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Panos J. Antsaklis Anthony N. Michel A Linear Systems Primer Birkh¨auser Boston • Basel • Berlin

Panos J. Antsaklis Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556 U.S.A. Anthony N. Michel Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556 U.S.A. Cover design by Mary Burgess. Mathematics Subject Classiﬁcation (2000): 34A30, 34H05, 93-XX, 93-01, 93Axx, 93A30, 93Bxx, 93B03, 93B05, 93B07, 93B10, 93B11, 93B12, 93B15, 93B17, 93B18, 93B20, 93B25, 93B50, 93B55, 93B60, 93Cxx, 93C05, 93C15, 93C35, 93C55, 93C57, 93C62, 93Dxx, 93D05, 93D15, 93D20, 93D25, 93D30 Library of Congress Control Number: 2007905134 ISBN-13: 978-0-8176-4460-4 e-ISBN-13: 978-0-8176-4661-5 Printed on acid-free paper. c2007 Birkh ¨auser Boston All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Birkh¨auser Boston, c/o Springer Science+Business Media LLC, 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 de- veloped 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. www.birkhauser.com (MP)

To our Families To Melinda and our daughter Lily and to my parents Dr. Ioannis and Marina Antsaklis —Panos J. Antsaklis To Leone and our children Mary, Kathy, John, Tony, and Pat —Anthony N. Michel And to our Students

Preface Brief Description The purpose of this book is to provide an introduction to system theory with emphasis on control theory. It is intended to be the textbook of a typical one-semester course introduction to systems primarily for ﬁrst-year graduate students in engineering, but also in mathematics, physics, and the rest of the sciences. Prerequisites for such a course include undergraduate-level diﬀer- ential equations and linear algebra, Laplace transforms, and modeling ideas of, say, electric circuits and simple mechanical systems. These topics are typi- cally covered in the usual undergraduate curricula in engineering and sciences. The goal of this text is to provide a clear understanding of the fundamental concepts of systems and control theory, to highlight appropriately the princi- pal results, and to present material suﬃciently broad so that the reader will emerge with a clear picture of the dynamical behavior of linear systems and their advantages and limitations. Organization and Coverage This primer covers essential concepts and results in systems and control the- ory. Since a typical course that uses this book may serve students with diﬀerent educational experiences, from diﬀerent disciplines and from diﬀerent educa- tional systems, the ﬁrst chapters are intended to build up the understanding of the dynamical behavior of systems as well as provide the necessary mathe- matical background. Internal and external system descriptions are described in detail, including state variable, impulse response and transfer function, polynomial matrix, and fractional representations. Stability, controllability, observability, and realizations are explained with the emphasis always being on fundamental results. State feedback, state estimation, and eigenvalue as- signment are discussed in detail. All stabilizing feedback controllers are also parameterized using polynomial and fractional system representations. The emphasis in this primer is on time-invariant systems, both continuous and

viii Preface discrete time. Although time-varying systems are studied in the ﬁrst chapter, for a full coverage the reader is encouraged to consult the companion book titled Linear Systems 1 that oﬀers detailed descriptions and additional mate- rial, including all the proofs of the results presented in this book. In fact, this primer is based on the more complete treatment of Linear Systems, which can also serve as a reference for researchers in the ﬁeld. This primer focuses more on course use of the material, with emphasis on a presentation that is more transparent, without sacriﬁcing rigor, and emphasizes those results that are considered to be fundamental in systems and control and are accepted as important and essential topics of the subject. Contents In a typical course on Linear Systems, the depth of coverage will vary de- pending on the goals set for the course and the background of the students. We typically cover the material in the ﬁrst three chapters in about six to seven weeks or about half of the semester; we spend about four to ﬁve weeks covering Chapters 4–8 on stability, controllability, and realizations; and we spend the remaining time in the course on state feedback, state estimation, and feedback control presented in Chapters 9–10. This book contains over 175 examples and almost 160 exercises. A Solutions Manual is available to course instructors from the publisher. Answers to selected exercises are given at the end of this book. By the end of Chapter 3, the students should have gained a good under- standing of the role of inputs and initial conditions in the response of systems that are linear and time-invariant and are described by state-variable inter- nal descriptions for both continuous- and discrete-time systems; should have brushed up and acquired background in diﬀerential and diﬀerence equations, matrix algebra, Laplace and z transforms, vector spaces, and linear transfor- mations; should have gained understanding of linearization and the generality and limitations of the linear models used; should have become familiar with eigenvalues, system modes, and stability of an equilibrium; should have an understanding of external descriptions, impulse responses, and transfer func- tions; and should have learned how sampled data system descriptions are derived. Depending on the background of the students, in Chapter 1, one may want to deﬁne the initial value problem, discuss examples, brieﬂy discuss existence and uniqueness of solutions of diﬀerential equations, identify methods to solve linear diﬀerential equations, and derive the state transition matrix. Next, in Chapter 2, one may wish to discuss the system response, introduce the impulse response, and relate it to the state-space descriptions for both continuous- and discrete-time cases. In Chapter 3, one may consider to study in detail the response of the systems to inputs and initial conditions. Note that it is 1 P.J. Antsaklis and A.N. Michel, Linear Systems, Birkh¨auser, Boston, MA, 2006.

