*Linear Systems Theory* by João Hespanha is primarily concerned with systems of the form

*x*' = *Ax* + *Bu*; *y* = *Cx* + *Du*.

Here *x* = *x*(*t)* is a vector-valued function of time. *A*, *B*, *C*, and *D* are matrices, or possibly matrix-valued functions of time. In the continuous-time case, *x*' is the derivative of *x* with respect to time. An initial condition *x*(0) = *x*_{0} must be specified to uniquely determine the solution.

It would be easy to underestimate the scope and usefulness linear system theory. Some possible objections would be as follows.

- How useful is a theory that only considers linear systems?
- Why limit your attention to first order derivatives?
- Aren't linear differential equations very well understood?

First, many systems of practical interest are non-linear, even highly non-linear, and strictly speaking linear systems theory does not apply to such systems. However, the local behavior of non-linear systems is often described well by linear approximations. Also, devices designed to control linear systems often work surprisingly well when used to control non-linear systems.

Second, systems of first order differential equations are more general than they seem. A system of nth order differential equations can easily be reduced to a (higher dimensional) system of first order differential equations by introducing additional variables.

Finally, while linear differential equations are indeed well understood, there are open problems in the *control* of linear systems. A course on differential equations would consider the function *u*(*t*) as given. Also, such a course would consider the equation *y* = *Cx* + *Du* irrelevant since it does not impact the solution *x*. However, control theory is concerned with choosing *u* to regulate *y*. For example, one may be interested in determining the input *u* that drives *y* to zero most quickly, subject to energy constraints on *u*.

In addition to systems of differential equations, *Linear Systems Theory* also considers difference equations of the form

*x*^{+} = *Ax* + *Bu*; *y* = *Cx* + *Du*.

where *x*^{+}(*t*) = *x*(*t*+1). The theory of such discrete-time systems is surprisingly similar to that of continuous-time systems involving differential equations.

*Linear Systems Theory* developed from a set of lecture notes, and in some ways still reads like a set of lecture notes. The book is self-contained and polished, but the pace is brisk. Also, each chapter corresponds to the material that might be covered in one lecture.

The book makes extensive use of linear algebra. It assumes familiarity with basic linear algebra but reviews more advanced aspects of the subject that are not always covered in an introductory course: positive definite matricies, Jordan canonical form, pseudo-inverses, etc. Notes on using *Matlab*® are sprinkled throughout the text. Readers who use *Matlab*® will appreciate these notes but others can easy overlook these notes without loss of continuity.

*Linear Systems Theory* covers a great deal of material in around 250 pages. An instructor using the book as a text would need to help students unpack some of the condensed presentation. Also, an instructor would need to provide applications; the book does not contain many applications until near the end.

John D. Cook is a research statistician at M. D. Anderson Cancer Center and blogs at The Endeavour.