Stabilisation and ℋ_∞ control for switched port-controlled Hamiltonian systems with unstable modes and actuator saturation

ABSTRACT

In this paper, the problems of stabilisation and ${\mathcal{H}}_{\mathrm{\infty }}$ control are addressed for switched port-controlled Hamiltonian (SPCH) systems with unstable modes and actuator saturation (AS) via energy-based multiple Lyapunov functions methods. Firstly, by a switching state feedback controller with truncation-inequality techniques, the considered system is transformed into an unforced SPCH system with asymptotically stable, stable and unstable modes. Then, by developing an alternative switching law with slow/fast mode-dependent average dwell time (MDADT) switching scheme, new sufficient conditions are obtained for stabilisation of the SPCH system. Secondly, by designing a new switching ${\mathcal{H}}_{\mathrm{\infty }}$ feedback controller, revealing the characteristics of alternative switching signals and defining an indicative function, a new criterion for ${\mathcal{H}}_{\mathrm{\infty }}$ control of the SPCH systems with unstable modes and AS is achieved via the slow/fast MDADT switching scheme. Moreover, based on the obtained results and the indicative function, ${\mathcal{H}}_{\mathrm{\infty }}$ control conditions for the corresponding SPCH system are also derived under a slow MDADT switching scheme and a traditional average dwell time scheme, respectively. Finally, some numerical examples are given to verify the effectiveness of the proposed methods.

Keywords:

• ${\mathcal{H}}_{\mathrm{\infty }}$ control
• energy-based multiple Lyapunov functions
• slow/fast mode-dependent average dwell time
• switched port-controlled Hamiltonian systems
• actuator saturation

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 A function $\kappa (\cdot ,\cdot ):[0,\mathrm{\infty })\times [0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is said to be class $\mathcal{K}\mathcal{L}$ if $\kappa (\cdot ,t)$ belongs to class $\mathcal{K}$ for every fixed t>0 and $\kappa (s,t)\to 0$ as $t\to \mathrm{\infty }$ for every fixed $s\ge 0.$

Additional information

Funding

The authors would like to express sincere appreciation to the Associate Editor and the anonymous reviewers for their valuable comments on improving the quality and presentation of the paper. This work is supported by National Natural Science Foundation [grant number 61573218], [grant number 61973188], [grant number 61722302], [grant number 61603227], [grant number 11701260], [grant number 61773235], the Research fund for the Taishan Scholar Project of Shandong Province of China, and the Foundation for Innovative Research Groups of National Natural Science Foundation of China [grant number 61821004].

Notes on contributors

Zi-Ming Wang

Zi-Ming Wang received the B.S. degree in mathematics from Bohai University, Jinzhou, China, in 2006, and the M.S. degree in applied mathematics from Tianjin Polytechnic University, Tianjin, China, in 2010. He is currently pursuing the Ph.D. degree in control theory and control engineering with Shandong University, Jinan, China. His research interests include switched systems, port-controlled Hamiltonian systems, and optimization analysis.

Airong Wei

Airong Wei received her M.S. degree from Shandong University of Technology in 1997 and her Ph.D. degree from School of Control Science and Engineering, Shandong University in 2006. From 2007 to 2009, she worked as a Postdoctoral Fellow in School of Mathematics, Shandong University. Now she is a professor with the School of Control Science and Engineering, Shandong University. Her research interests include switched nonlinear control systems with constrains, Hamiltonian systems, and control of multi-agent systems.

Xudong Zhao

Xudong Zhao received the B.S. degree in Automation from Harbin Institute of Technology in 2005 and the Ph.D. degree from Control Science and Engineering from Space Control and Inertial Technology Center, Harbin Institute of Technology in 2010. Dr. Zhao was a lecturer and an associate professor at the China University of Petroleum, China. From March 2013, he was with Bohai University, China, as a Professor. In 2014, Dr. Zhao worked as a postdoctoral fellow in the Department of Mechanical Engineering, the University of Hong Kong. Since December 2015, he has been with Dalian University of Technology, China, where he is currently a Professor. Dr. Zhao serves as associate editor for IEEE Transactions on Systems, Man and Cybernetics: Systems, Nonlinear Analysis: Hybrid Systems, Neurocomputing, IEEE Access, International Journal of General Systems, ACTA Automatica Sinica, Assembly Automation, and Journal of Aeronautics, etc. In addition, he has been granted as the outstanding reviewer for Automatica, IET Control Theory and Applications, and Journal of the Franklin Institute, etc. He received the 2017–2018 and 2019 Highly Cited Researcher award. His research interests include hybrid systems, positive systems, multi-agent systems, fuzzy systems and their applications.

Jinge Yang

Jinge Yang received the B.S. degree in mathematics from Bohai University, Jinzhou, China, in 2006, and the Ph.D. degree in basic mathematics from School of Mathematical Sciences, Dalian University of Technology, Dalian, China, in 2013. From 2013, he has been with Nanchang Institute of Technology, Nanchang, China, where he is currently an Associate Professor. His research interests include partial differential equation, stability analysis, and robust control.

Guangdeng Zong

Guangdeng Zong received his Ph. D. degree in control theory and application from Southeast University, Nanjing, China, in 2005. He is currently a full professor in School of Engineering at Qufu Normal University. His major research interests include time-delay systems, switched systems and robust control.