Canonical conservative state/signal shift realizations of passive discrete time behaviors

摘要

A passive linear discrete time invariant s/s (state/signal) system σ=(V;X,W) consists of a Hilbert (state) space X, a Kreǐn (signal) space W, a maximal nonnegative (generating) subspace V of the Kreǐn space K:=-X[{dot plus}]X[{dot plus}]W. The sets of trajectories (x(·);w(·)) generated by V on the discrete time intervals I?Z{double-struck} are defined by. (x(n+1);x(n);w(n))~∈V,n~∈I. This system is forward conservative, or backward conservative, or conservative if V~?V~([⊥]), V~([⊥]0V, or V~([⊥])=V, respectively. The set W_+~σ of all signal components w(·) of trajectories (x(·);w(·)) of σ on I=Z{double-struck}~+ with x(0)=0 and w(·)~(∈2)(Z{double-struck}~+;W) is called the future time domain behavior of σ. The Fourier transform ?_+~σ of W_+~σ is called the future frequency domain behavior of σ. This set is a maximal nonnegative right-shift invariant subspace in the Kreǐn space K~2(D{double-struck};W) that as a topological vector space coincides with the usual Hardy space H~2(D{double-struck};W), but has the indefinite Kreǐn space inner product inherited from W. A subspace of K~2(D{double-struck};W) with the above properties is called a passive future frequency domain behavior on W. It has been shown earlier by the present authors that every passive future frequency domain behavior ?+ on W may be realized as the future frequency domain behavior of some passive s/s system σ, and that it is possible to require, in addition, that σ is (a) controllable and forward conservative, (b) observable and backward conservative, or (c) simple and conservative. These three types of realizations are determined by ?+ up to unitary similarity. Canonical functional shift realizations of the types (a) and (b) have been obtained earlier by the present authors, and their connection to the classical de Branges-Rovnyak models have been discussed. Here we present analogous results for a realization of the type (c).