Structure-preserving model order reduction for K-power bilinear systems via Laguerre functions

ABSTRACT

This paper presents a series of structure-preserving model order reduction algorithms for K-power bilinear systems via Laguerre functions. The method first aims to rewritten the K-power bilinear system as a general bilinear system and calculate the approximate low-rank factors of the cross Gramian of the bilinear system by combining the idea of Laguerre functions expansion of the matrix exponential function. After that, the approximate balanced system of the K-power bilinear system is constructed by the corresponding projection transformation of each subsystem. In order to achieve the purpose of model order reduction, the states with smaller singular values are then truncated, so as to further obtain the reduced order model. For this approach, there is a disadvantage that unstable systems may be generated although the original one is stable. To alleviate the inadequacies of this approach, we have improved the model reduction procedure, which is based upon the dominant subspace projection method. In addition, we also carried out a correlation analysis on the stability of improved algorithms. Finally, numerical experiments are employed to substantiate the effectiveness of the presented algorithms.

KEYWORDS:

• Model order reduction
• K-power bilinear systems
• Laguerre functions
• cross Gramian

Disclosure statement

No potential conflict of interest was reported by the authors.

Data Availability Statement

Data available on request from the authors: the data that support the findings of this study are available from the corresponding author [Name: Zhi-Hua Xiao, Email: [email protected]], upon reasonable request.

Additional information

Funding

This work was supported by the Natural Science Foundation of China (NSFC) [grant numbers 62273059 and 61803046].

Notes on contributors

Hui Jin

Hui Jin received the Bachelor degree in Applied Mathematics from Yangtze University, Jingzhou, Hubei, China, in 2020. He is currently pursuing a Master's degree in Applied Mathematics from Yangtze University. His research interest covers model reduction and control theory.

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.

Qiu-Yan Song

Qiu-Yan Song received an MS degree in Applied Mathematics from Yangtze University in 2018. She is currently pursuing a Ph.D. degree in Control Science and Engineering from Shanghai University. Her research interests include model order reduction and control theory.

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.