Quadratic stabilisation of switched affine systems

Abstract

We deal with quadratic stabilisation for switched systems which are composed of a finite set of affine subsystems, where both subsystem matrices and affine vectors in the vector fields are switched independently, and no single subsystem has desired quadratic stability. We show that if a convex combination of subsystem matrices is Hurwitz and another convex combination of affine vectors is zero, then we can design a state-dependent switching law and an output-dependent switching law such that the entire switched system is quadratically stable at the origin. If the convex combination of affine vectors is not zero, we discuss the quadratic stabilisation to a convergence set defined by the convex combination of subsystem matrices and that of affine vectors. We extend the discussion to switched uncertain affine systems with norm bounded uncertainties, and establish a quadratically stabilising state-dependent switching law based on an ${\mathcal{H}}_{\mathrm{\infty }}$ norm condition for a combination of subsystems. Several numerical examples show effectiveness of the results.

Keywords:

• Switched affine systems (SAS)
• switched uncertain affine systems (SUAS)
• quadratic stabilisation
• convex combination
• state-dependent switching
• output-dependent switching
• convergence set
• LMIs
• norm bounded uncertainties

Acknowledgments

The authors would like to thank Prof. Yufang Chang and Prof. Bo Fu with Hubei University of Technology for valuable discussion.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research has been supported in part by National Science Foundation of Fujian Province of China under Grant 2017J01567, and the Fundamental Research Funds for the Central Universities under grant no. JBK190502; Japan Ministry of Education, Sciences and Culture under Grants-in-Aid for Scientific Research (C) 21560471.

Notes on contributors

Minqing Xiao

Minqing Xiao received the Ph.D. degree from Chongqing University, Chongqing, China, in 2008. He is currently a Professor with the College of Mathematics and Informatics, Fujian Normal University, Fuzhou, China. His current research interests include robust control/filter theory, delta operator systems, networked control systems, and switched systems.

Guisheng Zhai

Guisheng Zhai received his B.S. degree from Fudan University, China, in 1988, and he received his M.E. and Ph.D. degrees, both in system science, from Kobe University, Japan, in 1993 and 1996, respectively. After two years of industrial experience, Dr. Zhai moved to Wakayama University, Japan, in 1998, and then to Osaka Prefecture University, Japan, in 2004. He held visiting professor positions at University of Notre Dame from August 2001 to July 2002, and at Purdue University from March 2016 through February 2017. In April 2010, he joined the faculty board of Shibaura Institute of Technology, Japan, where he currently is a full Professor of Mathematical Sciences. His research interests include large scale and decentralised control systems, robust control, switched systems and switching control, networked control and multi-agent systems, neural networks and signal processing, etc. Dr. Zhai is on the editorial board of several academic journals including IET Control Theory & Applications, International Journal of Applied Mathematics and Computer Science, Journal of Control and Decision, and Frontiers of Mechanical Engineering. He is a Senior Member of IEEE, a member of ISCIE, SICE, JSST and JSME.

Chi Huang

Chi Huang received the M.S. degree in mathematics and the Ph.D. degree with the Department of Mathematics from the City University of Hong Kong, Kowloon, Hong Kong, in 2009 and 2012, respectively. He is currently an associate professor at the Department of Computer Science, Southwestern University of Finance and Economics, Chengdu, China. His research interests include networked systems, Boolean networks and machine learning. He has published more than 30 papers in refereed international journals. Dr. Huang was the recipient of the Best Paper Award in the Eighth Asian Control Conference in 2011.