Table Of ContentPanos J. Antsaklis
Anthony N. Michel
Linear Systems
Birkhauser
Boston • Basel • Berlin
Panos J. Antsaklis Anthony N. Michel
Dept. of Electrical Engineering Dept. of Electrical Engineering
University of Notre Dame University of Notre Dame
275 Fitzpatrick Hall 275 Fitzpatrick Hall
Notre Dame, IN 46556 Notre Dame, IN 46556
USA USA
Cover design by Mary Burgess.
AMS Subject Classification: 34A30, 34H05, 47N70, 65F05, 93-XX, 93-01, 93Axx, 93A30, 93Bxx, 93B03,
93B05, 93B07, 93B10, 93B11, 93B12, 93B15, 93B17, 93B18, 93B20, 93B25, 93B50, 93B52, 93B55, 93B60,
93Cxx, 93C05, 93C10, 93C15, 93C35, 93C55, 93C57, 93C62, 93Dxx, 93D05, 93D10, 93D15, 93D20, 93D25,
93D30
Library of Congress Cataloging-in-Publication Data
Antsaklis, Panos J.
Linear systems / Panos J. Antsaklis, Anthony N. Michel.
p. cm.
Originally published : New York : McGraw-Hill, ©1997.
Includes bibliographical references and index.
ISBN 0-8176-4334-2 (alk. paper) - ISBN 0-8176-4435-0 (e-ISBN) 1. Linear control
systems. 2. Control theory. 3. Signal processing. I. Michel, Anthony N. II. Title.
TJ220.A58 2005
629.8^32-dc22 2005053096
ISBN-10 0-8176-4434-2 elSBN 0-8176-4435-0 Printed on acid-free paper.
ISBN-13 978-0-8176-4434-5
©2006 Birkhauser Boston, 2nd Corrected Printing BirkhdUSer
Originally published by McGraw-Hill, Englewood Cliffs, NJ,1997.
All rights reserved. This work may not be translated or copied in whole or in part without the written permission
of the publisher (Birkhauser Boston, c/o Springer Sciences-Business Media Inc., 233 Spring Street, New York,
NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection
with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or
dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not
identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary
rights.
Printed in the United States of America. (KeS/IBT)
987654321
www. birkhauser.com
To Our Families
To To
Melinda and our daughter Lily Leone and our children
and to my parents Mary, Kathy, John,
Dr loannis and Marina Antsaklis Tony, and Pat
—Panos J. Antsaklis —Anthony N. Michel
Mechanics is the paradise of the mathematical
sciences because by means of it one comes to
the fruits of mathematics.
LEONARDO DA VINCI
1452-1519
CONTENTS
Preface
XV
Mathematical Descriptions of Systems 1
1.1 Introduction 2
A. Physical Processes, Models, and Mathematical
Descriptions, 2/B. Classification of Systems, 3 /
C. Finite-Dimensional Systems, 4/D. Chapter Description, 6/
E. Guidelines for the Reader, 7
1.2 Preliminaries 8
A. Notation, 8/B. Continuous Functions, 9
1.3 Initial-Value Problems 10
A. Systems of First-Order Ordinary Differential
Equations, 10/B. Classification of Systems of First-Order
Ordinary Differential Equations, 11 / C. nth-Order Ordinary
Differential Equations, 12
1.4 Examples of Initial-Value Problems 13
*1.5 More Mathematical Preliminaries 17
A. Sequences, 17 /B. Sequences of Functions, 18/C The
Weierstrass M-Test, 21
*1.6 Existence of Solutions of Initial-Value Problems 21
A. The Ascoli-Arzela Lemma, 22 /B. e-Approximate
Solutions, 23 / C. The Cauchy-Peano Existence Theorem, 25
*1.7 Continuation of Solutions 26
A. Zom's Lemma, 26/B. Continuable Solutions, 27/
C Continuation of Solutions to the Boundary ofD, 28
*1.8 Uniqueness of Solutions 29
A. The Gronwall Inequality, 29 /B. Unique Solutions, 30
*1.9 Continuous Dependence of Solutions on Initial
Conditions and Parameters 33
1.10 Systems of First-Order Ordinary Differential Equations 37
A. More Mathematical Preliminaries: Vector Spaces, 37/
B. Further Mathematical Preliminaries: Normed Linear
Spaces, 41 / C. Additional Mathematical Preliminaries:
Convergence, 44 /D. Solutions of Systems of First-Order
Ordinary Differential Equations: Existence, Continuation,
Uniqueness, and Continuous Dependence on Initial
Conditions, 45
