Second-order algorithm for optimal model order reduction

Abstract

A second-order algorithm is given for optimal model order reduction of continuous single-input/single-output systems. Given a transfer function of order n, it finds the transfer function of a model of order m<n that minimizes the integral-square difference between the impulse responses of the two systems. It is shown that this is exactly equivalent to two other problems: 1) minimizing the integral-square difference between the frequency responses of the two systems and 2) minimizing the mean-square difference between the statistical steady-state responses of the two systems to a white-noise input. The algorithm is based on the known explicit solution of the governing Lyapunov equation, which yields analytical expressions for the first and second derivatives of the performance index with respect to the residues and poles of the reduced-order model. These derivatives are then used in a Newton-Raphson algorithm. The algorithm is applied to find exact solutions for four examples, one of which has an unstable mode. The examples corroborate previous research indicating that Moore's truncated balanced realization is not always close to the optimal reduced order model. Matching step response instead of impulse response requires only a slight change in the algorithm.