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
This article deals with the equivalence of representations of behaviors of linear differential systems. In general, the behavior of a given linear differential system has many different representations. In this paper we restrict ourselves to kernel and image representations. Two kernel representations are called equivalent if they represent one and the same behavior. For kernel representations defined by polynomial matrices, necessary and sufficient conditions for equivalence are well known. In this paper, we deal with the equivalence of rational representations, i. e. kernel and image representations that are defined in terms of rational matrices. As the first main result of this paper, we will derive a new condition for the equivalence of rational kernel representations of possibly noncontrollable behaviors. Secondly we will derive conditions for the equivalence of rational representations of a given behavior in terms of the polynomial modules generated by the rows of the rational matrices. We will also establish conditions for the equivalence of rational image representations. Finally, we will derive conditions under which a given rational kernel representation is equivalent to a given rational image representation.
In this paper, we study the problem of robust stabilization for linear differential systems in the behavioral framework. We study the existence of controllers that regularly stabilize all plants in a given neighborhood around a certain nominal plant. We call such controllers robustly stabilizing controllers. This problem was studied in the behavioral framework by Trentelman, H.L et al. In contrast to their work, however, in the present paper, we study the problem considering neighborhoods that are defined entirely representation free. These neighborhoods are induced by different kinds of concepts of distance between behaviors. As one of our main results we obtain that a given controller regularly stabilizes all plants in one of these neighborhoods if and only if it regularly stabilizes all plants in all of the other neighborhoods.if G D P 1 Q is a left coprime factorization of G over ROE . Representations of B as in (4) are called rational image representations. For more details on rational image representations, we refer the reader to [15,17,18]. The number of columns of G in any rational image representation of B, with G full column rank, is equal to m.B/. The number of rows of Q G in rational kernel representation of B, with Q G full row rank, is equal to p.B/. DISTANCE BETWEEN BEHAVIORSIn this section, we review the relevant material on the notion of distance between behaviors. We refer to [15] for a more detailed treatment of this subject.
Abstract-This article deals with the equivalence of representations of behaviors of linear differential systems. In general, the behavior of a given linear differential system has many different representations. In this paper we restrict ourselves to kernel and image representations. Two kernel representations are called equivalent if they represent one and the same behavior. For kernel representations defined by polynomial matrices, necessary and sufficient conditions for equivalence are wellknown. In this paper, we deal with the equivalence of rational representations, i. e. kernel and image representations that are defined in terms of rational matrices. As the main result of this paper, we will derive a new condition for equivalence of rational kernel representations of possibly noncontrollable behaviors. This paper also deals with the equivalence of polynomial as well as rational image representations.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.