6. Geometric Theory of Output Regulation for Linear Distributed Parameter Systems

Abstract

Our first main objective is to present a brief introduction to the geometric theory of output regulation for distributed parameter systems. In this introduction we include references to some of the literature, as well as a survey of our recent work in this area. In particular, we describe our extension of the characterization, well known in finite-dimensional theory, of solvability of the state and error feedback regulator problems in terms of solvability of a pair of operator equations, referred to as the regulator equations. We present our main results for bounded input and output operators and finite-dimensional exosystems. Next we present an extension of these results to the class of regular linear systems with unbounded input and output operators obtained in our most recent work. We also present a result establishing that a class of boundary control systems governed by the heat equation on a bounded domain belongs to the well-known class of regular linear systems. Thus we provide a large class of systems for which our regulator theory applies. Next, the results for bounded input and output operators are applied to derive a simple formula for the solution of the regulator equations for retarded systems. Finally, we discuss several directions of future research in this area.