Near-optimal Frequency-weighted Interpolatory Model Reduction

TL;DR

This paper introduces a new interpolatory framework for weighted-H2 model reduction of MIMO systems, providing a tractable algorithm with competitive performance compared to existing methods.

Contribution

It develops a novel weighted-H2 inner product representation and derives new interpolatory conditions for optimal model reduction, connecting them with Riccati-based conditions.

Findings

The new method is effective for large systems.

It outperforms Frequency Weighted Balanced Truncation in examples.

It is competitive with the Weighted IRKA algorithm.

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.