Model reduction of large-scale systems by least squares

In this paper we introduce an approximation method for model reduction of large-scale dynamical systems. This is a projection which combines aspects of the SVD and Krylov based reduction methods. This projection can be efficiently computed using tools from numerical analysis, namely the rational Krylov method for the Krylov side of the projection and a low-rank Smith type iteration to solve a Lyapunov equation for the SVD side of the projection. For discrete time systems, the proposed approach is based on the least squares fit of the (r + 1)th column of a Hankel matrix to the preceding r columns, where r is the order of the reduced system. The reduced system is asymptotically stable, matches the first r Markov parameters of the full order model and minimizes a weighted error. The method is also generalized for moment matching at arbitrary interpolation points. Application to continuous time systems is achieved via the bilinear transformation. Numerical examples prove the effectiveness of the approach. The proposed method is significant because it combines guaranteed stability and moment matching, together with an optimization criterion.