Time-Domain Moment Matching for Second-Order Systems

TL;DR

This paper introduces a novel second-order time-domain moment matching framework for structure-preserving model reduction of high-dimensional dynamical systems, avoiding the need for first-order system conversion.

Contribution

It develops a second-order moment matching approach based on Sylvester equations, extending the Loewner framework, and addresses two-sided matching and derivatives.

Findings

Framework effectively reduces high-dimensional systems

Preserves second-order structure in reduced models

Demonstrated on vibrating systems

Abstract

The paper develops a second-order time-domain moment matching framework for the structure-preserving model reduction of second-order dynamical systems of high dimension, avoiding the first-order double-sized equivalent system. The moments of a second-order system are defined based on the solutions of second-order Sylvester equations, leading to families of parameterized second-order reduced models that match the moments of an original second-order system at selected interpolation points. Furthermore, a two-sided moment matching problem is addressed, providing a unique second-order reduced system that matches two distinct sets of interpolation points. We also construct the reduced second-order systems that match the moments of both the zero and first-order derivatives of the transfer function of the original second-order system. Finally, the Loewner framework is extended to second-order systems, where two parameterized families of models are presented that retain the second-order structure and interpolate sets of tangential data. The theory of the second-order time-domain moment matching is illustrated on vibrating systems.