Equivalence of Convolution Systems in a Behavioral Framework

Abstract

In this paper the problem of system equivalence is tackled for a rather general class of linear time-invariant systems. We consider AR-systems described by linear continuous shift-invariant operators with finite memory, acting on Frechet-signal spaces, containing the space {\cal E} ({\open R}) of infinitely differentiable functions on {\open R}. This class is in one–one correspondence with matrices of suitable sizes over the convolution algebra {\cal E}′ ({\open R}) of all compactly supported distributions. Using some deep results from the theory of Frechet spaces, various necessary and sufficient conditions for system equivalence and system inclusion are formulated. It is shown that a surjectivity demand on the system defining convolution operator matrix is necessary and sufficient for being able to translate the problem of system equivalence into division properties over the convolution algebra {\cal E}′({\open R}). This surjectivity condition is guaranteed if the system defining matrix over {\cal E}′({\open R}) has a right-inverse over {\cal D}′({\open R}), the space of all Schwartz distributions.