Panos 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.

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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

C O N T E N T S

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

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 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

2.9 Summary

2.10 Notes

2.11 References

2.12 Exercises

165 Contents

174

190

191

191

192

193

Controllability, Observability, and Special Forms 3.1 Introduction

A. Brief Introduction to Reachability and Observability, 215 / B. Chapter Description, 223 / C. Guidelines for the Reader, 225

214

215

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

A. Realizations Using Duality, 402 /B. Realizations in 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

402 Contents

5.5

5.6

5.7

5.8

Summary

Notes

References

Exercises

Stability 6.1 Introduction

A. Chapter Description, 433 /B. Guidelines for the Reader, 434

6.2 Matliematical Background Material A. Bilinear Functionals and Congruence, 435 /B. Euclidean Vector Spaces, 437/C Linear Transformations on Euclidean Vector Spaces, 441

PART 1 Lyapunov Stability

6.3 The Concept of an Equilibrium

6.4 Qualitative Characterizations of an Equilibrium

6.5 Lyapunov Stability of Linear Systems

6.6 Some Geometric and Algebraic Stability Criteria A. Some Graphical Criteria, 462 /B. Some Algebraic Criteria, 465

6.7 The Matrix Lyapunov Equation

6.8 Linearization

PART 2 Input-Output Stability of Continuous-Time Systems

6.9 Input-Output Stability

PART 3 Stability of Discrete-Time Systems

6.10 Discrete-Time Systems

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

6.12 Notes

424

424

425

425

432

432

434

445

445

447

452

461

468

477

481

481

489

489

508

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

PREFACE

This text is intended primarily for first-year graduate students and advanced under­graduates in engineering who are interested in control systems, signal processing, and communication systems. It is also appropriate for students in applied mathemat­ics, economics, and certain areas in the physical and biological sciences. Designed for a challenging, one-semester systems course, the book presents an introduction to systems theory, with an emphasis on control theory. It can also be used as supplemen­tary material for advanced systems and control courses and as a general reference on the subject.

The prerequisites for using this book are topics covered in a typical undergrad­uate curriculum in engineering and the sciences: undergraduate-level differential equations, linear algebra, Laplace transforms, and the modeling of electric circuits and simple mechanical systems.

The study of linear systems is a foundation for several disciplines, including control and signal processing. It is therefore very important that the coverage of linear systems be comprehensive and give readers sufficient breadth and depth in analysis and synthesis techniques of such systems. We believe that the best preparation for this is a firm understanding of the fundamentals that govern the behavior of complex systems. Indeed, only a thorough understanding of system behavior enables one to take full advantage of the various options available in the design of the best kinds of control systems and signal processors. Therefore, the primary aim of this text is to provide an understanding of these fundamentals by emphasizing mathematical descriptions of systems and their properties.

In writing this book, our goal was to clearly present the fundamental concepts of systems theory in a self-contained text. In addition to covering the fundamental prin­ciples, we provide sufficient background in analysis and algebra, to enable readers to move on to advanced topics in the systems area. The book is designed to highlight the main results and distinguish them from supporting results and extensions. Fur­thermore, we present the material in a sufficiently broad context to give readers a clear picture of the dynamical behavior of linear systems and the limitations of such systems.

The theory of linear systems is a mature topic, and there are literally thou­sands of scholarly papers reporting research on this subject. This book empha­sizes fundamental results that are widely accepted as essential to the subject. For those readers interested in further detail, the end-of-chapter material includes additional results in the exercise sections as well as pertinent references and notes.

The book covers both continuous-time and discrete-time systems, which may be time-varying or time-invariant. The material is organized in such a manner that it is possible to concentrate only on the time-invariant case, if desired. The time-invariant case is treated in separate sections, and the results are presented so that they can be developed independently of the time-varying case. This type of organization provides considerable flexibility in covering the material.

xvi Although the text is designed to serve primarily the needs of graduate students, it Preface should also prove valuable to researchers, and practitioners, we tried to make it easy

to use, and the book should prove valuable for self-study. Many simple examples are included to clarify the material and to encourage readers to actively participate in the learning process. The exercises at the end of each chapter introduce additional supporting concepts and results and encourage readers to gain additional insight by using what was learned. The exercises also encourage readers to comment, interpret, and visualize results (e.g., responses), making use of computer programs to aid in calculations and the generation of graphical results, when appropriate.

Over the past several years, the material has been class-tested in a first-year graduate-level course on linear systems, and its development has been influenced greatly by student feedback. Although there are many ways of using this book in a course, we suggest in the following several useful guidelines. Because any course on linear systems will most likely serve students with different educational experiences from a variety of disciplines and institutions. Chapters 1 and 2 provide necessary background material and develop certain systems fundamentals. Armed with this foundation, we develop essential results on controllability and observability (Chapter 3), on state observers and state feedback (Chapter 4), and on realization of systems (Chapter 5). Chapters 6 and 7 address basic issues concerning stabihty (Chapter 6) and the representation of systems using polynomial matrices and matrix fractions (Chapter 7). The appendix presents supplementary material (concerning numerical aspects).

How to use this book

At the beginning of each chapter is a detailed description of the chapter's con­tents, along with guidelines for readers. This material should be consulted when de­signing a course based on this book. In the following we give a general overview of the book's contents, with suggested topics for an introductory, one-semester course in linear systems.

