Model order reduction of linear and bilinear systems via low-rank Gramian approximation

Abstract

In this paper, we propose a series of model order reduction algorithms based on low-rank Gramian approximation for linear and bilinear systems. The main idea of the approach for linear systems is to use approximate low-rank factors of the controllability and observability Gramians to generate approximate balanced system for the large-scale system. Then, the reduced-order models are obtained by truncating the states corresponding to the smaller approximate Hankel singular values. The low-rank factors are constructed directly from the Laguerre functions expansion coefficient vectors of the matrix exponential functions by solving a recurrence formula instead of Lyapunov equations. In addition, the reduction procedure is modified with the idea of dominant subspace projection method to produce a stable reduced model under certain conditions. Furthermore, our algorithms are extended to bilinear systems successfully, with a series of corresponding algorithms for bilinear systems derived. Finally, numerical experiments are provided to demonstrate the effectiveness of the proposed methods.