Implicit higher-order moment matching technique for model reduction of quadratic-bilinear systems

Mian Mohammad Arsalan Asif,Mian Ilyas Ahmad,Peter Benner,Lihong Feng,Tatjana Stykel

doi:10.1016/j.jfranklin.2020.11.012

Abstract

We propose a projection based multi-moment matching method for model order reduction of quadratic-bilinear systems. The goal is to construct a reduced system that ensures higher-order moment matching for the multivariate transfer functions appearing in the input-output representation of the non-linear system. An existing technique achieves this for the first two multivariate transfer functions, in what is called the symmetric form of the multivariate transfer functions. We extend this framework to an equivalent and simplified form, the regular form, which allows us to show moment matching for the first three multivariate transfer functions. Numerical results for three benchmark examples of quadratic-bilinear systems show that the proposed framework exhibits better performance with reduced computational cost in comparison to existing techniques.

Highlights

We consider the problem of model order reduction for a single-input single-output (SISO) quadratic-bilinear descriptor system of the form ( ) = ( ) + ( ) ( ) + ( ) ⊗ ( ) + ( ), (1.1)
We propose a multi-moment matching technique that utilises the regular form of the multivariate transfer functions (2.8) and is, easy to be extended to higher multi-moments
It is a well-used example for model order reduction of nonlinear systems, where a non-linear RC circuit as shown in Figure 1 is considered for model reduction

Summary

Introduction

An extension to two-sided projection has been presented in Benner and Breiten (2015) that refines the quality of the approximations in term of accuracy It is observed in Ahmad, Benner and Jaimoukha (2016) that a simplified and equivalent representation of the multivariate transfer functions can be identified, for which it is relatively easy to extend the moment matching concept to higher multivariate transfer functions. The construction of the projection matrices and requires the use of the matrices and from the nonlinear terms, and will result in more accurate reduced-order system This is different from Benner and Breiten (2012), where multi-moment matching of the first two transfer functions is discussed.

Background

Multi-Moment Matching for the First Two Regular Transfer Functions

Multi-Moment Matching for the First Three Regular Transfer Functions

Numerical Examples

Non-linear RC Circuit

Conclusion