4. Model Reduction for Control Design for Distributed Parameter Systems

Abstract

4.1 Introduction During the past decades, considerable advances have been made in the numerical simulation of controlled distributed parameter systems (DPS). In the opinion of this author, this numerical sophistication has not been matched by the theoretical understanding of the approximation processes involved. The aim of this chapter is to shed a little light on some of the system theoretic properties which determine the suitability of an approximation scheme for control design of DPS. At the same time, a new robust control design is proposed which leads to robust, low-order controllers. It is shown that, at least for the class of exponentially stabilizable and detectable state linear systems with bounded and finite-rank input and output operators, this design always leads to a low-order controller which stabilizes not only the original system but also a large class of perturbations. This robustly stabilizing controller also guarantees bounds on the main performance indices. The class of systems considered in this chapter is that of the exponentially stabilizable and detectable state linear systems Σ(A, B, C) on the Hilbert space Z, where A is the infinitesimal generator of the strongly continuous semigroup T(t) on Z, and the operators B and C are finite-rank and bounded; B ∈ ℒ(ℂm, Z), C ∈ ℒ(Z,ℂk). The basic properties required of a finite-dimensional controller for this system are: (P1) The controller stabilizes Σ(A, B, C). (P2) The controller is robust so that it will have a chance of stabilizing the actual physical plant.