (y, t) + 6n (y, t)v,

(2.115)

where an(0,t) __ 0 for all t > 0, and v(t) E R' is a new input, v E U, such that the closed loop system is exponentially passive in the region 12, and Vn, (x, t) = Vn, (x, t), for x E , fl 122 where Vn : 12 x R+ -> R+ is a storage function ensuring exponential passivity in 12. The given definitions show that passivity can be considered as a control design goal. In the terminology of [50] we seek for the conditions of equivalence via static state (output) feedback of a given system to a passive system. As one can see from definitions here we consider the case of a static (memoryless) feedback. First we will present a solution of the posed problem for the case of linear systems and then our goal is to extend the result to the case of nonlinear systems. The following theorem can be considered as an immediate consequence of the Feedback Kalman-Yakubovich lemma (Lemma 2.4). Theorem 2.13 [103] Consider the linear system x=Ax+Bu, y=Cx,

(2.116)

where x(t) E W1, y(t) E R'", u(t) E R n, m < n, rankB = m. Then the following three statements are equivalent: 1. The system (2.116) can be made state strictly passive by means of the linear output feedback u = Ky + Lv

(2.117)

for some K, L, where v (t) E 1R' is a new input, det L $ 0.

2. The system (2.116) can be made state strictly passive by means of the linear state feedback u = Mx + Lv

(2.118)

for some M, L, where v(t) E Rm is a new input, det L # 0. 3. The system (2.116) is minimum phase and rankCB = m.

It is a nontrivial fact that the state feedback strict passifiability is equivalent to the output feedback strict passifiability and as we will see the similar property holds also for nonlinear systems.

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The mathematics of nonlinear control

Since in practice the new input is often an inaccessible disturbance affecting the system, the matrix function /3 in the passifying feedback (matrix L in (2.117), (2.118)) cannot be changed in this case by the system designer and should be considered as given. It is worth mentioning that the necessary and sufficient condition of existence of a strictly passifying feedback in this case is positive definiteness of the matrix CBL (compare this with the condition r = rT > 0 in lemma 2.4). To present a nonlinear version of Theorem 2.13 we need some definitions. Previously for the case of single-input-single-output systems we introduced the concept of relative degree. Recall that the relative degree r is exactly equal to the number of times one has to differentiate the output in order to have the value of input explicitly appearing in the equation describing evolution of y(r) (t). For multiple-input-multiple-output square systems (m = 1) one can define the relative degree as an m-dimensional vector (see [152]). For our purposes it is sufficient to say that the system x = f(x) + g(x)u, y = h(x),

(2.119)

where x(t) E R', y(t) E Rm, u(t) E Rm has relative degree (1,1, ... ,1)T at the point x = 0 if the matrix L9h(0) is nonsingular. If, additionally, the system (2.119) has relative degreet one at each point xo E R" we will say that the system (2.119) has a uniform relative degree. Assume that the system (2.119) has relative degree one at the origin. If, additionally, the distribution spanned by the columns of the matrix g(x) is involutive, then it is possible to find such a coordinate transformation that in new coordinates the system equations are written in the so-called normal form q(z,y) a(z,y) + b(z,y)u

(2.120)

and this model is valid in the neighborhood of the origin. Throughout this book, when dealing with passive systems we will frequently use the form (2.120). It should be noted that although this form is valid only in the neighborhood of the origin there exist coordinate-free conditions which ensure the existence of a globally defined diffeomorphism which transforms the original system (2.119) into the normal form (2.120) ([49], Theorem 5.5). The reason for our interest in systems with relative degree one is tSometimes for the sake of brevity we will say that a system has relative degree one if each entry of the vector relative degree is equal to unity

4

Feedback stabilization and passivity

81

that under some mild regularity conditions any passive system has relative degree one as it is seen from the following theorem. Theorem 2 .14 [50] Suppose system (2.119) is passive with a twice differentiable storage function V : R" -> R+ which is positive definite. Assume that rankg(0) = m. Suppose that either rankL9h(x) = const in the neighborhood of the origin or that V is nondegenerate. Then L9h(0) is nonsingular, the system (2.119) has relative degree (1, 1 .... 1)T at the origin, the zero dynamics of (2.119) locally exists and the system (2.119) is weakly minimum phases. Notice that if the system (2.119) is written in the normal form (2.120), then the equation z = q(z,0 )

(2.121)

provides the zero dynamics of the system (2.119). In [50] the converse theorem was established: with some mild regularity assumptions the system (2.119) is locally equivalent via state static feedback to a passive system if it is weakly minimum phase and has relative degree one. Below we will study feedback equivalence to an exponentially passive system. From the previous theorems we know that for linear systems there exists a state or output feedback which makes the system state strictly passive if and only if the system is minimum-phase (zero dynamics system has an asymptotically stable zero solution) and has relative degree one (rankCB = m). Since for linear time-invariant systems strict passivity is equivalent to exponential passivity, it makes plausible a conjecture that the nonlinear system (2.119) can be made state strictly passive by means of state and/or output feedback if it is exponentially minimum phase and has relative degree one. Indeed, this is the case for systems with factorized high-frequency gain as one can see from the following theorem. Theorem 2 .15 [103] Suppose the system (2.119) has globally defined normal form (2.120) with the factorized high-frequency gain b(z,y) - bo(z)bi(y) (2.122) §Note that the definition of weak minimum phasenes in [50] is formally stronger than ours: it requires existence of time-independent Lyapunov function instead of Lyapunov stability of the zero equilibrium of the zero dynamics system.

