Feedback Systems: Input-Output Properties

Feedback Systems

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Barry C. Arnold, N. Balakrishnan, and H. N. Nagaraja, A First Course in Order Statistics Charles A . Desoer and M . Vidyasagar, Feedback Systems: Input-Output

Properties

Feedback Systems Input-Output Properties

Charles A. Desoer University of California at Berkeley Berkeley, California

M. Vidyasagar Tata Consultancy Services Hyderabad, India

SIAM Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2009 by the Society for Industrial and Applied Mathematics This SIAM edition is an unabridged republication of the work first published by Academic Press, Inc., 1975. 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. Library of Congress Cataloging-in-Publication Data Desoer, Charles A. Feedback systems : input-output properties / Charles A. Desoer, M. Vidyasagar. p. cm. -- (Classics in applied mathematics ; 55) Originally published: New York : Academic Press, 1975. Includes bibliographical references and index. ISBN 978-0-898716-70-2 1. Feedback control systems. I. Vidyasagar, M. (Mathukumalli), 1947- II. Title. TJ216.D42 2008 629.8'3--dc22 2008039189

is a registered trademark.

CONTENTS

Preface to the Classics Edition

xi

Preface

xiii

Acknowledgments

xv

Note to the Reader

xvii

List of Symbols

xix

I

Memoryless Nonlinearities

1 Sector conditions ... ... 2 Linear feedback around a nonlinearity (memoryless case) 3 Multiple nonlinearities ... Notes and References vii

... ...

2 4 8 9

viii

CONTENTS

II 1 2 3 4 5 6 7 8

III 1 2 3 4 5 6 7 8

IV

Norms Norms: definitions and examples Equivalent norms Relations between normed spaces . Geometric interpretation of norms Induced norms of linear maps Two examples Norms and spectral radius ... The measure of a matrix Notes and references

11 13 15 18 20 23 27 29 35

General Theorems Setting of the problem Small gain theorem Small gain theorem: incremental form A boundedness result An existence and uniqueness theorem Loop transformation theorem ... Stability General feedback formula Notes and references

37 40 44 46 47 50 52 54 55

Linear Systems

0 Introduction 1 Linear feedback systems with rational transfer functions 2 Necessary and sufficient conditions: factorization method 3 Linear feedback systems with dynamics in the feedback path (rational transfer functions case) 4 Convolution feedback systems 5 Graphical test 6 Discrete-time systems 7 Linear time-varying systems 8 Slowly varying systems 9 Linearization Notes and references

56 57 62 73 82 92 98 108 125 128 135

CONTENTS

V 1 2 3 4 5 6 7 8 9

VI

ix

Applications of the Small Gain Theorem Continuous-time systems—LP stability L2 Stability—circle criterion Exponential weighting—L00 stability Discrete-time systems—LP stability Slowly-varying linear systems Nonlinear circuit example Existence of periodic solutions Popov criterion ... ... ... Instability Notes and references

...

...

...

... ...

137 139 143 146 147 149 151 155 157 167

Passivity

0 1 2 3 4 5 6 7 8 9 10

Introduction Motivation from circuit theory Scalar products Formal framework Passive systems: definition and examples Passivity theorem The Popov criterion Discrete-time case Average logarithmic variation criterion Multiplier theory Relation between the passivity theorem and the small gain theorem 11 Invertibility of I + H 12 Instability theorems Notes and references

168 169 171 172 173 181 186 191 194 199 215 219 224 226

Appendixes A

Integrals and Series

A.l Regulated functions A. 2 Integrals ... A.3 Series

229 232 234

CO N T E N T S

X

B

Fourier Transforms

B.l L 1 Theory B.2 L2 Theory B.3 Laplace transform

235 236 237

C

Convolution

C.l C.2 C.3 C.4

Introduction Convolution of functions Convolution of a measure and a function Convolution of sequences

