Frequency domain integrals for stability preservation in Galerkin-type projection-based model order reduction

TL;DR

This paper presents a frequency domain integral approach to ensure stability preservation in Galerkin-type projection-based model order reduction of high-dimensional linear dynamical systems, addressing instability issues in reduced models.

Contribution

It introduces a novel frequency domain integral method for stability-preserving transformations in Galerkin-type model reduction, including a regularization extension for differential-algebraic systems.

Findings

The method effectively preserves stability in reduced models.

Numerical results demonstrate the approach's efficiency for high-dimensional systems.

Quadrature-based approximations solve high-dimensional Lyapunov inequalities efficiently.

Abstract

We investigate linear dynamical systems consisting of ordinary differential equations with high dimensionality. Model order reduction yields alternative systems of much lower dimensions. However, a reduced system may be unstable, although the original system is asymptotically stable. We consider projection-based model order reduction of Galerkin-type. A transformation of the original system ensures that any reduced system is asymptotically stable. This transformation requires the solution of a high-dimensional Lyapunov inequality. We solve this problem using a specific Lyapunov equation. Its solution can be represented as a matrix-valued integral in the frequency domain. Consequently, quadrature rules yield numerical approximations, where large sparse linear systems of algebraic equations have to be solved. We analyse this approach and show a sufficient condition on the error to meet the Lyapunov inequality. Furthermore, this technique is extended to systems of differential-algebraic equations with strictly proper transfer functions by a regularisation. Finally, we present results of numerical computations for high-dimensional examples, which indicate the efficiency of this stability-preserving method.