$\mathcal{H}_2$-gap model reduction for stabilizable and detectable systems

TL;DR

This paper introduces a novel $\\mathcal{H}_2$-gap based model reduction method for stabilizable and detectable systems, enabling efficient reduced models that accurately approximate unstable systems without full-order computations.

Contribution

It develops an interpolatory model reduction approach using the $\mathcal{H}_2$-gap metric, avoiding the need for full closed-loop system solutions.

Findings

Effective reduced models for unstable systems demonstrated.

Numerical examples show accurate approximation of unstable PDEs.

Algorithm avoids full-order closed-loop system evaluations.

Abstract

We formulate here an approach to model reduction that is well-suited for linear time-invariant control systems that are stabilizable and detectable but may otherwise be unstable. We introduce a modified $\mathcal{H}_2$-error metric, the $\mathcal{H}_2$-gap, that provides an effective measure of model fidelity in this setting. While the direct evaluation of the $\mathcal{H}_2$-gap requires the solutions of a pair of algebraic Riccati equations associated with related closed-loop systems, we are able to work entirely within an interpolatory framework, developing algorithms and supporting analysis that do not reference full-order closed-loop Gramians. This leads to a computationally effective strategy yielding reduced models designed so that the corresponding reduced closed-loop systems will interpolate the full-order closed-loop system at specially adapted interpolation points, without requiring evaluation of the full-order closed-loop system nor even computation of the feedback law that determines it. The analytical framework and computational algorithm presented here provides an effective new approach toward constructing reduced-order models for unstable systems. Numerical examples for an unstable convection diffusion equation and a linearized incompressible Navier-Stokes equation illustrate the effectiveness of this approach.