Optimal simulation of deterministic dynamic systems using stable solutions of special reduction problems

Abstract

Consider two linear deterministic dynamic systems (U) and (A) that transform the input signal x(t) into the output signal y(t), 0 <t ___ T (Fig. 1). In this paper, we study the simulation of system (U) by system (A) combined with some (specially chosen) linear deterministic system (Z) so that the new system (B) (see Fig. 2) produces in response to an arbitrary admissible input signal x an output signal y which is the best approximation of the response of system (U) to the same input x. System (Z) in this setup is a control block for system (A) simulating the behavior of system (U). We start with a mathematical formulation of the problem. Let X, Y, and W be reflexive Banach spaces of the functions x(t), y(t), and w(t), 0 ___ t _ T (the case T = +oo is allowed), and ~ (W, Y), ,,~(X, IV)~ q.~(X, Y) reflexive Banach spaces of linear bounded operators acting from W to Y, from X to W, and from X to Y, respectively. Norms in ~ , ~ , and °2/are denoted as I1" 114, I1" II ~: , and. II" ll~z----II" tl. These operator norms are not necessarily defined in the form II Ao II -sup{ II Aox I1: II x II =1 }, but they are naturally assumed to satisfy the standard norm axioms and are consistent with the corresponding norms in the spaces X, Y, and W. The norm in o2/is moreover assumed multiplicative: II A ~A 2 II <~ IIA ~ II ~" IIA~ I I for any A~6~, As6,~. In what follows, the operators U~v~/, Z 6 ~ and A E ~ are interpreted as linear deterministic dynamic systems (U), (Z), and (A), respectively; the input signals for systems (U) and (Z) are the functions x(t) E X, while the input signals for (A) are the functions w(t) E W. The output signals are generated by the following rules: y = Ux, w = Zx, y = Aw. The sought simulating system is described by an operator of the form AZ E ~ . Our problem of optimal simulation of system (U) by system (A) with a special choice of an appropriate system (Z) is mathematically formulated as the following extremal problem: find an operator R E ~ , such that