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LINEAR SYSTEM
THEORY AND DESIGN
Third Edition
Chi-Tsong Chen
State University of New York at Stony Brook
New York Oxford
OXFORD UNIVERSITY PRESS
1999
OXFORD UNIVERSITY PRESS
Oxford New York
Athens Auckland Bangkok Bogot
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and associated companies in
Berlin Ibadan
Copyright © 1999 by Oxford University Press, Inc.
Published by Oxford University Press, Inc.
198 Madison Avenue, New York, New York 10016
Oxford is a registered trademark of Oxford University Press
All rights reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted,
in any form or by any means, electronic, mechanical,
photocopying, recording, or otherwise, without the prior
permission of Oxford University Press.
Library of Congress Cataloging-in-Publication Data
Chen, Chi-Tsong
Linear system theory and design / by Chi-Tsong Chen. — 3rd ed.
p. cm. — (The Oxford series in electrical and computer engineering)
Includes bibliographical references and index.
ISBN 0-19-511777-8 (cloth).
1. Linear systems. 2. System design. I. Title. II. Series.
QA402.C44 1998
629.8'32—dc21 97-35535
CIP
Printing (last digit): 98765432 1
Printed in the United States of America
on acid-free paper
Contents
Preface xi
Chapter 1: Introduction 1
1.1 Introduction 1
1.2 Overview 2
Chapter 2: Mathematical Descriptions of Systems 5
2.1 Introduction 5
2.1.1 Causaliity and Lumpedness 6
2.2 Linear Systems 7
2.3 Linear Time-Invariant (LTI) Systems 11
2.3.1 Op-Amp Circuit Implementation 16
2.4 Linearization 17
2.5 Examples 18
2.5.1 RLC Networks 26
2.6 Discrete-Time Systems 31
2.7 Concluding Remarks 37
Problems 38
Chapter 3: Linear Algebra 44
3.1 Introduction 44
3.2 Basis, Representation, and Orthonormalization 45
3.3 Linear Algebraic Equations 48
3.4 Similarity Transformation 53
3.5 Diagonal Form and Jordan Form 55
3.6 Functions of a Square Matrix 61
3.7 Lyapunov Equation 70
3.8 Some Useful Formulas 71
3.9 Quadratic Form and Positive Definiteness 73
3.10 Singular-Value Decomposition 76
3.11 Norms of Matrices 78
Problems 78
vii
viii CONTENTS
Chapter 4: State-Space Solutions and Realizations 86
4.1 Introduction 86
4.2 Solution of LTI State Equations 87
4.2.1 Discretization 90
4.2.2 Solution of Discrete-Time Equations 92
4.3 Equivalent State Equations 93
4.3.1 Canonical Forms 97
4.3.2 Magnitude Scaling in Op-Amp Circuits 98
4.4 Realizations 100
4.5 Solution of Linear T ime-Varying (LTV) Equations 106
4.5.1 Discrete-Time Case 110
4.6 Equivalent Time-Varying Equations 111
4.7 Time-Varying Realizations 11 5
Problems 11 7
Chapter 5: Stability 121
5.1 Introduction 121
5.2 Input–Output Stability of LTI Systems 121
5.2.1 Discrete-Time Case 126
5.3 Internal Stability 129
5.3.1 Discrete-Time Case 131
5.4 Lyapunov Theorem 132
5.4.1 Discrete-Time Case 135
5.5 Stability of LTV Systems 137
Problems 140
Chapter 6: Controllability and Observability 143
6.1 Introduction 143
6.2 Controllability 144
6.2.1 Controllablity Indices 150
6.3 Observability 153
6.3.1 Observability Indices 157
6.4 Canonical Decomposition 158
6.5 Conditions in Jordan-Form Equations 164
6.6 Discrete-Time State Equations 169
6.6.1 Controllability to the Origin and Reachability 171
6.7 Controllability After Sampling 172
6.8 LTV State Equations 176
Problems 180
Chapter 7: Minimal Realizations and Coprime Fractions 184
7.1 Introduction 184
7.2 Implications of Coprimeness 185
7.2.1 Minimal Realizations 189
Contents ix
7.3 Computing Coprime Fractions 192
7.3.1 QR Decomposition 195
7.4 Balanced Realization 197
7.5 Realizations from Markov Parameters 200
7.6 Degree of Transfer Matrices 205
7.7 Minimal Realizations—Matrix Case 207
7.8 Matrix Polynomial Fractions 209
7.8.1 Column and Row Reducedness 212
7.8.2 Computing Matrix Coprime Fractions 214
7.9 Realizations from Matrix Coprime Fractions 220
7.10 Realizations from Matrix Markov Parameters 225
7.11 Concluding Remarks 227
Problems 228
Chapter 8: State Feedback and State Estimators 231
8.1 Introduction 231
8.2 State Feedback 232
8.2.1 Solving the Lyapunov Equation 239
8.3 Regulation and Tracking 242
8.3.1 Robust Tracking and Disturbance Rejection 243
8.3.2 Stabilization 247
8.4 State Estimator 247
8.4.1 Reduced-Dimensional State Estimator 251
8.5 Feedback from Estimated States 253
8.6 State Feedback—Multivariable Case 255
8.6.1 Cyclic Design 256
8.6.2 Lyapunov-Equation Method 259
8.6.3 Canonical-Form Method 260
8.6.4 Effect on Transfer Matrices 262
8.7 State estimators—Multivariable Case 263
8.8 Feedback from Estimated States—Multivariable Case 265
Problems 266
Chapter 9: Pole Placement and Model Matching 269
9.1 Introduction 269
9.1.1 Compensator Equations–Classical Method 271
9.2 Unity-Feedback Configuration–Pole Placement 273
9.2.1 Regulation and Tracking 275
9.2.2 Robust Tracking and Disturbance Rejection 277
9.2.3 Embedding Internal Models 280
9.3 Implementable Transfer Functions 283
9.3.1 Model Matching–Two-Parameter Configuration 286
9.3.2 Implementation of Two-Parameter Compensators 291
9.4 Multivariable Unity-Feedback Systems 292
9.4.1 Regulation and Tracking 302
9.4.2 Robust Tracking and Disturbance Rejection 303
9.5 Multivariable Model Matching–Two-Parameter
Configuration 306
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