IEEE Transactions on Automatic Control, June 1999.

Linear Systems - Panos J. Antsaklis and Anthony N. Michel (New York, NY: McGraw-Hill, 1997). Reviewed by Theodore E. Djaferis, Electrical and Computer Engineering, University of Massachusetts, Amherst MA 01003-4410, Phone: (413)-545-3561, Fax:(413)-545-1993, Email: djaferis@ecs.umass.edu

1.Introduction

Scientists and engineers have found over the years that a "systems approach" is crucial for the solution of a wide range of problems. In particular, they have discovered that many natural and man-made phenomena can be effectively studied by considering them as systems described by differential or difference equations. Furthermore, this approach has also been demonstrated to be essential for the solution of a multitude of automatic control problems. Consequently, over the last three decades a great deal of effort has been directed towards laying down the foundations of system theory. Its cornerstone is linear system theory. There is no doubt that any serious educational or research endeavor in systems and automatic control must be based on the fundamental understanding of linear systems.

In view of the importance of linear system theory it is not surprising to see that dozens of textbooks have been published on this topic. Some of the most well known are [1,2,3,4,5]. One of the most recent books is entitled: Linear Systems, by Panos J. Antsaklis and Anthony N. Michel. It was published by McGraw-Hill in 1997 and is the topic of this book review. There are three words that characterize this work: thoroughness, completeness and clarity. The authors are congratulated for taking the time to write an excellent linear systems textbook!

2. Book Contents

The book contains seven chapters with material presented that covers continuous-time systems as well as discrete-time systems. Even though the book is devoted to the theory of linear systems, on a number of occasions, topics are presented that make connections to nonlinear systems. Chapter 1 presents background material on nonlinear differential equations and initial value problems, existence and uniqueness of solutions, properties of solutions and linearization. Mathematical descriptions of systems are given in both state-space form as well as input-output. Chapter 2 first introduces important concepts from linear algebra and then focuses on the theory of linear differential equations and difference equations. Chapter 3 deals with the fundamental concepts of controllability and observability for both continuous-time and discrete-time systems. Chapter 4 discusses state feedback, pole placement and the construction of observers. Chapter 5 presents a number of realization algorithms. Chapter 6 deals with both Lyapunov as well as input-output stability of linear systems. Chapter 7 focuses on polynomial matrix descriptions and matrix fractional descriptions of time-invariant systems.

3. Coverage of Material

Throughout the book the authors follow the practice of first presenting required background material, which is then used to develop the results. Relevant material from real analysis is included in Chapter 1 and linear algebra topics in Chapter 2. This improves readability and makes the presentation ``self-contained" to a large extent. In addition, at the end of each chapter a comprehensive list of references is provided so that the interested reader can explore in more depth. Furthermore, the book integrates very successfully material from state-space descriptions and differential operator representations. The range of topics covered can be very appropriately labeled as ``standard" for a first course on linear systems.

The coverage of topics is very thorough and complete. Let me illustrate this point by considering the presentation of the concept of controllability which is introduced in Chapter 3. Reachability (controllability-from-the-origin) and controllability (to-the-origin) are defined for time-varying continuous-time systems. In this context the notions are shown to be equivalent. Specific inputs that accomplish the transfer are constructed using the reachability (controllability) Grammian. Many criteria are developed and proved. The time invariant case is developed separately where connections to system zeros are made. This is followed by material on canonical forms of time-invariant systems. A treatment of reachability/controllability is also carried out for discrete-time systems showing that the notions are not always equivalent. These results are used in Chapter 4 for pole placement and observer design, and in the construction of realizations and a discussion about minimality in Chapter 5. In Chapter 7 connections are made between controllability and polynomial matrix descriptions.

The book is written for a first-year graduate student. Very few undergraduates would have the necessary background and mathematical sophistication to appreciate this book. Not only can the book be used as a textbook but engineers and scientists will frequently consult it as a reference.

A great variety of exercises are found at the end of each chapter for which a separate ``solutions manual" is available. The availability of a solutions manual and the great flexibility in assigning exercises is of tremendous help to the prospective instructor.

4. Personal Experience with the Book

At our graduate program we have a first-semester, first-year course on linear systems. My choice of textbook for this course on previous occasions was Chen ([2]). However, last time I used Antsaklis/Michel and was quite happy with the change. There is no question that there is more material contained in the textbook than one can expect to cover in a semester at a typical graduate school. Inevitably, the instructor must make a choice of what material to emphasize. The authors make some suggestions in the introduction on how this can be done. My approach was to cover only the continuous-time case. We first focused on the theory of linear differential equations in state-space form (existence and uniqueness of solutions to initial value problems) the transition matrix and the variation-of-constants formula. We then briefly recalled linear algebra concepts, discussed state-space and input output descriptions. Next, we studied reachability/controllability, observability, Grammians with special emphasis on the time-invariant case and canonical forms. We then used state feedback for pole placement (multivariable case) and constructed full-state observers (separation property). Next, we considered some realization algorithms and then discussed Lyapunov stability (in the context of nonlinear systems) and input-output stability. Finally, we briefly introduced polynomial matrix descriptions and interconnected systems.

5. Conclusions

The authors have used their mastery of the subject to produce a textbook that very effectively presents the theory of linear systems as it has evolved over the last thirty years. The result is a comprehensive, complete and clear exposition that serves as an excellent foundation for more advanced topics in system theory and control.

References

[1] Brockett, R. W., Finite Dimensional Linear Systems, Wiley, New York, 1970.

[2] Chen, C.-T., Linear System Theory and Design, Holt, Rinehart, and Winston, New York, 1984.

[3] Kailath, T., Linear Systems, Prentice Hall, Englewood Cliffs, New Jersey, 1980.

[4] Wolovich, W. A., Linear Multivariable Systems, Springer-Verlag, New York, 1974.

[5] Wonham, W. M., Linear Multivariable Control: A Geometric Approach, 3/E, Springer-Verlag, New York, 1985.