Robust stability analysis of uncertain switched linear systems with unstable subsystems

Abstract

The problem of robust stability for switched linear systems with all the subsystems being unstable is investigated. Unlike the most existing results in which each switching mode in the system is asymptotically stable, the subsystems may be unstable in this paper. A necessary condition of stability for switched linear systems is first obtained with certain hypothesis. Then, under two assumptions, sufficient conditions of exponential stability for both deterministic and uncertain switched linear systems are presented by using the invariant subspace theory and average dwell time method. Moreover, we further develop multiple Lyapunov functions and propose a method for constructing multiple Lyapunov functions for the considered switched linear systems with certain switching law. Several examples are included to show the effectiveness of the theoretical findings.

Keywords:

• switched systems
• unstable systems
• multiple Lyapunov functions
• robust stability
• average dwell time

Acknowledgements

This work is supported partially by the National Natural Science Foundation of China (Grant No. 11171197), by the Fundamental Research Funds for the Central Universities of China (Grant No. GK201102023). The authors would like to thank the anonymous referees and the editors for their valuable comments and helpful suggestions.

Additional information

Notes on contributors

Qiang Yu

Qiang Yu received his BS degree from the Department of Mathematics, Shi He Zi University in 2003, and the MS degree from Shaanxi Normal University, China, in 2009. From 2009 to 2011, he was with Heng Shui University as an instructor. He is currently a PhD candidate at the College of Mathematics and Information Science, Shaanxi Normal University, China. His research interests are stability of switched systems, robust control and time-delay systems.

Baowei Wu

Baowei Wu was born in 1963. He received the PhD degree from Xi'an Jiaotong University in 1998. He is now a full professor at the College of Mathematics and Information Science, Shaanxi Normal University, China. His main research interests include linear systems, robust control and time-delay systems.