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

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. Furthermore, we extend this technique to systems of differential-algebraic equations with strictly proper transfer functions by a regularisation. Finally, results of numerical computations are presented for high-dimensional examples.