Near-optimal Frequency-weighted Interpolatory Model Reduction

Abstract

This paper develops an interpolatory framework for weighted-\(\mathcal{H}_2\) model reduction of MIMO dynamical systems. A new representation of the weighted-\(\mathcal{H}_2\) inner products in MIMO settings is introduced and used to derive associated first-order necessary conditions satisfied by optimal weighted-\(\mathcal{H}_2\) reduced-order models. Equivalence of these new interpolatory conditions with earlier Riccati-based conditions given by Halevi is also shown. An examination of realizations for equivalent weighted-\(\mathcal{H}_2\) systems leads then to an algorithm that remains tractable for large state-space dimension. Several numerical examples illustrate the effectiveness of this approach and its competitiveness with Frequency Weighted Balanced Truncation and an earlier interpolatory approach, the Weighted Iterative Rational Krylov Algorithm.