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.

Although the text is designed to serve primarily the needs of graduate students, it 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 stability (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 1.10A to 1.10C is included for review and to establish some needed notation. This material should not require formal class time. In Subsection 1.10D [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, respectilively). Section 2.5 (concerned with linear periodic systems) may be omitted without 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. Konstantopoulos, 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 staffof McGraw-Hill, especially with Lynn Cox, the Electrical Engineering Editor. Finally, we are both indebted 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).

Panos J. Antsaklis

Anthony N. Michel