On the pole of non-square transfer function matrix Moore–Penrose pseudo-inverses

Abstract

An essential step in many controller design approaches is computing the inverse of the plant. For a square plant, its inverse is stable if the plant is minimum phase (MP). Nevertheless, this conclusion does not hold for a non-square plant. In this paper, the pole feature of the Moore–Penrose pseudo-inverse of a non-square transfer function matrix is analysed. Instead of complicated advanced mathematical tools, only basic results of polynomial theory and the Binet–Cauchy theorem are used in the analysing procedure. The condition for testing the stability of the Moore–Penrose pseudo-inverse of an MP non-square transfer function matrix is given. This condition implies that the Moore–Penrose pseudo-inverse of a non-square transfer function matrix cannot be directly used as the optimal controller. Numerical examples are provided to illustrate the correctness of the condition.

Keywords:

• Moore–Penrose pseudo-inverse
• non-square transfer function matrix
• stability
• pole
• Binet–Cauchy theorem

Acknowledgements

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

Additional information

Funding

This work is supported by the National Science Foundation of China [grant number 61025016], [grant number 11072144], [grant number 61034008], [grant number 61221003].

Notes on contributors

Wei Zhang

Wei Zhang received the BS degree from the Department of Mathematics, Shaanxi Normal University, Xi'an, China, in 2003 and MS degree in operation research and cybernetics from Northwestern University, Shenyang, China in 2008. He is currently a PhD student of the Department of Automation, Shanghai Jiao Tong University, Shanghai, China. His research interests focus on robust control, optimal control, decoupling control, and their applications in industry and missile control.

Linlin Ou

Linlin Ou was born in Suzhou, China, on 28 November 1980. She received the BS degree in vacuum technology and equipment and the MS degree in fluid measurement and control from Hefei University of Technology, Hefei, China, in 2000 and 2003, respectively, and the PhD degree in control theory and engineering from Shanghai Jiao Tong University, Shanghai, China, in 2006. She is currently an assistant professor in the Department of Automation, Zhejiang University of Technology, Hangzhou, China. Her research interests focus on theoretical aspects of time-delayed systems and control, cooperative control of multi-agent system. Her research interests focus on theoretical aspects of time-delayed systems and control.

Xing He

Xing He received his BS and MS degrees from Northwestern Polytechnical University, Xi'an, China in 1990 and 1993, respectively and his PhD degree from Shanghai Jiao Tong University, Shanghai, China in 1995. Since 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 PhD degrees from Zhejiang University, Hangzhou, China in 1990, 1993, and 1996, respectively. He worked in 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.