Linear System Theory, Second Edition

LINEAR SYSTEM THEORY Second Edition

WILSON J. RUGH Department of Electrical and Computer Engineering The Johns Hopkins University

PRENTICE HALL, Upper Saddle River, New Jersey 07458 MSmUTO K eiETWTKNICA E ENERGM USP BtftUOTECA Prof, Fonseco Telfes

Library of Congress Cataloging-in-Publlcatlon Data Rugh, Wilson J. Linear system theory / Wilson J. Rugh. —2nd ed. p. cm. — (Prentice-Hall information and system sciences series) Includes bibliological references and index. ISBN: 0-13-441205-2 1, Control theory. 2. Linear systems. I. Title, n. Series. QA402.3R84 1996 O03'.74-dc20 95-21164 CIP

Acquisitions editor: Tom Robbing Production editor: Rose Kernan Copy editor: Adrienne Rasmussen Cover designer: Karen Salzbach Buyer Donna Sullivan Editorial assistant: Phyllis Morgan

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© 1996 by Prentice-Hall, Inc. Simon & Schuster/A Viacom Company Upper Saddle River, NJ 07458

All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs.

Printed in the United States of America 10 9 8 7 6 5 4 3 2

ISBN D - L 3 - M M 1 E D S - E 90000>

9 I 780134»412054 I Prentice-Hall International (UK) Limited, London Prentice-Hall of Australia Pty. Limited, Sydney Prentice-Hall Canada Inc., Toronto Prentice-Hall Hispanoamericana, S.A., Mexico Prentice-Hall of India Private Limited, New Delhi Prentice-Hall of Japan, Inc., Tokyo Simon & Schuster Asia Pte. Ltd., Singapore Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro

-

To Terry, David, and Karen

PRENTICE HALL INFORMATION AND SYSTEM SCIENCES SERIES Thomas Kailath, Editor

ANDERSON & MOORE ANDERSON & MOORE ASTROM & WITTENMARK BASSEVILLE & N1KIROV BOYD & BARRATT DICKINSON FREEDLAND GARDNER GRAY & DAVISSON GREEN & LIMEBEER HAYKIN HAYKIN JAIN JOHANSSON JOHNSON KAILATH KUNG KUNG, WHTTEHOUSE, & KAILATH, EDS. KWAKERNAAK & SIVAN LANDAU LJUNG LJUNG & GLAD MACOVSKI MOSCA NARENDRA & ANNASWAMY RUGH RUGH SASTRY & BODSON SOLIMAN & SRINATH SOLO & KONG SRINATH, RAJASEKARAN, & VISWANATHAN VISWANADHAM & NARAHARI WILLIAMS

Optimal Control: Linear Quadratic Methods Optimal Filtering Computer-Controlled Systems: Theory and Design, 2/E Detection of Abrupt Changes: Theory & Application Linear Controller Design: Limits of Performance Systems: Analysis, Design and Computation Advanced Control System Design Statistical Spectral Analysis: A Nonprobabilistic Theory Random Processes: A Mathematical Approach for Engineers Linear Robust Control Adaptive Filter Theory Blind Deconvolution Fundamentals of Digital Image Processing Modeling and System Identification Lectures on Adaptive Parameter Estimation Linear Systems VLSI Array Processors VLSI and Modern Signal Processing Signals and Systems System Identification and Control Design Using P.I.M. + Software System Identification: Theory for the User Modeling of Dynamic Systems Medical Imaging Systems Stochastic and Predictive Adaptive Control Stable Adaptive Systems Linear System Theory Linear System Theory, Second Edition Adaptive Control: Stability, Convergence, and Robustness Continuous and Discrete-Time Signals and Systems Adaptive Signal Processing Algorithms: Stability & Performance Introduction to Statistical Signal Processing with Applications Performance Modeling of Automated Manufacturing Systems Designing Digital Filters

