An extended inner–outer factorisation algorithm based on the structure of a transfer function matrix inverse

Abstract

In this paper, the structure feature of the inverse of a multi-input/multi-output square transfer function matrix is explored. Instead of complicated advanced mathematical tools, we only use basic results of complex analysis in the analysing procedure. By employing the Laurent expression, an elegant structure form of the expansion is obtained for the transfer function matrix inverse. This expansion form is the key of deriving an analytical solution to the inner–outer factorisation for both stable plants and unstable plants. Different from other computation algorithm, the obtained inner–outer factorisation is given in an analytical form. The solution is exact and without approximation. Numerical examples are provided to verify the correctness of the obtained results.

Keywords:

• transfer function matrix inverse
• inner–outer factorisation
• Laurent expansion
• standard series expansion
• complex analysis

Acknowledgements

We thank the anonymous reviewers for their constructive comments and helpful suggestions.

Additional information

Funding

This paper was partly supported by the National Science Foundation of China [grant number 61025016], [grant number 61034008]， [grant number 61221003]; Program of Shanghai Subject Chief Scientist [grant number 14XD1402400].

Notes on contributors

Wei Zhang

Wei Zhang received his BS degree from the Department of Mathematics, Shaanxi Normal University, Xi'an, China, in 2003, MS degree in operation research and cybernetics from Northwestern University, Shenyang, China, in 2008, and Ph.D. degree in control theory and engineering from Shanghai Jiao Tong University, Shanghai, China, in 2014. He is currently an assistant professor with the School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai, China. His research interests focus on robust control, optimal control, decoupling control and their applications in industry.

Chunhui Zhao

Chunhui Zhao (M’2012) received her BEng, MSc and Ph.D. degrees from the Department of Automation, Northeastern University, Shenyang, China, in 2003, 2006 and 2009, respectively. She was a recipient of the National Top 100 Excellent Doctor Thesis Nomination Award, New Century Excellent Talents in University, China. She was a postdoctoral fellow (January 2009–January 2012) at Hong Kong University of Science and Technology, and University of California, Santa Barbara, Los Angeles, USA. Since January 2012, she has been working as a distinguished professor with the Department of Control Science and Engineering, Zhejiang University, Hangzhou, China. Her research interests include statistical process monitoring, fault diagnosis and glucose control. She has published over 100 papers in peer-reviewed international journals and conferences.

Xing He

Xing He received his BS and MS degrees from Northwestern Polytechnical University, Xi'an, China, in 1990 and 1993, respectively, and his Ph.D. degree from Shanghai Jiao Tong University, Shanghai, China, in 1995. In 1995, he joined the Department of Automation, Shanghai Jiao Tong University, where he is currently an associate professor. His research interests include distributed control, robust control and networked control.

Weidong Zhang

Weidong Zhang received his BS, MS and Ph.D. degrees from Zhejiang University, Hangzhou, China, in 1990, 1993 and 1996, respectively. He worked with the National Key Laboratory of Industrial Control Technology as a post-doctor before joining Shanghai Jiao Tong University, Shanghai, China, in 1998 as an associate professor. Since 1999, he has been a professor at the Department of Automation, Shanghai Jiao Tong University. He served as the deputy dean of the Department of Automation, Shanghai Jiao Tong University. He has authored or co-authored over 200 journal and conference papers. His research interests include robust control, distributed control and their applications in industry.