Linear System Theory: The State Space Approach

This is a classic text written by two pioneers in the field. It was first published in 1963; the current edition from Dover is an unabridged reprint of the original. The authors’ motivation was to produce a book that captured a couple of decade’s worth of scientific and technological development in the field. They also were acknowledging the influence (just emerging at the time) of large-scale digital computing. They also understood that engineers would need tools to develop algorithms and programs to manage complex control systems.

While the book represents a perspective from an earlier era, it is by no means outdated. Many of the ideas they describe remain part of a critical core that many engineering students continue to see today. Yet it is a pioneering work and students might find it overwhelming with its level of detail and elaboration. The book was written with graduate students in mind, and is appropriate at that level, but much of the material is now taught in a simpler and more direct form to undergraduate engineers.

One especially significant feature that distinguishes this book from its predecessors is its emphasis on a state-space approach. Before they take up the conventional material in differential equations, Laplace and Fourier transforms and the like, the authors introduce the idea of state at the beginning and then construct a more general system-theoretic framework in which they can treat linear and nonlinear differential or difference equations.

Basic language, concepts, and notations are introduced in the first two chapters. The authors do this more elaborately and in greater detail than is common in more current texts. They are particularly careful to establish a basis for rigorous discussion of the concepts of state, system and system equivalence. Here they also begin to describe a standard way of describing systems via diagrams. The third chapter offers a similarly detailed discussion of linearity and time invariance.

After these preliminaries we get to the core material. It begins with a general treatment of state equations for (not necessarily linear) time-invariant differential systems. This is followed by separate treatments of linear time-invariant and linear time-varying differential systems. The next two chapters investigate stability questions for differential systems and then consider stability in the context of systems defined by their impulse response.

The transfer function, one of the most important engineering tools for the analysis and design of time-invariant systems, is treated next. After a discussion of realization of a system whose matrix transfer function is given, the authors consider constraints imposed on transfer functions by physical characteristics of linear systems; these include stability, causality, and dispersion of the impulse response. Part of the objective here is to present methods for handling the stability problem for single- and multi-loop feedback systems using only data based on sinusoidal steady-state measurements.

The final two chapters briefly discuss two additional topics: discrete time systems and then the dual concepts of controllability and observability.

Prerequisites include at least undergraduate courses in ordinary differential equations and linear algebra. The authors use the language of linear differential operators and bits of functional analysis (e.g. the Riesz representation theorem) also pop up occasionally.

This book would not be an ideal choice for a student’s first exposure to linear system theory. The treatment at times is more elaborate than warranted in an introduction and there are no exercises. More current books (such as Hespanha’s Linear Systems Theory) convey the basic material in a more streamlined form with more examples and plenty of exercises.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.