CONTENTS

PREFACE

xiii

CHAPTER DEPENDENCE CHART

xv

1

MATHEMATICAL NOTATION AND REVIEW Vectors 2 Matrices 3 Quadratic Forms 8 Matrix Calculus 10 Convergence 11 Laplace Transform 14 z-Transform 16 Exercises 18 Notes 21

1

2

STATE EQUATION REPRESENTATION Examples 24 Linearization 28 State Equation Implementation 34 Exercises 34 Notes 38

23

3

STATE EQUATION SOLUTION Existence 41 Uniqueness 45 Complete Solution 47 Additional Examples 50 Exercises 53 Notes 55

40

VII

viii

Contents

4

TRANSITION MATRIX PROPERTIES Two Special Cases 58 General Properties 61 State Variable Changes 66 Exercises 69 Notes 73

58

5

TWO IMPORTANT CASES Time-Invariant Case 74 Periodic Case 81 Additional Examples 87 Exercises 92 Notes 96

74

6

INTERNAL STABILITY Uniform Stability 99 Uniform Exponential Stability 101 Uniform Asymptotic Stability 106 Lyapunov Transformations 107 Additional Examples 109 Exercises 110 Notes 113

99

7 LYAPUNOV STABILITY CRITERIA Introduction 114 Uniform Stability 116 Uniform Exponential Stability 117 Instability 122 Time-Invariant Case 123 Exercises 125 Notes 129

114

8

ADDITIONAL STABILITY CRITERIA Eigenvalue Conditions 131 Perturbation Results 133 Slowly-Vary ing Systems 135 Exercises 138 Notes 140

131

9

CONTROLLABILITY AND OBSERVABILITY Controllability 142 Observability 148 Additional Examples 150 Exercises 152 Notes 155

142

Contents 10

11

12

REALIZABILITY Formulation 159 Readability 160 Minimal Realization Special Cases 164 Time-Invariant Case Additional Examples Exercises 177 Notes 180

ix 158

162 169 175

MINIMAL REALIZATION Assumptions 182 Time-Varying Realizations 184 Time-Invariant Realizations 189 Realization from Markov Parameters Exercises 199 Notes 201

182

194

INPUT-OUTPUT STABILITY Uniform Bounded-Input Bounded-Output Stability Relation to Uniform Exponential Stability 206 Time-Invariant Case 211 Exercises 214 Notes 216

203 203

13 CONTROLLER AND OBSERVER FORMS Controllability 219 Controller Form 222 Observability 231 Observer Form 232 Exercises 234 Notes 238

218

14

LINEAR FEEDBACK Effects of Feedback 241 State Feedback Stabilization 244 Eigenvalue Assignment 247 Noninteracting Control 249 Additional Examples 256 Exercises 258 Notes 261

240

15

STATE OBSERVATION Observers 266 Output Feedback Stabilization Reduced-Dimension Observers

265 269 272

x

Contents Time-Invariant Case 275 A Servomechanism Problem Exercises 284 Notes 287

280

16

POLYNOMIAL FRACTION DESCRIPTION Right Polynomial Fractions 290 Left Polynomial Fractions 299 Column and Row Degrees 303 Exercises 309 Notes 310

290

17

POLYNOMIAL FRACTION APPLICATIONS Minimal Realization 312 Poles and Zeros 318 State Feedback 323 Exercises 324 Notes 326

312

18

GEOMETRIC THEORY Subspaces 328 Invariant Subspaces 330 Canonical Structure Theorem 339 Controlled Invariant Subspaces 341 Controllability Subspaces 345 Stabilizability and Detectability 351 Exercises 352 Notes 354

328

19 APPLICATIONS OF GEOMETRIC THEORY Disturbance Decoupling 357 Disturbance Decoupling with Eigenvalue Assignment Noninteracting Control 367 Maximal Controlled Invariant Subspace Computation Exercises 377 Notes 380 20

