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

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 QP, 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-l, 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. Key words, pseudo-rational impulse responses, delay-differential systems, matrix fractional representation, shift realization, canonical realization, approximate left coprimeness AMS(MOS) subject classifications. 93B 15, 93C05, 93C20