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The mathematics of nonlinear control

where bo(z),b1(y) are smooth matrices, bo(z) = bo(z)T > 0 and bl(y) is invertible on Rm. Then the following three statements are equivalent: 1. The system (2.120) is semiglobally output feedback exponentially passive. 2. The system (2.120) is globally state feedback exponentially passive. 3. The system (2.120) is globally exponentially minimum phase.

This theorem solves the design problem of an output static feedback which makes the system exponentially passive. It follows from the proof that the semiglobally passifying output feedback can be taken in the form u = -[bi(y)]-1[bo(0)-1a(O,y) +'Yy] + [bi(y)]-1v,

ry > 0 (2.123)

or u = -[bi(y)]-1['yy - v], y > 0.

(2.124)

As in the case of linear systems if one looks for the output passifying feedback with fixed ,9, e.g., /3(y) = Im, then a more strong condition should be imposed: for example one can require that b(z, y) = b(z, y)T > 0 for all z, Y.

Theorem 2.15 shows that the system is semiglobally output feedback exponentially passive if it is exponentially minimum phase. Thus one may wonder whether it is possible to relax the conditions of the theorem, namely, whether the minimum phaseness (asymptotic stability of the zero dynamics) can provide the output feedback strict passivity? In general the answer is negative, as one can see from the following example. Example 2 .12 [49] Consider the relative degree one minimum phase system -z3 + y, z+u.

Its zero dynamics is described by the following equation

The zero dynamics has globally asymptotically stable zero equilibrium, but this equilibrium is not exponentially stable.

Feedback stabilization and passivity

83

The control law u = -yy yields a closed loop system whose Jacobian matrix at the origin has the characteristic polynomial A2+7A-1=0 and thus eigenvalues are in the right -half plane for any ry, that is, no C'smooth output feedback can locally stabilize the system at the origin and therefore there is no output feedback which makes the closed loop strictly passive. A

2.4.4

Input-to-state stability

In the previous sections we discussed the concept of dissipativity and the related concepts of passivity and strict passivity. The starting point of our consideration was the generalization of classical Lyapunov conditions to certain classes of dissipation inequalities. These inequalities characterize the energy flow from the input to the output. Introducing different types of dissipation inequalities it is possible to characterize different properties of the systems with input and output. In this section we will study the concept of input-to-state stability (ISS) which plays an important role in the nonlinear control theory. Before giving a rigorous definition we present the following motivating example.

Example 2.13 Consider the following first order system: x = -x + u with the input u and output y - x. function candidate

(2.125)

Examining the following storage

V(x) = x2/2 we readily obtain that V(x) = -x2 + ux. Therefore the system (2.125) is state strictly passive and the storage function satisfies the following dissipation inequality:

V(x(t)) - V(x(0)) =

J0 t (u(s)x(s ) - x2(s))ds

The mathematics of nonlinear control

84

Simple calculations give V(x) _ -x2 + ux = - 4x2 - 4x2 + ux - u2 + u2 l l 2 +u2 4x2-(2-u) < u2 - 4x2 The last inequality may be interpreted as the dissipation inequality with the supply rate w (u, x) = u2 - 3/4x2 , that is, the system (2.125) is Ww. A dissipative with the supply rate Based on this example one may define an ISS property as W-dissipativity with the storage function V : R' x Il8+ -p )(S+, satisfying ryi(Ixl) < V(x, t) < 72(lxl) for some k,,. functions yi, y2 with the supply rate w(u, x) = pi (lul)P2(lxl) for some /Cc,,, functions pi, p2 . It is convenient to give an equivalent definition which describes in an intuitively clear way the evolution of an ISS system affected by the input u. The following definition is an extension of Sontag 's definition [287] to the time-varying systems [168]. Definition 2.28

The system (2.126)

x = F(x, u, t)

where F : R' x R' x R+ -4 R" is piecewise continuous in t and locally Lipshitz in x, u uniformly in t, is said to be input-to-state stable (ISS) if there exist a class KG function Q and a class K function ry, such that, for any x (O) and for any admissible input u E U the solution exists for all t > 0 and satisfies

Ix(t)I 2, x(t) escapes to infinity in finite time. A This example shows that even in the case when the input tends to zero and the uncontrolled system u(t) = 0 is globally asymptotically stable (this property will be referred to as the 0-GAS property) the solution of the overall system may tend to infinity when t approaches some limit value Tu,xo. Therefore one may conjecture that if the system is forward complete and 0-GAS, then it has a CICS property. This is not the case at least for non-autonomous systems as seen from the next example.

