Distance between Behaviors and Rational Representations

In this paper we study notions of distance between behaviors of linear differential systems. We introduce four metrics on the space of all controllable behaviors which generalize existing metrics on the space of input-output systems represented by transfer matrices. Three of these are defined in terms of gaps between closed subspaces of the Hilbert space $\mathcal{L}_2(\mathbb{R})$. In particular we generalize the "classical" gap metric. We express these metrics in terms of rational representations of behaviors. In order to do so, we establish a precise relation between rational representations of behaviors and multiplication operators on $\mathcal{L}_2(\mathbb{R})$. We introduce a fourth behavioral metric as a generalization of the well-known $\nu$-metric. As in the input-output framework, this definition is given in terms of rational representations. For this metric, however, we establish a representation-free, behavioral characterization as well. We make a comparison between the four metrics and compare the values they take and the topologies they induce. Finally, for all metrics we make a detailed study of necessary and sufficient conditions under which the distance between two behaviors is less than one. For this, both behavioral as well as state space conditions are derived in terms of driving variable representations of the behaviors. [PUBLICATION ABSTRACT]