A brief review and insights into matrix inequalities for H_∞ static-output-feedback control and a local optimal solution

ABSTRACT

This paper first gives a brief review of four kinds of matrix inequalities and two open problems about $H_{\mathrm{\infty }}$ static-output-feedback (SOF) control of continuous-time systems. Then, by clarifying the relationships among those matrix inequalities, the two open problems are proved to be essentially equivalent to a problem of choosing a coordinate transformation matrix (CTM). A new two-step optimisation method is established to solve this open problem. In the first step, a cone complementarity linearisation (CCL) method is utilised to solve the SOF-stabilisation problem so as to obtain a proper initial CTM. In the second step, a T-K iteration strategy is developed to optimise the choice of the CTM to locally solve the optimal continuous-time $H_{\mathrm{\infty }}$ SOF control problem. Numerical examples are given to demonstrate the effectiveness and benefits of the proposed method.

KEYWORDS:

• $H_{\mathrm{\infty }}$ control
• static output feedback (SOF)
• coordinate transformation matrix (CTM)

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [Grant Nos 61503290 and 51805394] and partly by the Open Research Fund of Fujian Engineering Research Center of Motor Control and System Optimal Schedule (Huaqiao University) [FERC007].

Notes on contributors

Zhi-Yong Feng

Zhi-Yong Feng received B.S. and M.S. in Control Science and Engineering from Central South University, Changsha, China, in 2003 and 2008, respectively, and Ph.D. degree in Engineering from Akita Prefectural University, Akita, Japan, in 2012. Since January 2013, he has been Faculty Member with Wuhan University of Technology, where he is currently an Associate Professor in the School of Automotive Engineering. His current research interests include static-output-feedback control, robust control of multidimensional systems, automotive control.

Jinhua She

Jinhua She received the B.S. degree in engineering from Central South University, Changsha, Hunan, China, in 1983, and the M.S. and Ph.D. degrees in engineering from the Tokyo Institute of Technology, Tokyo, Japan, in 1990 and 1993, respectively. In 1993, he joined the Department of Mechatronics, School of Engineering, Tokyo University of Technology. In 2008, he transferred to the University’s School of Computer Science, where he is currently a Professor. He received the Control Engineering Practice Paper Prize of the International Federation of Automatic Control in 1999 ( jointly with M. Wu and M. Nakano). His research interests include the application of control theory, repetitive control, process control, Internet-based engineering education, and robotics.

Li Xu

Li Xu received the B. Eng. degree from Huazhong University of Science and Technology, Wuhan, China, in 1982, and the M. Eng. and Dr. Eng. degrees from Toyohashi University of Technology, Toyohashi, Japan, in 1990 and 1993, respectively. From April 1993 to March 1998, he was an Assistant Professor at the Department of Knowledge-Based Information Engineering, Toyohashi University of Technology. From April 1998 to March 2000, he was a Lecturer at the Department of Information Management, Asahi University, Gifu, Japan. Since April 2000, he has been with the Faculty of Systems Science and Technology, Akita Prefectural University, Akita, Japan, where he is currently a Professor at the Department of Electronics and Information Systems. His research interests include multidimensional system theory, signal processing and the applications of computer algebra to system theory. Dr. Xu was an Associate Editor from April 2000 to November 2014, and has been an Editorial Board Member since December 2014 for the international journal of Multidimensional Systems and Signal Processing (MSSP).