D

Algebras

D.l D.2 D.3

Algebras

E

Ideals Inverses in B e l l m a n - G r o n w a l l Lemma

A

238 239 242 243

245 248 249 252

References

255

Index

261

PREFACE TO T H E C L A S S I C S EDITION

It is difficult to believe that more than 30 years have passed since Charlie and I published Feedback Systems: Input-Output Properties. In spite of the passage of time, the book continues to get cited, primarily because there are many results in this book that are simply not found anywhere elseMathematics (or control theory) experiences its own winds of fashion, just as any other subject does. Other than The Analysis of Feedback Systems by Jan Willems, our book is perhaps the only one to contain the foundational material on feedback stability. As mentioned in the preface to the original edition, Charlie had finished large parts of the manuscript by the summer of 1973 but was finding it difficult to finish the rest, owing to a paucity of time. A t that time I was a (no doubt brash) 25-year-old, newly married, with lots of time on my hands. So Charlie invited me to Berkeley to finish up the manuscript, which we did. W h e n I went to Berkeley, I had already embarked on my other book-writing project, namely Nonlinear Systems Analysis (now also a SIAM Classic). However, I was a little apprehensive about trying to write a book on my own. The experience of writing a book with Charlie was so pleasurable and such a boost to my self-confidence that there has been no holding me back ever since. Shortly I will enter "double figures" in the matter of book writing. I leave it to the public at large to determine whether Charlie is to be congratulated or castigated for his role in all this. -M. Vidyasagar

XI

PREFACE

During the last 15 years the subject of nonlinear feedback systems has undergone considerable development. On the one hand, the theory of Lyapunov functions has evolved rapidly: It is, in fact, the subject of three authoritative treatises by Zubov, Yoshizawa, and Hahn, respectively. For this reason, this book does not cover the theory of Lyapunov functions. On the other hand, the techniques of functional analysis, pioneered by Sandberg and Zames, have developed equally rapidly and generated a large number of results concerning the input-output properties of nonlinear feedback systems. The principal advantage of the latter technique is that it covers distributed systems almost as easily as lumped systems and that it deals with multiinput-multioutput systems in the same framework as the single-input-single-output systems. This is also true for the continuous-time and discrete-time systems. The thrust of this book is, therefore, on the multiinput-multioutput feedback system made of distributed subsystems with attention principally focused on continuous-time systems. Our purpose has been to develop the main techniques that led to the basic input-output properties of feedback systems; we have not even attempted to present an encyclopedic coverage of all problems solved by such methods. Rather, we have tried to maintain a happy medium

xiii

PREFACE

between illustrating the techniques by treating simple cases and developing the best and most esoteric results. Our audience consists of mathematically inclined engineers interested in feedback systems. Most of the subject matter in this book has been taught at Berkeley to engineering students at the Master's level. Experience has shown that engineering students with a good junior course in " mathematical analysis " and an undergraduate course in "control" experience little difficulty with the material covered here. With such an audience in mind, the first two chapters slowly develop special material which is constantly used in the remaining four chapters. Useful mathematical facts whose proofs would take us too far from our main path are collected in the five short appendixes. On the other hand, our audience allows us to motivate many approaches by analogies from conservation of energy and reference to standard reasonings of circuit theory and control theory. This audience would also understand without further explanation terms such as "positive feedback," "feedforward," "transfer function," and "impulse response." In the first chapter we develop a few simple facts about feedback systems and exhibit simple examples of nonlinear systems which illustrate the important distinction that has to be maintained between the questions of existence, uniqueness, continuous dependence, and boundedness, roughly, in the sense of bounded-input-bounded-output. The second chapter develops a number of useful properties of norms and induced norms and of normed spaces, which are used later. Several theorems are presented in Chapter III at a very general and abstract level; to compensate for this, numerous exercises point to useful applications. Chapter IV covers the main results concerning linear systems. Chapter V uses these results to illustrate the use of the small gain theorem: Each of the nine sections applies the small gain theorem to a different class of systems. Finally, Chapter VI develops the framework necessary to discuss passivity and the applications of the passivity theorem. In short, our purpose is to present the main results and the main techniques so that the reader can easily follow the current literature on the subject. A few words about the nature of collaborative effort. Desoer laid down the plan of the book and had the first three chapters and the appendixes almost complete by the winter of 1973. At that time rough drafts of the remaining three chapters were rendered obsolete by recent developments. Vidyasagar, who had indicated an interest in writing a book of this sort, joined in the effort to revise the first three chapters and to rewrite completely the last three chapters. This was done at Berkeley during the summer of 1973. Both authors are responsible for any errors.