DISCRETE TIME: STATE EQUATIONS Examples 384 Linearization 387 State Equation Implementation 390 State Equation Solution 391 Transition Matrix Properties 395 Additional Examples 397 Exercises 400 Notes 403

357 362 376

383

Contents 21

xi

DISCRETE TIME: TWO IMPORTANT CASES Time-Invariant Case Periodic Case 412 Exercises 418 Notes 422

406

406

22

DISCRETE TIME: INTERNAL STABILITY Uniform Stability 423 Uniform Exponential Stability 425 Uniform Asymptotic Stability 431 Additional Examples 432 Exercises 433 Notes 436

423

23

DISCRETE TIME: LYAPUNOV STABILITY CRITERIA

437

Uniform Stability 438 Uniform Exponential Stability Instability 443 Time-Invariant Case 445 Exercises 446 Notes 449 24

440

DISCRETE TIME: ADDITIONAL STABILITY CRITERIA

450

. Eigenvalue Conditions 450 Perturbation Results 452 Slowly-Varying Systems 456 Exercises 459 Notes 460

j

r

25

DISCRETE TIME: REACHABILITY AND OBSERVABILITY Reachability 462 Observability 467 Additional Examples 470 Exercises 472 Notes 475

462

26

DISCRETE TIME: REALIZATION

477

Readability 478 Transfer Function Realizability 481 Minimal Realization 483 Time-Invariant Case 493 Realization from Markov Parameters Additional Examples 502 Exercises 503 Notes 506

498

Contents

XII

27

DISCRETE TIME: INPUT-OUTPUT STABILITY Uniform Bounded-Input Bounded-Output Stability Relation to Uniform Exponential Stability 511 Time-Invariant Case 517 Exercises 519 Notes 520

508

508

28

DISCRETE TIME: LINEAR FEEDBACK Effects of Feedback 523 State Feedback Stabilization 525 Eigenvalue Assignment 532 Noninteracting Control 533 Additional Examples 541 Exercises 543 Notes 544

521

29

DISCRETE TIME: STATE OBSERVATION Observers 547 Output Feedback Stabilization 550 Reduced-Dimension Observers 553 Time-Invariant Case 556 A Servomechanism Problem 562 Exercises 565

546

Notes

567

AUTHOR INDEX

569

SUBJECT INDEX

573

PREFACE

A course on linear system theory at the graduate level typically is a second course on linear state equations for some students, a first course for a few, and somewhere between for others. It is the course where students from a variety of backgrounds begin to acquire the tools used in the research literature involving linear systems. This book is my notion of what such a course should be. The core material is the theory of time-varying linear systems, in both continuous- and discrete-time, with frequent specialization to the timeinvariant case. Additional material, included for flexibility in the curriculum, explores refinements and extensions, many confined to time-invariant linear systems. Motivation for presenting linear system theory in the time-varying context is at least threefold. First, the development provides an excellent review of the time-invariant case, both in the remarkable similarity of the theories and in the perspective afforded by specialization. Second, much of the research literature in linear systems treats the timevarying case—for generality and because time-varying linear system theory plays an important role in other areas, for example adaptive control and nonlinear systems. Finally, of course, the theory is directly relevant when a physical system is described by a linear state equation with time-varying coefficients. Technical development of the material is careful, even rigorous, but not fancy. The presentation is self-contained and proceeds step-by-step from a modest mathematical base. To maximize clarity and render the theory as accessible as possible, I minimize terminology, use default assumptions that avoid fussy technicalities, and employ a clean, simple notation. The prose style intentionally is lean to avoid beclouding the theory. For those seeking elaboration and congenial discussion, a Notes section in each chapter indicates further developments and additional topics. These notes are entry points to the literature rather than balanced reviews of so many research efforts over the years. The continuous-time and discrete-time notes are largely independent, and both should be consulted for information on a specific topic. xiii

