Frequency domain integrals for stability preservation in Galerkin-type projection-based model order reduction
For engineers and scientists using model order reduction, this work provides a stability-preserving technique that addresses a known bottleneck (instability in reduced models) with a novel method.
The authors propose a transformation of high-dimensional linear dynamical systems that ensures any Galerkin-type projection-based reduced-order model is asymptotically stable. The method solves a Lyapunov inequality via frequency-domain integrals and quadrature, and numerical results on high-dimensional examples demonstrate its efficiency.
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.