New results on the equivalence of rational representations of behaviors

This article deals with the equivalence of representations of behaviors of linear differential systems. In general, the behavior of a given linear differential system has many different representations. In this paper we restrict ourselves to kernel and image representations. Two kernel representations are called equivalent if they represent one and the same behavior. For kernel representations defined by polynomial matrices, necessary and sufficient conditions for equivalence are well-known. In this paper, we deal with the equivalence of rational representations, i. e. kernel and image representations that are defined in terms of rational matrices. As the main result of this paper, we will derive a new condition for equivalence of rational kernel representations of possibly noncontrollable behaviors. This paper also deals with the equivalence of polynomial as well as rational image representations.