xiv

ACKNOWLEDGMENTS

In writing a book that presents a unified view of a large number of results obtained by many researchers over a decade or more, the authors faced the difficult problem of acknowledging the contributions of individual researchers. At one time Desoer attempted to collect a reasonably complete bibliography, but soon the size of the list became so overwhelming that the attempt was abandoned. We finally settled on referring specific results to those papers which we believed taught us a particular approach or solved a particular problem. Thus when we refer a particular item to a specific author, we do not purport to settle claims of priority. We leave such questions of priority and of distinguishing individual contributions .to the professional historians. If in the process we have committed some injustices, we apologize and assure the authors involved that the mistake was unintentional. The preparation of this book involved a great amount of preliminary research that would have been impossible without support and encouragement from several agencies and institutions. The first author gratefully acknowledges the National Science Foundation, the National Aeronautics and Space Administration, the Joint Services Electronics Program, and the University of California. It is a pleasure to acknowledge the contributions of the

XV

ACKNOWLEDGMENTS

many discussions, clarifications, and hard work of colleagues, visitors, and former students in the seminar on nonlinear feedback systems. In the words of E. A. Guillemin, he " cannot name one without naming them all, and he cannot name them all because he can't be sure that he won't miss one or two." They are not nameless, because they have made a name for themselves. The second author gratefully acknowledges the generous policies of Sir George Williams University (Concordia University), which made possible his stay at Berkeley, as well as research support from the National Research Council of Canada.

xvi

NOTE TO THE

READER

The chapter number and chapter title are shown at the top of each left page, e.g., "II Norms." The section number and the section title are shown at the top of each right page, e.g., " 2 Equivalent Norms." Within each section, comments, definitions, equations, facts, theorems, remarks, and so forth, are numbered by means of a numeral appearing in the left-hand margin. These numerals constitute within each section a strictly increasing sequence. From time to time, the numerals are not consecutive: This will not cause problems since these numerals are used like street numbers for houses; they merely identify, within the sequence, one specific item. In referring to an item within the same section, we use only the item number, e.g., "Definition (3)," "Fact (11)." In references within the same chapter but referring to an item in another section, we use the section and item number, e.g., "Definition (2.3)," "Theorem (7.1)." In referring to items in another chapter we use the chapter, section, and item number, e.g., "Eq. (II.3.2)," "Theorem (III.2.1)." The figures are numbered consecutively within a chapter; e.g., "Fig. III. 1" means the first figure of Chapter III. Every technical term is printed boldface in its defining sentence. For convenience the abbreviations u.t.c. for "under these conditions" and s.t. for "so that" are used throughout. xvii

LIST OF S Y M B O L S

A convolution algebra A2 two-sided convolution algebra [a, b] closed interval: {x e U | a < x < b] (a, b] interval: {x e U \ a < x < b} boldface denotes operators, typically maps from Le -> Le B(x; r) open ball of center x and radius r {x\ \\x\\ < r} B(x; r) closed ball of center x and radius r C field of complex numbers C + closed right half plane, i.e., {z e C | Re z > 0} Cn linear space of ordered ^-tuples in C C"x m ring of matrices with n rows and m columns with elements in C H inner product function space {/: 6T -> i^\ < / | / > < 00} He extended inner product function space L normed function space {/: 2T -> "T\ ||/|| < 00} Le extended normed function space LP(U), (LP(U+)) space of functions {/: U -> U} such that / h^ | f(t) \p is integrable over U (U+ , resp.); typically p = 1, 2, 00 lp space of sequences z = (z 0 , zu ...) such that £ j | zk \p is summable xix

L I S T OF S Y M B O L S

R field of real numbers R+ set of nonnegative real numbers { x e l : x > 0} R linear space of ordered ^-tuples in R R n X m ring of matrices with n rows and m columns with elements in U R[s] commutative ring of polynomials in s R(s) commutative field of rational functions in s sgn "signum" function: sgn x is 1, 0, or — 1, resp., according as x > 0, x =0, x [ R w i t h 0(0) = 0. U.t.c. the following four statements are equivalent: (i) kx < (e)le < k2, (ii) kte2 < e(j)(e) < k2 e2, (iii) [0(e) - kxe][4>{e) -k2e] (iii) 0 > 0 0 - k,e2\e - k2 e2) = e2{ - k^ty

- k2 e)

Hence (iii) follows since e2 > 0. (iii) => (ii) Multiply (iii) by e2 and note that (for e ^ 0) k2e2 > kxe2\ hence e(p(e) is intermediate between these two numbers. (i) => ( v ) With kx > 0, (i) implies that <j)(e)le > 0, Me # 0. Invert (i) and multiply by (/>2(