Reduced-order dynamic compensation using the Hyland-Bernstein optimal projection equations

Gradient-based homotopy algorithms have previously been developed for synthesizing H₂ optimal reduced-order dynamic compensators. These algorithms are made efficient and avoid high-order singularities along the homotopy path by constraining the controller realization to a minimal parameter basis. However the resultant homotopy algorithms sometimes experience numerical ill-conditioning or failure due to the minimal parameterization constraint. This paper presents a new homotopy algorithm which is based on solving the optimal projection equations, a set of coupled Riccati and Lyapunov equations that characterize the optimal reduced-order dynamic compensator. Path following in the proposed algorithm is accomplished using a predictor/corrector scheme that computes the prediction and correction steps by efficiently solving a set of four Lyapunov equations coupled by relatively low rank linear operators. The algorithm does not suffer from ill-conditioning due to constraining the controller basis and often exhibits better numerical properties than the gradient-based homotopy algorithms