Pseudo-Rational Input/Output Maps and Their Realizations: A Fractional Representation Approach to Infinite-Dimensional Systems

Abstract

This paper studies matrix fractional representation for impulse responses of a certain class of infinite-dimensional linear systems which contains, in particular, delay-differential systems. Such impulse responses are called pseudo-rational in this paper. This fractional representation is effectively used to derive concrete function space models from the abstract shift realizations. Given a fractional representation $Q^{ - 1} * P$, a standard observable realization, analogous to Fuhrmann realizations for finite-dimensional systems, is associated to it. A new notion of coprimeness called approximate left coprimeness is introduced, and it is shown that the standard observable realization associated to the representation $Q^{ - 1} * P$ is canonical if and only if Q and P are approximately left coprime. Some examples are discussed to illustrate the relationships among various coprimeness concepts that have appeared in the literature.