Model Reduction of Descriptor Systems by Interpolatory Projection Methods

TL;DR

This paper develops a modified interpolatory projection method for descriptor systems that ensures bounded approximation errors, extending IRKA for optimal interpolation point selection, with practical numerical demonstrations.

Contribution

It introduces new subspace conditions for descriptor systems and extends IRKA to improve model reduction accuracy and stability.

Findings

Modified subspace conditions guarantee bounded errors.

Extended IRKA for descriptor systems achieves near-optimal interpolation points.

Numerical examples validate theoretical improvements.

Abstract

In this paper, we investigate interpolatory projection framework for model reduction of descriptor systems. With a simple numerical example, we first illustrate that employing subspace conditions from the standard state space settings to descriptor systems generically leads to unbounded H2 or H-infinity errors due to the mismatch of the polynomial parts of the full and reduced-order transfer functions. We then develop modified interpolatory subspace conditions based on the deflating subspaces that guarantee a bounded error. For the special cases of index-1 and index-2 descriptor systems, we also show how to avoid computing these deflating subspaces explicitly while still enforcing interpolation. The question of how to choose interpolation points optimally naturally arises as in the standard state space setting. We answer this question in the framework of the H2-norm by extending the Iterative Rational Krylov Algorithm (IRKA) to descriptor systems. Several numerical examples are used to illustrate the theoretical discussion.