Rational representations of behaviors: interconnectability and stabilizability

This paper presents an analysis of representation and stability properties of dynamical systems whose signals are assumed to be square summable sequences. Systems are defined as families of trajectories with no more structure than linearity and shift invariance. We depart from the usual input-output and operator theoretic setting and view relationships among system variables as a more general starting point for the study of dynamical systems. Parametrizations of two model classes are derived in terms of analytic functions which define kernel and image representations of dynamical systems. It is shown how state space models are derived from these representations. Uniqueness and minimality of these representations are completely characterized. Elementary properties like stabilizability, regularity, and interconnectability of dynamical systems are introduced and characterized in this set theoretic framework.