Finite‐dimensional approximation for optimal fixed‐order compensation of distributed parameter systems

Abstract

In controlling distributed parameter systems it is often desirable to obtain low-order, finitedimensional controllers in order to minimize real-time computational requirements. Standard approaches to this problem employ model/controller reduction techniques in conjunction with LQG theory. In this paper we consider the finite-dimensional approximation of the infinite-dimensional Bernstein/Hyland optimal projection theory. Our approach yields fixed-finite-order controllers which are optimal with respect to high-order, approximating, finite-dimensional plant models. We illustrate the technique by computing a sequence of first-order controllers for one-dimensional, single-input/single-output, parabolic (heat/diffusion) and hereditary systems using spline-based, Ritz-Galerkin, finite element approximation. Our numerical studies indicate convergence of the feedback gains with less than 2% performance degradation over full-order LQG controllers for the parabolic system and 10% degradation for the hereditary system.

Optimal Fixed-Finite-Dimensional Compensator for Burgers’ Equation with Unbounded Input/Output Operators

“…The possibility of applying this approach to distributed parameter systems was first suggested by Johnson in [9] and Pearson [13]. The idea of fixing the order of the finite-dimensional compensator, while retaining the distributed parameter model was expanded and developed by Bernstein and Hyland in [1] and [2]. The method extends the full order LQG case to an "optimal fixed-finite-order compensator" characterized by four equations; two modified Riccati equations and two modified Lyapunov equations, coupled by an oblique projection whose rank is precisely equal to the order of the compensator.…”

“…that (1) is satisfied for almost every t > 0 where u(t) is given by (14), and that w(t) E 3 ?NC is continuously differentiable and satisfies (13) where y(t) is given by (2).…”

“…To actually use these equations to compute the optimal fixed-finite-order compensator, a finite dimensional approximation is needed (see [2] for details).…”