Preface ix possible to start the coverage of the material with Chapter 3 going back to Chapters 1 and 2 as the need arises. A convenient way to decide the particular topics from each chapter that need to be covered is by reviewing the Summary and Highlights sections at the end of each chapter. The Lyapunov stability of an equilibrium and the input/output stability of linear time-invariant systems, along with stability, controllability and ob- servability, are fundamental system properties and are covered in Chapters 4 and 5. Chapter 6 describes useful forms of the state space representations such as the Kalman canonical form and the controller form. They are used in the subsequent chapters to provide insight into the relations between input and output descriptions in Chapter 7. In that chapter the polynomial matrix rep- resentation, an alternative internal description, is also introduced. Based on the results of Chapters 5–7, Chapter 8 discusses realizations of transfer func- tions. Chapter 9 describes state feedback, pole assignment, optimal control, as well as state observers and optimal state estimation. Chapter 10 character- izes all stabilizing controllers and discusses feedback problems using matrix fractional descriptions of the transfer functions. Depending on the interest and the time constraints, several topics may be omitted completely without loss of continuity. These topics may include, for example, parts of Section 6.4 on controller and observer forms, Section 7.4 on poles and zeros, Section 7.5 on polynomial matrix descriptions, some of the realization algorithms in Section 8.4, sections in Chapter 9 on state feedback and state observers, and all of Chapter 10. The appendix collects selected results on linear algebra, ﬁelds, vector spaces, eigenvectors, the Jordan canonical form, and normed linear spaces, and it addresses numerical analysis issues that arise when computing solu- tions of equations. Simulating the behavior of dynamical systems, performing analysis us- ing computational models, and designing systems using digital computers, although not central themes of this book, are certainly encouraged and often required in the examples and in the Exercise sections in each chapter. One could use one of several software packages speciﬁcally designed to perform such tasks that come under the label of control systems and signal processing, and work in diﬀerent operating system environments; or one could also use more general computing languages such as C, which is certainly a more te- dious undertaking. Such software packages are readily available commercially and found in many university campuses. In this book we are not endorsing any particular one, but we are encouraging students to make their own informed choices. Acknowledgments We are indebted to our students for their feedback and constructive sugges- tions during the evolution of this book. We are also grateful to colleagues

x Preface who provided useful feedback regarding what works best in the classroom in their particular institutions. Special thanks go to Eric Kuehner for his expert preparation of the manuscript. This project would not have been possible without the enthusiastic support of Tom Grasso, Birkh¨auser’s Computational Sciences and Engineering Editor, who thought that such a companion primer to Linear Systems was an excellent idea. We would also like to acknowledge the help of Regina Gorenshteyn, Associate Editor at Birkh¨auser. It was a pleasure writing this book. Our hope is that students enjoy reading it and learn from it. It was written for them. Notre Dame, IN Spring 2007 Panos J. Antsaklis Anthony N. Michel

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