Dimension reduction for k-power bilinear systems using orthogonal polynomials and Arnoldi algorithm

Abstract

In this paper, a dimension reduction method via general orthogonal polynomials and multiorder Arnoldi algorithm is proposed, which focuses on the topic of structure-preserving for k-power bilinear systems. The main procedure is using a series of expansion coefficient vectors of each state variables in the space spanned by general orthogonal polynomials that satisfy a recurrence formula to generate a projection based on multiorder Arnoldi algorithm. The resulting reduced-order model not only matches a desired number of expansion coefficients of the original output but also retains the topology structure. Meanwhile, the stability is well preserved under some certain conditions and the error bound is also given. Finally, two numerical simulations are provided to illustrate the effectiveness of our proposed algorithm in the views of accuracy and computational cost.

Keywords:

• Dimension reduction
• k-power bilinear systems
• orthogonal polynomials
• stability
• function approximation

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Natural Science Foundation of China (NSFC) [grant numbers 11871393 and 61803046], Research Project of Hubei Provincial Department of Education [grant number Q20181305], a key project of the International Science and Technology Cooperation Program of Shaanxi Research $\mathrm{\&}$ Development Plan [grant number 2019KWZ-08], Special Scientific Research Project of Shaanxi Education Department [grant number 19JK0839] and Basic Research Plan of Natural Science of Shaanxi Province [grant number 2020JQ-569].

Notes on contributors

Zhen-Zhong Qi

Zhen-Zhong Qi received the PhD degree from Xi'an Jiaotong University (XJTU), China. He is currently a lecturer in department of mathematics, Northwest University, China. He focuses on applying mathematics and computation to solve science and engineering problems. His research interests include model order reduction, control theory and circuit simulation.

Yao-Lin Jiang

Yao-Lin Jiang is a full professor (Changjiang Scholar in China, too) at the School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, China. He has published 4 books and about 230 papers in journals. His research interests include theoretical studies of engineering problems, model order reduction, waveform relaxation, numerical solutions of differential equations, dynamics of non-linear systems, circuit simulation and parallel processing.

Zhi-Hua Xiao

Zhi-Hua Xiao received his PhD degree from Xi'an Jiaotong University, Shaanxi, China, in 2015. He is an associate professor in the School of Information and Mathematics at Yangtze University. He has published about 20 papers in journals. His research interests include theoretical studies of control systems, model order reduction and numerical linear algebra.