Distance between Behaviors and Rational Representations

Abstract

Abstract. 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 L 2 (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 L 2 (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, 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. 1. Introduction. This paper deals with notions of distance between systems. In the context of linear systems with inputs and outputs, several concepts of distance have been studied in the past. Perhaps the most well-known distance concept is that of gap metric introduced by Zames and El-Sakkary in [28] and extensively used by Georgiou and Smith in the context of robust stability in [7]. The distance between two systems in the gap metric can be calculated, but the calculation is by no means easy and requires the solution of an H ∞ optimization problem; see [6]. A distance concept which is equally relevant in the context of robust stability is the so-called ν-gap, introduced by Vinnicombe in [22], [21]. Computation of the ν-gap between two systems is much easier than that of the ordinary gap and basically requires computation of the winding number of a certain proper rational function, followed by computation of the L ∞ -norm of a given proper rational matrix. A third distance concept is that of L 2 -gap, which is the most easy to compute but which is not at all useful in the context of robust stability, as shown in [21]. More recently an alternative notion of gap for linear input-output systems was introduced by Ball and Sasane in [13], allowing also nonzero initial conditions of the system. In this paper we will put the above four distance concepts into a more general, behavioral context, extending them to a framework in which the systems are not necessarily identified with their representations (e.g., transfer matrices), but in which, instead, their behaviors, i.e., the spaces of all possible trajectories of the systems, form

Rational representations and robust stabilization in the behavioral framework

“…As a second result, we show that for neighborhoods in the gap, the problem of robust stabilization cannot be solved in the sense that given any controller that stabilizes a nominal plant, in any neighborhood around this nominal plant there exists a plant that is not stabilized by this controller. We then study the problem for neighborhoods defined using the V gap, the behavioral version of Vinnicombe's ν gap . Finally, we show that a controller regularly stabilizes all plants in one of the neighborhoods defined in this paper if and only if it is a regularly stabilizing controller for all plants in any of the other neighborhoods defined.…”

“…We then have , where and ∥ G Δ − G 1 ∥ ∞ ≤ γ . Because , from Lemma 6.2, , there exists a constant orthogonal matrix W such that G 1 W = G 2 . Further, as G Δ =( G Δ W ) W T , again using Lemma 6.2, , we have .…”

“…Representations of as in are called rational image representations . For more details on rational image representations, we refer the reader to .…”

“…In this section, we review the relevant material on the notion of distance between behaviors. We refer to for a more detailed treatment of this subject.…”

“…We call M G the multiplication operator with symbol G . We refer the reader to for more results on the topic of multiplication operators in the context of behaviors.…”