xiv

Preface

Over 400 exercises are offered, ranging from drill problems to extensions of the theory. Not all exercises have been duplicated across time domains, and this is an easy source for more. All exercises in Chapter 1 are used in subsequent material. Aside from Chapter 1, results of exercises are used infrequently in the presentation, at least in the more elementary chapters. But linear system theory is not a spectator sport, and the exercises are an important part of the book. In this second edition there are a number of improvements to material in the first edition, including more examples to illustrate in simple terms how the theory might be applied and more drill exercises to complement the many proof exercises. Also there are 10 new chapters on the theory of discrete-time, time-varying linear systems. These new chapters are independent of, and largely parallel to, treatment of the continuous-time, time-varying case. Though the discrete-time setting often is more elementary in a technical sense, the presentation occasionally recognizes that most readers first study continuous-time systems. Organization of the material is shown on the Chapter Dependence Chart. Depending on background it might be preferable to review mathematical topics in Chapter I as needed, rather than at the outset. There is flexibility in studying either the discrete-time or continuous-time material alone, or treating both, in either order. The additional possibility of caroming between the two time domains is not shown in order to preserve Chart readability. In any case discussions of periodic systems, chapters on Additional Stability Criteria, and various topics in minimal realization are optional. Chapter 13, Controller and Observer Forms, is devoted to time-invariant linear systems. The material is presented in the continuous-time setting, but can be entered from a discrete-time preparation. Chapter 13 is necessary for the portions of chapters on State Feedback and State Observation that treat eigenvalue assignment. The optional topics for time-invariant linear systems in Chapters 16-19 also require Chapter 13, and also are accessible with either preparation. These topics are the polynomial fraction description, which exhibits the detailed structure of the transfer function representation for multi-input, multi-output systems, and the geometric description of the fine structure of linear state equations.

Acknowledgments I wrote this book with more than a little help from my friends. Generations of graduate students at Johns Hopkins offered gentle instruction. Colleagues down the hall, around the continent, and across oceans provided numerous consultations. Names are unlisted here, but registered in my memory. Thanks to all for encouragement and valuable suggestions, and for pointing out obscurities and errors. Also I am grateful to the Johns Hopkins University for an environment where I can freely direct my academic efforts, and to the Air Force Office of Scientific Research for support of research compatible with attention to theoretical foundations.

WJR Baltimore, Maryland, USA

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If x is complex, then the transpose of .v must be replaced by conjugate transpose, also known as Hermitian transpose, and thus written XH, throughout the above discussion.

Matrices

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Overbar denotes the complex conjugate, A~ , when transpose is not desired. For scalar x either is correctly construed as complex conjugate, and U* is the magnitude of x.

Matrices For matrices there are several standard concepts and special notations used in the sequel. The m x n matrix with all entries zero is written as Om x,,, or simply 0 when dimensional emphasis is not needed. For square matrices, m = n, the zero matrix sometimes is written as 0,,, while the identity matrix is written similarly as /„ or /. We reserve the notation ek for the ^'''-column or £'''-row, depending on context, of the identity matrix. The notions of addition and multiplication for conformable matrices are presumed to be familiar. Of course the multiplication operation is more interesting, in part because it is not commutative in general. That is, AB and BA are not always the same. If A is square, then for nonnegative integer k the power Ak is well defined, with A° = I. If there is a positive k such that Ak = 0, then A is called nilpotent. Similar to the vector case, the transpose of a matrix A with entries atj is the matrix AT with /J-entry given by a,-,-. A useful fact is (AB)T = BTAT. For a square « x n matrix A, the trace is the sum of the diagonal entries, written

trA = i>,.

(4)

i=l

If B also is n x n, then //• [AB] = tr [BA]. A familiar scalar-valued function of a square matrix A is the determinant. The determinant of A can be evaluated via the Laplace expansion described as follows. Let Cjj denote the cofactor corresponding to the entry a-,j. Recall that c// is (- 1)'+J times the determinant of the ( H - l ) x ( « - I ) matrix that results when the /'''-row and j'1'column of A are deleted. Then for any fixed /, 1