Generalized state space realizations of non-proper rational transfer functions

Abstract

In this paper we propose an algebraic realization theory for polynomial transfer functions based on classical module theory. Realizations are given by linear systems whose generalized state space has a natural module structure over the ring of formal power series K [ z −1 ]. Existence and uniqueness of minimal realizations are proved and their properties are described. The dimension of the minimal realization is called generalized degree of the polynomial transfer function and it is proved to be equal to the rank of an associated generalized Hankel matrix. Finally, a correspondence between realizations and fractional representations over K [ z −1 ]of polynomial matrices is established and some properties of such representations are investigated.