5. A Synthesis of Time and Frequency Domain Methods for the Control of Infinite-Dimensional Systems: A System Theoretic Approach

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Abstract
5.1. Introduction. In 1988 I was approached to write an article on frequency domain methods for distributed parameter systems, but as the chapter title reveals, this is not what I have written. Although the finite-dimensional systems and control community has in the past had its state-space advocates and its frequency domain advocates, from the recent research literature it is clear that the “twain” have finally met [86], [84], [64], [100]. This is certainly not the case in the area of infinite-dimensional systems and the wide gap between the books [35] and [77] has not really been bridged by the monograph [5]. I believe that this is now possible and this article has been written in an attempt to show how a system theoretic approach can lead to an integrated time and frequency domain treatment of control problems for infinite-dimensional linear systems. Indeed, experience with finite-dimensional control theory indicates that this is a necessary prerequisite to be able to cope successfully with robustness and other performance objectives in more realistic problems of control design. Although the class of linear infinite-dimensional systems is too large to be covered by one meta theory, it is very useful to identify subclasses which have common mathematical and system theoretic properties and to develop a theory for an appropriate subclass. The most successful class is the class of real, proper, rational transfer functions or the class of linear, time-invariant, finite-dimensional systems usually represented by the matrix quadruple (A, B, C, D). In this chapter, the focus is on two classes of linear infinite-dimensional systems, both with finite rank inputs and outputs; the Callier—Desoer class has a frequency domain description, while the Pritchard—Salamon has a state-space description. These two special classes serve to illustrate the advantages of a synthesis of time and frequency domain methods and to suggest ways of extending this approach to more general classes of infinite-dimensional systems.

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