An efficient numerically robust homotopy algorithm for H2 model reduction using the optimal projecton equations

Abstract

Homotopy approaches have previously been developed for synthesizing #2 optimal reduced-order models. Some of the previous homotopy algorithms were based on directly solving the optimal projection equations, a set of two Lyapunov equations mutually coupled by a nonlinear term involving a projection matrix r, that characterize the optimal reduced-order model. These algorithms are numerically robust but suffer from the curse of large dimensionality. Subsequently, gradient-based homotopy algorithms were developed. To make these algorithms efficient and to eliminate singularities along the homotopy path, the basis of the reduced-order model was constrained to a minimal parameterization. However, the resultant homotopy algorithms sometimes experienced numerical ill-conditioning or failure due to the minimal parameterization constraint. This paper presents a new homotopy approach to solve the optimal projection equations for #2 model reduction. The current algorithm avoids the large dimensionality of the previous approaches by efficiently solving a pair of Lyapunov equations coupled by low rank linear operators