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

In controlling distributed parameter systems it is often desirable to obtain low‐order, finite‐dimensional 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 a 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.