State-space models for infinite-dimensional systems

Abstract

Distributed effects are present in almost all physical systems. In some cases these can be safely ignored but there are many interesting problems where these effects must be taken into account. Most infinite dimensional systems which are important in control theory are specifiable in terms of a hite number of pa- rameters and hence are, in principle, amenable to identification. The state-space theory of infinite dimensional systems has advanced greatly in the last few years and is now at a point where real applica- tions can be contemplated. The realizability criteria provided by this work can be employed effectively in the first step of the identifi- cation procedure, i.e., in the selection of an appropriate infinite dimensional model. We show that there exists a natural classsca- tion of nonrational transfer functions, which is based on the char- acter of their singularities. This classiiication has important im- plications for the problem of finite dimensional approximations of infinite dimensional systems. In addition, it reveals the class of transfer functions for which there exist models with spectral proper- ties closely reflecting the properties of the singularities of the trans- fer functions. The study of models with iniinitesimal generators having a connected resolvent sheds light on some open problems in classical frequency response methods. Finally, the methods used here allow one to see the finite dimensional theory itself more clearly as the result of placing it in the context of a larger theory.