Mathematical Systems Theory I
(subtitled "Modelling, State Space Analysis, Stability and Robustness") provides a detailed and rigorous mathematical development of finite-dimensional, time-invariant linear systems. One way to think of this book is to see it as the rigorous mathematical foundation underlying the "linear systems" course common to many engineering curricula. A second volume is evidently in the works; it will concentrate on control theory.
The intended audience for this text is advanced undergraduates and first or second year graduate students. The authors suggest that several courses based on their book are possible depending on the selection of chapters. Indeed, the content ranges from rigorous proofs of basic results in linear systems theory to new results on the research frontier. In that sense, the book also serves as a valuable research reference.
The first two chapters — discussing mathematical modeling and basic state space theory — are introductory, and nicely self-contained. The chapter on modeling includes a broad collection of examples ranging from population dynamics and economics to switching networks in digital systems. The remaining three chapters make up more than 70% of the text; this is significantly more demanding material and is intended to prepare the reader for research in systems theory. Chapter 3 focuses on stability theory, emphasizing Lyapunov stability. System perturbations are discussed in Chapter 4; both polynomial and matrix perturbation techniques are presented. The last chapter, on uncertain systems, includes new tools developed by the authors to deal with model uncertainty (imperfectly known parameters or dynamics and incomplete models).
It is a little disappointing not to see any mention of the interaction of systems theory with data — as, for example, in the standard Kalman filter. Students approaching systems theory from the mathematical side are often surprised by the difficulties that real data can introduce. It would have useful to see some discussion of this both in the introductory chapters and in the chapter on uncertain systems.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.