Krylov subspace methods for model reduction of quadratic‐bilinear systems

Mian Ilyas Ahmad,Peter Benner,Imad Jaimoukha

doi:10.1049/iet-cta.2016.0415

Abstract

The authors propose a two sided moment matching method for model reduction of quadratic-bilinear descriptor systems. The goal is to approximate some of the generalised transfer functions that appear in the input–output representation of the non-linear system. Existing techniques achieve this by utilising moment matching for the first two generalised transfer functions. In this study, they derive an equivalent representation that simplifies the structure of the generalised transfer functions. This allows them to extend the idea of two sided moment matching to higher subsystems which was difficult in the previous approaches. Numerical results are given for some benchmark examples of quadratic-bilinear systems.

Highlights

Consider a multi-input multi-output (MIMO) quadratic-bilinear descriptor system of the form Σ: ̇ ( ) ( ) = ( ) . (1)In this paper, we discuss the approximation of QBDAEs by constructing reduced order models
Projection based moment matching techniques [2,3,4] have been used in the literature to construct such reduced order systems, in contrast to trajectorybased methods such as proper orthogonal decomposition (POD) [5], the reduced basis method, POD with discrete empirical interpolation method [6], and, the trajectory piecewise linear method [7]
In the remaining part of this paper, our goal is to identify the regular form of the multivariate transfer functions that can hopefully simplify the moment matching concept and allows us to use this new framework for third and higher subsystems

Summary

Introduction

Consider a multi-input multi-output (MIMO) quadratic-bilinear descriptor system of the form. Projection based (generalised-) moment matching techniques [2,3,4] have been used in the literature to construct such reduced order systems, in contrast to trajectorybased methods such as proper orthogonal decomposition (POD) [5], the reduced basis method, POD with discrete empirical interpolation method [6], and, the trajectory piecewise linear method [7]. We utilise the known connection between different forms of the generalised transfer functions in order to identify an equivalent representation of the generalised transfer functions which simplifies the structure of the moments This allows us to use two sided projection based moment matching techniques for higher dimensional transfer functions.

Background

Regular form of multivariate transfer functions

Multimoment-matching with the regular form

Non-linear RC circuit

Conclusion