Stability and ℓ₁-gain analysis for positive 2-D Markov jump systems

Abstract

This paper investigates the problems of stability and ${\ell }_1$-gain analysis for positive 2-dimensional (2-D) Markov jump systems. The mathematical model of 2-D Markov jump systems is established based on the Roesser model. Necessary and sufficient condition for stability and sufficient condition for ${\ell }_1$-gain computation are derived. Furthermore, the stability and ${\ell }_1$-gain conditions are extended to Markov jump systems with partially known transition probabilities. The effectiveness of the obtained theoretical findings is verified through two numerical examples.

Keywords:

• 2-dimensional systems
• ${\ell }_1$-gain
• Markov jump systems
• positive systems
• stability analysis

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Chang Zhao http://orcid.org/0000-0003-4234-6547

James Lam http://orcid.org/0000-0002-0294-0640

Additional information

Funding

This work was partially supported by GRF HKU (17205815, 17206818), NTUT-SZU Joint Research Program (2019004) and Tencent “Rhinoceros Birds”-Scientic Research Foundation for Young Teachers of Shenzhen University.

Notes on contributors

Yong Chen

Yong Chen was born in Changsha, Hunan province, China, in 1988. He received his B.Eng. degree in Automation (ranking No. 10 among the 126 students of the major) from Harbin Institute of Technology (HIT), Harbin, Heilongjiang province, China, in 2010 and the Ph.D. degree in Mechanical Engineering from the University of Hong Kong in June, 2016. From 2016 to 2017, he was a Research Associate in the Department of Mechanical Engineering, the University of Hong Kong. Since 2018, he is an Assistant Professor in the School of Automation, Central South University, Changsha, China. His current research interests include switched systems, stochastic systems, periodic systems, fuzzy systems, positive systems in control theory and micro quadrotor in control engineering.

Chang Zhao

Chang Zhao received her B.E. degree in Detection Guidance and Control Technology from Harbin Institute of Technology, Harbin, China in July, 2015 and the Master degree in Control Science and Engineering from Harbin Institute of Technology, Harbin, China in July, 2017. She is currently working toward her Ph.D. degree in Mechanical Engineering at the University of Hong Kong. Her research interests include two dimensional systems, cyber physical systems, state estimation and robust control.

James Lam

Professor J. Lam received a B.Sc. (1st Hons.) degree in Mechanical Engineering from the University of Manchester, and was awarded the Ashbury Scholarship, the A.H. Gibson Prize and the H. Wright Baker Prize for his academic performance. He obtained the M.Phil. and Ph.D. degrees from the University of Cambridge. He is a Croucher Scholar, Croucher Fellow, and Distinguished Visiting Fellow of the Royal Academy of Engineering. Prior to joining the University of Hong Kong in 1993 where he is now Chair Professor of Control Engineering, he was a lecturer at the City University of Hong Kong and the University of Melbourne. Professor Lam is a Chartered Mathematician, Chartered Scientist, Chartered Engineer, Fellow of Institute of Electrical and Electronic Engineers, Fellow of Institution of Engineering and Technology, Fellow of Institute of Mathematics and Its Applications, Fellow of Institution of Mechanical Engineers, and Fellow of Hong Kong Institution of Engineers. He is Editor-in-Chief of IET Control Theory and Applications and Journal of The Franklin Institute, Subject Editor of Journal of Sound and Vibration, Editor of Asian Journal of Control, Senior Editor of Cogent Engineering, Associate Editor of Automatica, International Journal of Systems Science, Multidimensional Systems and Signal Processing and Proc. IMechE Part I: Journal of Systems and Control Engineering. He is a member of the Engineering Panel (Joint Research Scheme), Research Grant Council, HKSAR. His research interests include model reduction, robust synthesis, delay, singular systems, stochastic systems, multidimensional systems, positive systems, networked control systems and vibration control. He is a Highly Cited Researcher in Engineering (2014, 2015, 2016, 2017, 2018) and Computer Science (2015).

Yukang Cui

Yukang Cui received his B.Eng. degree in Automation from the Harbin Institute of Technology in July, 2012 and the Ph.D. degree in Mechanical Engineering from the University of Hong Kong in March, 2017. During 2017, he was a Research Associate within the Department of Mechanical Engineering, the University of Hong Kong. Since 2018, he is an Assistant Professor in the College of Mechatronics and Control Engineering, Shenzhen University, Shenzhen, China. His current research interests include positive systems, singular systems, time-delay systems, robust control and multi-agent systems.

Ka-Wai Kwok

Ka-Wai Kwok has served as Assistant Professor in Department of Mechanical Engineering, The University of Hong Kong (HKU), since 2014. He obtained a Ph.D. at Hamlyn Centre for Robotic Surgery, Department of Computing, Imperial College London in 2012, where he continued research on surgical robotics as a postdoctoral fellow. In 2013, he was awarded the Croucher Foundation Fellowship, which supported his research jointly supervised by advisors in The University of Georgia, and Brigham and Women's Hospital – Harvard Medical School. His research interests focus on intra-operative medical image processing, surgical robotics and their control techniques. His multidisciplinary work has been recognised by various scientific communities through his international conference/journal paper awards, e.g. the Best Conference Paper Award of ICRA 2018, which is the largest conference ranked top in the field of robotics, as well as TPEL 2018, RCAR 2017, ICRA 2017, ICRA 2014, IROS 2013 and FCCM 2011, Hamlyn 2012 and 2008, and Surgical Robot Challenge 2016. He also became the recipient of the Early Career Awards 2015/16 offered by Research Grants Council (RGC) of Hong Kong.