Introduction to Mathematical Systems Theory

The authors have taught a course on systems theory, control and identification for several years in the Netherlands. This book, now in its second edition, developed from that experience. Their students came from mathematics, computer science, economics and business mathematics. In the U.S. many engineering students are required to take a course like this, but mathematics students would usually see the material only in courses specifically dealing with control theory.

 

The principal goals of system theory include prediction, control and identification of systems. Usually, this requires the development of a system model.  The book begins with dynamical systems and a discussion of the types of models that can be used to model dynamical processes. It emphasizes the importance of the careful selection of a model and its parameters. Since values of the parameters are usually not known exactly, knowing how to estimate them from available data is critical for forecasting and control. The authors restrict themselves to discrete systems throughout the text to avoid requiring more background in deterministic and stochastic differential equations.

 

The setting is somewhat different from the usual way mathematicians think about dynamical systems. The authors introduce them via input-output systems and then fairly quickly describe dual views of such systems in the time and frequency domains. A discrete-time input-output map \( T(u)=y \) has the representation in \( R^{n} \)  as \( x(t + 1) = Ax(t) + Bu(t) \), and \( y(t + 1) = Cx(t) + Du(t) \) with \( x(0) = 0 \) where \( x \) represents the state of the system, \( u \) is the input, \( y \) is the output and \( A \), \( B \), \( C \) and \( D \) are matrices of appropriate dimensions. In the frequency domain the transfer function describes an equivalent relationship between input and output

 

The purpose of a control algorithm is to identify inputs \( u(t) \) that lead to a target state. Predictive algorithms seek to determine the state \( x(t) \) at future times. System identification is different because it deals with systems that are not known a priori and must be estimated from data. These are the most common, and they often require a cycle of model development and reformulation.

 

The authors treat both deterministic and stochastic discrete systems, and their book includes most of the basic topics of control theory. They offer a fairly quick overview of optimal control where an optimization algorithm is used to choose how to minimize costs associated with the evolution of the system. The special case of a linear system with a quadratic cost function is treated in more detail. Filtering and prediction, including a discussion of the Kalman filter, are also discussed in some detail.  The book is somewhat unusual in its choice of examples and applications. The very first examples explore business and economic questions of supply and demand of goods produced in a market and with issues of macroeconomic policy instead of traditional control systems in an industrial environment. These signal a broader approach to the subject than typical engineering texts.

 

The overall treatment tends toward the theoretical and includes many theorems and proofs. There are a few examples throughout, but one might have wished for more. The authors touch quickly on many different topics in less than two hundred pages; their goal seems to be to provide an overview of systems theory without teaching the more practical aspects of implementation. This would be a challenging book for a novice and is probably better suited to readers with at least some prior background in systems theory.  

 

Prerequisites include calculus, linear algebra, probability and statistics, and some elementary results from Fourier series. Exercises are provided via the book website.

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.