H∞ filtering in a behavioral framework

In this paper we treat the problem of H∞ filtering. This problem has been studied in the context of H∞ control for linear time invariant systems before. Without exception, earlier papers on this subject assume that the plant whose signals we try to estimate in the presence of disturbances, is given by equations in the usual state space format. One of the basic philosophies of the b havioral approach is, that in the analysis and synthesis of control systems, one shouldnot consider the system to be identical to the set of equations by which it happens to be given. Instead, one should identify the system with the set of all possible timetrajectoriesthat are compatible with these equations. This set of trajectories is called the behaviorof the system. The idea is, of course, that the set of equations describing a system is not unique, so that there is an obvious arbitrariness in the choice of representation. Our point of view adcocates that in a theory of analysis and synthesis, one should try, as much as possible, to work with the behavior of the system, and not with one of its particular representations. Obviously, results obtained by applying this point of view will be more general than those obtained for just one particular representation. In fact, one big advantage of the behavioral approach is, that the results will apply to anyrepresentation in which a certain plant happens to be given. In the present paper, we will illustrate this point of view by setting up a theory of H∞ filtering in a behavioral framework. We will arrive at necessary and sufficient conditions for the existence of a filter. In line with the basic philosophy explained in the previous paragraph, these conditions will not be in terms of a particular representation of the plant, but in terms of properties of its behavior. An important role will be played by two-variable polynomial matrices, qudratic differential forms and the concept of dissipative system. It will be shown that the existence of an H∞ filter is equivalent to certain dissipativity properties of the system. Furthermore, we formulate a theorem which states that under certain assumptions on the plant, the filter can be implemented as an input/output processor with a proper transfer matrix. The outline of this paper is as follows. In section 2 we give a brief review of linear differential systems, which is the class of systems that we deal with in this paper. In section 3 we review some material on quadratic differential forms, two-variable polynomial matrices, and dissipative systems. In section 4, we formulate the H∞ filtering problem and give necessary and sufficient conditions for the existence of a filter. Furthermore, we formulate a theorem which states that under certain assumptions on the plant, the filter can be implemented as an input/output processor with a proper transfer matrix. Due to space limitations, we have omitted the proofs. For these we refer to [11]. Some words on notation. We use the standard notation Rn,Rn1×n2 , etc., for finite-dimensional vectors and matrices. When the dimension is not specified (but, of course, finite), we writeR•,Rnו,R•×•, etc. The space of infinitely differentiable functions with domain R and co-domainRn is denoted byC∞(R,Rn), and its subspace consisting of the elements with compact support by D(R,Rn). In order to avoid convergence issues, we frequently restrict attention to compact support elements of a behavior. For this reason, we introduce the notationB ∩D = B ∩D(R,R•).