The mathematics of nonlinear control

86

Example 2 .15 [208, 238] Consider the following equation i = - (a - sin ln (t + 1) - cos ln(t + 1)) x + u(t). (2.130) The solution of the homogeneous equation u(t) - 0 has the following form: x(t) = x(0) eae-(a -sinln( t+1))(t+1) therefore x(t) --40 as t -* oo and the system is 0-GAS whenever a > 1 and u(t) - 0.

Further let u(t) = e-a(t+1) and 1 < a < 1+e-"/2. Obviously, u(t) -+ 0. The solution of Eq. (2.130) with zero initial conditions in this case is given by

t x(t)

= e-(a-sinln ( t+1))(t+1) f e-(s+') sinln (s+1)ds.

0

Let to = e(2n + 1/2)a, n = 1, 2, .... If the-" < s < the -2"/3, then -1 < sin In s < -1/2 and we have t„-1 t e-" e-(3+1)sinln( s+l)ds >

e-ssinlnsds

t„e-* t„e-2x/3

>

J

e9'2ds

te-x

let"e -x/2 (et°( e-

2x/3-

a -x)/2

On the other hand, (a - sin In tn ) tn = (a - 1 )tn. Therefore, since to -+ 00 and a < 1 + e-"/2, we have x(tn-1 ) > 2exp

( 2 (1 +

e-n - a) to i x (

12th (e-'" - e-") I - 1 I ,

exp \\\ Eq. (2.130 ) with zero and, therefore x(ttn- 1 ) --+ oo. Thus Jthe soluof

initial conditions is unbounded and exists for any t > 0 in spite of u(t) --+ 0. IL

Although the ISS property is a sufficient but not necessary condition ensuring the CICS property it is much more convenient in practice since for any given bounded u it is possible to find an ultimate upper bound of IIxi , which depends on uThere exists a useful result which helps to check the ISS property.

Speed gradient algorithms 87

Theorem 2 . 16 ([287], see also Theorem C.2 in [168]) Suppose that for the system (2.125) there exists a function V : Rn x II8+ -+ R.. such that for all xElR anduERm,

7i(IxI) < V(x,t) < 72(IXI) IxI < p(I u1) at (x, t) + (VV(x, t))TF(x, u, t) < -73(1x1) where -yl,72 and p are class Ko. functions and 73 is a class K function. Then the system (2.125) is ISS. The function V satisfying all conditions of Theorem 2.16 is referred to as an ISS-Lyapunov function. One of the most interesting properties of an ISS system is that it is feedback equivalent via state feedback to a 0-GAS system [289]. This statement means that any 0-GAS system can be made ISS by an appropriate state feedback.

2.5 Speed gradient algorithms In this section we will present a unified approach to solving nonlinear control problems which has been developed in the 70s. This approach is closely related to both stability and passivity. It has been called the speed gradient method [92]. To make the SG method applicable the control goal should be reformulated via some goal function Q which is employed to construct a Lyapunov function for the closed loop system. The procedure of the SG method is very simple and does not require much knowledge in control theory which is important for the interdisciplinary readership of this book.

2.5.1

Goal- oriented formulation of the control problem

Consider a plant with a model given by the following ordinary differential equation: = F(x, u, t),

(2.131)

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The mathematics of nonlinear control

where x(t) E R' is the state vector, u(t) E R' is the control input. We will assume that the vector function F : R' x R'' x R+ -+ Rn is defined for all x E R", u E R', t > 0, piecewise continuous in t and continuously differentiable in x and u.

The general control problem can be defined as finding the control law u(t) = U[x(s), u(s) : 0 < s < t] which ensures the control objective: Qt < 0

when

t > t,,,

(2.132)

where Qt is the value of some nonnegative objective functional calculated at the instant t, A is some prespecified tolerance and t. is the time instant at which the control objective is achieved. The objective (2.132) can be formulated also as lim Qt < A,

(2.133)

t 00

which in fact slightly differs from (2.132). For the special case 0 = 0 we get the goal Qt -+0 as t ->oo. (2.134) The goal (2.134) is closely related to asymptotic stability, if Qt is a kind of positive definite functional. The control design problem as formulated above encompasses a fairly wide class of practical problems arising in science and technology. To understand better the idea of the goal-oriented formulation let us consider some examples. Suppose we need to handle the tap in such a way that the flow rate achieves the desired level y,,. The simplest model of the "tap" system is as follows (see Eq. (1.4)): (Tp + 1)y(t) = a + 13u( t)

(2.135)

where p = d/dt as usual . The model (2.135) can be rewritten in the form -T+Tu+T.

(2.136)

It is quite clear that ( 2.136 ) is a special case of (2.131) for x( t) = y(t) E 1[81, u(t) E 1181 and F(x,u,t) =T-1(-y+,13u+a).

Since our initial goal is to make the value y(t) close to the desired level y„ for sufficiently large time t > 0 we may formalize it as (y (t) - y*1 < d

i a.

Speed gradient

algorithms

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for t > t., where d > 0 is the given constant while t. > 0 is some uncertain time instant. Hence the control goal takes the form (2.132) where the goal function can be chosen, for example