State-space models for infinite-dimensional systems

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 finite number of parameters 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 applications can be contemplated. The realizability criteria provided by this work can be employed effectively in the first step of the identification procedure, i.e., in the selection of an appropriate infinite dimensional model. We show that there exists a natural classification of nonrational transfer functions, which is based on the character of their singularities. This classification has important implications 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 properties closely reflecting the properties of the singularities of the transfer functions. The study of models with infinitesimal 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.