^ 1.11 Systems of Linear First-Order Ordinary Differential
Contents Equations 47
A. Linearization, 48 /B. Examples, 52
1.12 Linear Systems: Existence, Uniqueness,
Continuation, and Continuity witli Respect to Parameters
of Solutions 54
1.13 Solutions of Linear State Equations 55
1.14 State-Space Description of Continuous-Time
Systems 58
1.15 State-Space Description of Discrete-Time Systems 60
1.16 Input-Output Description of Systems 65
A. External Description of Systems: General
Considerations, 65 /B. Linear Discrete-Time Systems, 68/
C. The Dirac Delta Distribution, 72 /D. Linear
Continuous-Time Systems, 76
1.17 Summary 79
1.18 Notes 80
1.19 References 80
1.20 Exercises 81
2 Response of Linear Systems 94
2.1 Introduction 94
A. Chapter Description, 94/B. Guidelines for the Reader, 96
1,1 Background Material 96
A. Linear Sub spaces, 97 /B. Linear Independence, 97 /
C. Bases, 99/D. Linear Transformations, 100/
E. Representation of Linear Transformations by
Matrices, 104/ "^F. Some Properties of Matrices, 107/
*G. Determinants of Matrices, 111 /H. Solving Linear
Algebraic Equations, 115/1. Equivalence and
Similarity, 116/J. Eigenvalues and Eigenvectors, 121 /
K. Direct Sums of Linear Subspaces, 126/L. Some
Canonical Forms of Matrices, 127/M. Minimal
Polynomials, 132 /N. Nilpotent Operators, 134/0. The
Jordan Canonical Form, 135
2.3 Linear Homogeneous and Nonhomogeneous Equations 138
A. The Fundamental Matrix, 139 /B. The State Transition
Matrix, 143 /C. Nonhomogeneous Equations, 145/D. How
to Determine ^(t^to), 146
lA Linear Systems with Constant Coefficients 148
A. Some Properties ofe^^, 148/B. How to Determine
e^^, 150/C Modes and Asymptotic Behavior of
Time-Invariant Systems, 156
*2.5 Linear Periodic Systems 161
2.6 State Equation and Input-Output Description of
Continuous-Time Systems 165 Contents
A. Response of Linear Continuous-Time Systems, 165 /
B. Transfer Functions, 168/C. Equivalence of Internal
Representations, 170
2.7 State Equation and Input-Output Description of
Discrete-Time Systems 174
A. Response of Linear Discrete-Time Systems, 174/B. The
Transfer Function and the z-Transform, 177/C. Equivalence
of Internal Representations, 180/D. Sampled-Data
Systems, 182 /E. Modes and Asymptotic Behavior of
Time-Invariant Systems, 186
2.8 An Important Comment on Notation 190
2.9 Summary 191
2.10 Notes 191
2.11 References 192
2.12 Exercises 193
Controllability, Observability, and Special Forms 214
3.1 Introduction 215
A. Brief Introduction to Reachability and Observability, 215 /
B. Chapter Description, 223 / C. Guidelines for the
Reader, 225
PART 1 Controllability and Observability 226
3.2 Reachability and Controllability 226
A. Continuous-Time Time-Varying Systems, 227/
B. Continuous-Time Time-Invariant Systems, 235 /
C Discrete-Time Systems, 241
3.3 Observability and Constructibility 247
A. Continuous-Time Time-Varying Systems, 248 /
B. Continuous-Time Time-Invariant Systems, 252 /
C Discrete-Time Systems, 257
PART 2 Special Forms for Time-Invariant Systems 263
3.4 Special Forms 263
A. Standard Forms for Uncontrollable and Unobservable
Systems, 263/B. Eigenvalue/Eigenvector Tests for
Controllability and Observability, 272 / C Relating
State-Space and Input-Output Descriptions, 275 /
D. Controller and Observer Forms, 278
*3.5 Poles and Zeros 298
3.6 Summary 308
xii 3.7 Notes 310
Contents 3.8 References 311
3.9 Exercises 311
4 State Feedback and State Observers 321
4.1 Introduction 322
A. A Brief Introduction to State-Feedback Controllers and
State Observers, 322 /B. Chapter Description, 325 /
C. Guidelines for the Reader, 326
4.2 Linear State Feedback 326
A. Continuous-Time Systems, 326/B. Eigenvalue
Assignment, 328 / C. The Linear Quadratic Regulator (LQR):
Continuous-Time Case, 342 /D. Input-Output
Relations, 345 /E. Discrete-Time Systems, 348/E The
