L₁-gain analysis and control synthesis of positive switched systems

Abstract

This paper investigates the problems of L₁-gain analysis and control synthesis of positive switched systems. Linear supply rates and L₁-gain notations are introduced to analyse the performance of the underlying systems. Stability with a weighted L₁-gain for autonomous systems are solved by using multiple linear copositive Lyapunov functions incorporated with the average dwell time approach. Then, state-feedback and output-feedback controllers are designed to guarantee the stabilisation with a weighted L₁-gain for non-autonomous systems. All present conditions are solvable in terms of linear programming. Finally, a practical example is given to illustrate the validity of theoretical findings.

Keywords:

• positive switched systems
• multiple linear co-positive Lyapunov functions
• linear programming
• state-feedback controller
• output-feedback controller

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions which have improved the quality of this paper.

Additional information

Funding

This work was supported partly by the National Natural Science Foundation of the People’s Republic of China [grant number 61074003], [grant number 61374077], [grant number 61203123] [grant number 11202121]; partly by the Fundamental Research Funds for the Central Universities, China [grant number 11CX04044A].

Notes on contributors

Junfeng Zhang

Junfeng Zhang received his BS and MS degrees in College of Mathematics and Information Science from Henan Normal University in 2007 and 2011, respectively. He is currently pursuing his Ph.D. in School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University. He was a recipient of Outstanding Master Degree Thesis Award from Henan Province, China, in 2011. His research interests include differential inclusions, switched systems, and positive systems.

Zhengzhi Han

Zhengzhi Han was born in 1947 and received his BS degree from East China Normal University in 1982, and his major is Mathematics. He received his Ph.D. degree from the East China University of Chemical Technology and Engineering in 1988 with the major of industrial automation. From 1988 to 1990, he worked in Shanghai Jiao Tong University as a postdoctoral fellow. From 1990, he becomes a faculty member of Shanghai Jiao Tong University. In 1992, he was promoted as a full professor. From 2000 to 2007, he was the chief editor of Shanghai Jiao Tong University Press. His current research interests include the design theory of nonlinear control systems and the theory of differential inclusion system.

Fubo Zhu

Fubo Zhu was born in 1979 and received the BS and MS degrees from Mathematic Department of Jiangsu Normal University, Xuzhou, China, in 2002 and 2006, respectively. He is now pursuing his Ph.D. degree in School of Electronic Information and Electrical Engineering at Shanghai Jiao Tong University. His current research interests include linear and nonlinear stochastic control systems, Markovian jump stochastic systems, and stochastic stability.