A state space realization of linear distributed parameter system (DPS) transfer operators

Abstract

We present in this paper a state space realization of transfer operators of linear time-invariant distributed systems. Our method is, on the one hand, the frequency domain analog of the universal state space representation of an input-output relation of A. V. Balakrishnan [1]; and on the other hand is based on the operator model theory of Nagy and Foias [2].

We shall discuss both discrete-time and continuous-time systems.

Given a discrete-time transfer operator {θ(z), H₁,H₂} which is analytic and bounded in ‖z‖<1, and is a map from ‖z‖<1 to the linear bounded operators from a Hilbert Space H₁ to another Hilbert Space H₂. Thus, we have the frequency domain input-output relation

$$v(z) = \theta (z)u(z)$$

((1))

where v(z) ε H²(H₂), u(z) ∈ H²(H₁). H²(H) is the space of power series f(x)=$\mathop \sum \limits_{n = 0}^\infty$f_nzⁿ, ∈ H and $\mathop \sum \limits_{n = 0}^\infty \parallel f_n \parallel _H^2 < \infty$. Our problem is to construct a state space for which θ(z) admits the canonical relization

$$\theta (z) = D + zC[I - zA]^{ - 1} B$$

((2))

where A, B, C, D are operators on intermediate spaces.

Clearly in (1) we only consider (the transforms of) those outputs in the future which resulted entirely from inputs defined over the same time interval. Let M be the closure of the set of these outputs, then a state space for the system can be taken to be the orthogonal complement (in H²(H₂)] M^┴ of M. Clearly M^┴ does contain those outputs which resulted from inputs in the past. These outputs actually represent the “controllable“ part of the system. Thus the system is controllable if these outputs are dense in M^┴. It will be shown that the operators A, B, C, D in (2) are expressible in terms of the compressed shift operator T and its adjoint T* on M^┴. T is in the Nagy-Foias Theory the canonical model of Hilbert Space contraction operators. It will be shown that the realization (1) will be canonical (controllable and observable) when the operators A, B, C, D are restricted to the cyclic subspaces of T and T*. These subspaces are contained in M^┴ and will be specifically characterized.

For the continuous-time case we consider a given transfer operator {θ(p),H₁,H₂} which is now analytic and bounded in Re. p>0, and its values are linear bounded operators from H₁ to H₂. In this case we associate with θ(p) the transform input and output spaces H²(π⁺,H₁) and H²(π⁺,H₂) which are the spaces of Laplace Transforms of L²(0,∞,H₁) and L²(0,∞,H₂) respectively. H²(π⁺,H_1,2) are of course related to their boundary spaces L²(iω, dω/2π, H_1,2). These latter spaces are in turn the Fourier transforms F L²(0,∞,H_1,2).

The state space relization of θ(p) is now

$$\theta (p) = D + C[pI - A]^{ - 1} B$$

where A, B, C, D are in this case related to the cogenerator of the (continuous) shift semigroup.

In both the discrete and continuous time cases we are required to consider the associated frequency operators θ(e^it) and θ(iω) — which are boundary functions of θ(z) and θ(p) respectively. A decomposition of these operators w.r.t. the orthogonal decompositions L ²₋ (H_1,2) ⊕ L ²₊ (H_1,2) and F L²(−∞, 0, H_1,2) ⊕ F L²(0, ∞, H_1,2) respectively, will automatically give the controllable part of the system. Also, the realization in both cases will be ‘reduced’ that is observable as well.

Work supported by the National Science Foundation USA under Grant #ENG 75-11876