Linear Quadratic Regulator (LQR): Discrete-Time Case, 348
4.3 Linear State Observers 350
A. Eull-Order Observers: Continuous-Time Systems, 350/
B. Reduced-Order Observers: Continuous-Time
Systems, 355 / C. Optimal State Estimation:
Continuous-Time Systems, 357 /D. Eull-Order Observers:
Discrete-Time Systems, 358/E. Reduced-Order Observers:
Discrete-Time Systems, 362 / E Optimal State Estimation:
Discrete-Time Systems, 362
4.4 Observer-Based Dynamic Controllers 363
A. State-Space Analysis, 364/B. Transfer Function
Analysis, 367
4.5 Summary 370
4.6 Notes 370
4.7 References 371
4.8 Exercises 372
5 Realization Theory and Algorithms 383
5.1 Introduction 384
A. Chapter Description, 384/B. Guidelines for the
Reader, 384
5.2 State-Space Realizations of External Descriptions 385
A. Continuous-Time Systems, 385 /B. Discrete-Time
Systems, 388
5.3 Existence and Minimality of Realizations 389
A. Existence of Realizations, 390 /B. Minimality of
Realizations, 394 / C. The Order of Minimal
Realizations, 397/D. Minimality of Realizations:
Discrete-Time Systems, 401
5.4 Realization Algoritlims 402
A. Realizations Using Duality, 402 /B. Realizations in Contents
Controller/Observer Form, 404 / C. Realizations with Matrix
A Diagonal, 417/D. Realizations with Matrix A in Block
Companion Form, 418/E. Realizations Using Singular Value
Decomposition, 423
5.5 Summary 424
5.6 Notes 424
5.7 References 425
5.8 Exercises 425
Stability 432
6.1 Introduction 432
A. Chapter Description, 433 /B. Guidelines for the
Reader, 434
6.2 Matliematical Background Material 434
A. Bilinear Functionals and Congruence, 435 /B. Euclidean
Vector Spaces, 437/C Linear Transformations on Euclidean
Vector Spaces, 441
PART 1 Lyapunov Stability 445
6.3 The Concept of an Equilibrium 445
6.4 Qualitative Characterizations of an Equilibrium 447
6.5 Lyapunov Stability of Linear Systems 452
6.6 Some Geometric and Algebraic Stability Criteria 461
A. Some Graphical Criteria, 462 /B. Some Algebraic
Criteria, 465
6.7 The Matrix Lyapunov Equation 468
6.8 Linearization 477
PART 2 Input-Output Stability of Continuous-Time
Systems 481
6.9 Input-Output Stability 481
PART 3 Stability of Discrete-Time Systems 489
6.10 Discrete-Time Systems 489
A. Preliminaries, 489/B. Lyapunov Stability of an
Equilibrium, 492 / C. Linear Systems, 495 /D. The
Schur-Cohn Criterion, 498/E. The Matrix Lyapunov
Equation, 499/F Linearization, 503 /G. Input-Output
Stability, 505
6.11 Summary 508
6.12 Notes 509
xiv 6.13 References 510
Contents 6.14 Exercises 511
7 Polynomial Matrix Descriptions and Matrix
Fractional Descriptions of Systems 517
7.1 Introduction 518
A. A Brief Introduction to Polynomial and Fractional
Descriptions, 518/B. Chapter Description, 522/
C. Guidelines for the Reader, 523
PART 1 Analysis of Systems 524
7.2 Background Material on Polynomial Matrices 524
A. Rank and Linear Independence, 524 /B. Unimodular and
Column (Row) Reduced Matrices, 526/C. Hermite and
Smith Forms, 531 /D. Coprimeness and Common
Divisors, 535/E. The Diophantine Equation, 540
7.3 Systems Represented by Polynomial Matrix Descriptions 553
A. Equivalence of Representations, 554/B. Controllability,
Observability, Stability, and Realizations, 560/
C. Interconnected Systems, 568
PART 2 Synthesis of Control Systems 589
7.4 Feedback Control Systems 589
A. Stabilizing Feedback Controllers, 589 /B. State Feedback
Control and State Estimation, 605 / C. Stabilizing Feedback
Controllers Using Proper and Stable MFDs, 611 /D. Two
Degrees of Freedom Feedback Controllers, 622
7.5 Summary 634
7.6 Notes 635
7.7 References 636
7.8 Exercises 638
Appendix Numerical Considerations 645
A.l Introduction 645
A.2 Solving Linear Algebraic Equations 646
A.3 Singular Values and Singular Value Decomposition 648
A.4 Solving Polynomial and Rational Matrix Equations
Using Interpolation Methods 653
A.5 References 659
Index 661
