Interpolatory methods for $\mathcal{H}_\infty$ model reduction of multi-input/multi-output systems

TL;DR

This paper introduces an efficient interpolatory approach for $ abla_ ext{H}_ ext{infty}$ model reduction of large-scale MIMO systems, outperforming traditional methods in quality and computational cost.

Contribution

It extends an existing SISO $ abla_ ext{H}_ ext{infty}$ reduction method to MIMO systems using interpolatory techniques and data-driven approximations, reducing computational effort.

Findings

Produces high-quality $ abla_ ext{H}_ ext{infty}$ reduced models

Outperforms balanced truncation in accuracy and cost

Often matches or exceeds optimal Hankel norm models

Abstract

We develop here a computationally effective approach for producing high-quality $\mathcal{H}_\infty$-approximations to large scale linear dynamical systems having multiple inputs and multiple outputs (MIMO). We extend an approach for $\mathcal{H}_\infty$ model reduction introduced by Flagg, Beattie, and Gugercin for the single-input/single-output (SISO) setting, which combined ideas originating in interpolatory $\mathcal{H}_2$-optimal model reduction with complex Chebyshev approximation. Retaining this framework, our approach to the MIMO problem has its principal computational cost dominated by (sparse) linear solves, and so it can remain an effective strategy in many large-scale settings. We are able to avoid computationally demanding $\mathcal{H}_\infty$ norm calculations that are normally required to monitor progress within each optimization cycle through the use of "data-driven" rational approximations that are built upon previously computed function samples. Numerical examples are included that illustrate our approach. We produce high fidelity reduced models having consistently better $\mathcal{H}_\infty$ performance than models produced via balanced truncation; these models often are as good as (and occasionally better than) models produced using optimal Hankel norm approximation as well. In all cases considered, the method described here produces reduced models at far lower cost than is possible with either balanced truncation or optimal Hankel norm approximation.