AbstractIn 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% degrad