Realization theoretic solution of two analytic matrix equations with application to stabilization of infinite-dimensional systems

Abstract

A realization theoretic approach is proposed for the solutions of two matrix analytic equations arising in the control of infinite-dimensional systems: the Bezout type equationQX + RY = φ(which arises in the stabilization of infinite-dimensional systems) and the skew matrix equation QY + XR φ (which arises in the tracking/ stabilization in the presence of disturbances), where Q, R and φ are given analytic matrices. The realization proposed here is based on the operator theoretic machinery described by Fuhrmann (1981). With this Form of realization, we extend to infinite-dimensional systems the solutions given by Emre (1980) and Emre and Silverman (1981) for finite-dimensional systems. A systematic characterization of all H² matrix solutions of these equations is given. When (Q, R) satisfies the Bezout (Corona) condition, explicit formulae for the H² solutions of the Bezout-type equation are given in terms of a realization and the Hankel map of the given system. H² solutions are shown to be an intermediate step for H^∞ solutions of such equations. This provides a parametrization of all H^∞, H² solutions, and solutions with y polynomials. All H² solutions of the skew matrix equation are given. All H^∞ solutions and solutions with X polynomials of the skew matrix equation can also be obtained under certain conditions. A new criterion is developed for Bezout type equations to have H^∞ solutions in the cases where the Corona condition (Fuhrmann 1981) is not necessarily satisfied. Approximate polynomial solutions for Bezout type equations are also given. This approach provides an explicit link between the state space descriptions of the open-loop systems and stabilizing compensators.