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 L2(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 L2(R). We introduce a fourth behavioral metric as a generalization of the well-known ν-metric. As in the input-output framework, this definition is given in terms of rational representations. For this metric we however 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 deirved in terms of driving variable representations of the behaviors.