A Complement on Elimination and Realization in Rational Representations

In this paper we study a number of problems in the context of rational representations of behaviors. In that context, a given proper real rational matrix can represent three behaviors. In the first place it can represent an input-output behavior. Secondly it can represent the kernel behavior of the rational ’differential operator’ associated with the rational matrix. Thirdly, it can represent the image behavior asociated with the rational matrix. On the other hand, every proper real rational matrix admits a realization as a finite dimensional linear state space system. Such realization can represent three system behaviors: an input-state-output behavior, an output nulling behavior, or a driving variable behavior. In this paper we will study the relation between the three external behaviors of these state representations, and the behaviors given by the three rational representations associated with the underlying rational matrix. Preliminary results from [5] will be complemented to obtain necessary and sufficient conditions such that the respective external behaviors are equal.