Stability preservation in Galerkin-type projection-based model order reduction

TL;DR

This paper presents a method to ensure stability in reduced-order models of high-dimensional linear and nonlinear dynamical systems by transforming the original system through an approximate Lyapunov equation solution.

Contribution

It introduces an efficient iterative approach to approximate the Lyapunov solution, enabling stability preservation in Galerkin-type projection-based model order reduction.

Findings

The proposed method effectively preserves stability in high-dimensional linear systems.

Numerical results demonstrate computational feasibility for large-scale problems.

The approach extends to nonlinear systems for stability of stationary solutions.

Abstract

We consider linear dynamical systems consisting of ordinary differential equations with high dimensionality. The aim of model order reduction is to construct an approximating system of a much lower dimension. Therein, the reduced system may be unstable, even though the original system is asymptotically stable. We focus on projection-based model order reduction of Galerkin-type. A transformation of the original system guarantees an asymptotically stable reduced system. This transformation requires the numerical solution of a high-dimensional Lyapunov equation. We specify an approximation of the solution, which allows for an efficient iterative treatment of the Lyapunov equation under a certain assumption. Furthermore, we generalise this strategy to preserve the asymptotic stability of stationary solutions in model order reduction of nonlinear dynamical systems. Numerical results for high-dimensional examples confirm the computational feasibility of the stability-preserving approach.