From Chapter 1, covering a first course in linear systems should include the fol­lowing: all the material on systems (Section 1.1); the material on initial-value prob­lems (Sections 1.3 and 1.4); the material on systems of linear first-order ordinary differential equations (Sections 1.11, 1.12, and 1.13); the material on state equation descriptions of continuous-time systems (Section 1.14) and discrete-time systems (Section 1.15); and the material on input-output descriptions of systems (Section 1.16). The mathematical background material in Sections 1.2, 1.5 and Subsections I.IOA to 1.IOC is included for review and to establish some needed notation. This material should not require formal class time. In Subsection 1.1 OD [dealing with existence, continuation, uniqueness, and continuous dependence (on initial condi­tions and parameters) of solutions of initial-value problems], the coverage should emphasize the results and their implications rather than the proofs of those results.

From Chapter 2, a first course in linear systems should include essentially all the material from the following sections: Section 2.3 (dealing with systems of lin­ear homogeneous and nonhomogeneous first-order ordinary differential equations); Section 2.4 (dealing with systems of linear first-order ordinary differential equa­tions with constant coefficients); and Sections 2.6 and 2.7 (which address the state equation description, the input-output description, and important properties, such

as asymptotic stability of continuous-time and discrete-time linear systems, respec- xvii tively). Section 2.5 (concerned with linear periodic systems) may be omitted with- Preface out any loss of continuity. As in Chapter 1, the mathematical background material in Section 2.2 (dealing with linear algebra and matrices) is included for review and to establish some important notation and should not require much formal class time.

For Chapter 3, it is best to consider the material in two parts. From Part 1 include Section 3.1 (where the concepts of controllability and observability are introduced); and Subsections 3.2B and 3.3B (where these concepts are developed in greater de­tail for continuous-time time-invariant systems). From Part 2 include Subsections 3.4A and 3.4D (where special forms of system descriptions are considered); and Subsection 3.4B (where an additional controllability and observability test is pre­sented). Similarly, the course should include the following material from Chapter 4: Section 4.1 (where state feedback and state observers are introduced); Subsections 4.2A and 4.2B (linear state feedback and eigenvalue assignment by state feedback are treated in detail); and Subsection 4.3A (where the emphasis is on identity ob­servers); and Section 4.4 (where observer-based controllers are developed). Also, the course should include material from Chapter 5: Section 5.2 (where realization theory is introduced); Section 5.3 (where the existence, minimality, and the order of minimal realizations are developed); and Subsections 5.4A, 5.4B, 5.4C, and 5.4E (where realization algorithms are presented).

The material outlined above constitutes the major portion of a first course in linear systems. The course is rounded out, if time permits with selected topics from Chapter 6 (stability) and Chapter 7 (polynomial matrix system descriptions and frac­tional representations of transfer function matrices of linear time-invariant systems). The choice of these topics, and where they are presented throughout the course, de­pends on the interests of the instructor and the students.

We have been using this textbook in a one-semester first-year graduate course in electrical engineering. Typically, we spend the first half of the course on Chapter 1, Chapter 2, and Part 1 of Chapter 3. The second half of the course is devoted to Part 2 of Chapter 3 and Chapters 4 and 5. Selected topics from stability theory and matrix fractional descriptions of systems from Chapters 6 and 7 are included, as needed. Detailed coverage of Chapters 1 and 2, with only selective coverage of Chapters 3, 4, and 5, would be appropriate in a course that emphasizes mathematical systems theory.

Chapter 6 can also be used as an introduction to a second-level graduate course on nonlinear systems and stability. Similarly, Chapter 7 stands alone and can be used in an advanced linear systems course or as an introduction to a multi-input/ multi-output linear control course. There is enough material in Chapters 3 through 7 for courses taught at several levels in a graduate program.

Acknowledgments

We are indebted to our students for their feedback and constructive suggestions during the evolution of this book. In particular, we would like to thank B. Hu, I. Kon-stantopoulos, and X. Koutsoukos and also Dr. K. Wang for their help and suggestions during the final editing of the manuscript. Special thanks go to Clarice Staunton for her patience in typing the many versions of our manuscript. We are also very ap­preciative of our excellent working relation with the staff of McGraw-Hill, especially

xviii with Lynn Cox, the Electrical Engineering Editor. Finally, we are both indebted Preface to many individuals who have shaped our views of systems theory, in particular.

Bill Wolovich, Brown University; Boyd Pearson, Rice University; David Mayne, Imperial College of the University of London (England); Sherman Wu, Marquette University; and Wolfgang Hahn, the Technical University of Graz (Austria).

The present printing of Linear Systems, by Birkhauser, is in response to many requests by colleagues from the US and around the world, who wanted to start using or continue using the book, which was out of print. Colleagues who had used the book in the classroom or as a reference, as well as our own students, identified several items that needed to be changed. These corrections, which mostly constituted minor typos, have been incorporated in the current printing of the book. This publication of Linear Systems was made possible primarily because of Tom Grasso, Birkhauser's Computational Sciences and Engineering Editor, whom we would like to thank for his professionalism, encouragement and support.

A new companion book entitled A Linear Systems Primer is forthcoming. The Primer-bdiscd on the more complete treatment of the present Linear Systems book-contains a treatment of linear systems that focuses primarily on the time-invariant case, using streamlined presentation of the material with less formal and more intuitive proofs without sacrificing rigor.

Panos J. Antsaklis and Anthony N. Michel Notre Dame, Indiana

August 2005