On the existence of minimal realizations of linear dynamical systems over Noetherian integral domains

Abstract

This paper studies the problem of obtaining minimal realizations of linear input/output maps defined over rings. In particular, it is shown that, contrary to the case of systems over fields, it is in general impossible to obtain realizations whose dimension equals the rank of the Hankel matrix. A characterization is given of those (Noetherian) rings over which realizations of such dimensions can be always obtained, and the result is applied to delay-differential systems.