Input-tailored system-theoretic model order reduction for quadratic-bilinear systems

TL;DR

This paper introduces a univariate frequency-based moment matching method for quadratic-bilinear systems, improving efficiency and flexibility over multivariate approaches by leveraging tensor structures and accommodating more general input relations.

Contribution

It presents a more rigorous, general, and efficient univariate moment matching approach for quadratic-bilinear systems, incorporating tensor structures and broader input relations.

Findings

The method reduces interpolation frequencies needed for model reduction.

It exploits low-rank tensor structures for computational efficiency.

It allows for more general input relations in the state equations.

Abstract

In this paper we suggest a moment matching method for quadratic-bilinear dynamical systems. Most system-theoretic reduction methods for nonlinear systems rely on multivariate frequency representations. Our approach instead uses univariate frequency representations tailored towards user-pre-defined families of inputs. Then moment matching corresponds to a one-dimensional interpolation problem, not to multi-dimensional interpolation as for the multivariate approaches, i.e., it also involves fewer interpolation frequencies to be chosen. Comparing to former contributions towards nonlinear model reduction with univariate frequency representations, our approach shows profound differences: Our derivation is more rigorous and general and reveals additional tensor-structured approximation conditions, which should be incorporated. Moreover, the proposed implementation exploits the inherent low-rank tensor structure, which enhances its efficiency. In addition, our approach allows for the incorporation of more general input relations in the state equations - not only affine-linear ones as in existing system-theoretic methods - in an elegant way. As a byproduct of the latter, also a novel modification for the multivariate methods falls off, which is able to handle more general input-relations.