Sufficient and necessary conditions for discrete-time nonlinear switched systems with uniform local exponential stability

ABSTRACT

In this paper, we investigate sufficient and necessary conditions of uniform local exponential stability (ULES) for the discrete-time nonlinear switched system (DTNSS). We start with the definition of T-step common Lyapunov functions (CLFs), which is a relaxation of traditional CLFs. Then, for a time-varying DTNSS, by constructing such a T-step CLF, a necessary and sufficient condition for its ULES is provided. Afterwards, we strengthen it based on a T-step Lipschitz continuous CLF. Especially, when the system is time-invariant, by the smooth approximation theorem, the Lipschitz continuity condition of T-step CLFs can further be replaced by continuous differentiability; and when the system is time-invariant and homogeneous, due to the extension of Weierstrass approximation theorem, T-step continuously differentiable CLFs can even be strengthened to be T-step polynomial CLFs. Furthermore, three illustrative examples are additionally used to explain our main contribution. In the end, an equivalence between time-varying DTNSSs and their corresponding linearisations is discussed.

KEYWORDS:

• Exponential stability
• T-step common Lyapunov functions
• approximation theorem

Acknowledgements

The authors deeply thank the anonymous reviewers for their numerous detailed suggestions and comments on improving the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Nature Science Foundation of China [grant number 11290141], [grant number 11371047], [grant number 11422111] and the SKLSDE [grant number 2015ZX-06].

Notes on contributors

Junjie Lu

Junjie Lu received her BSc degree from Henan Normal University, China, in 2010. She is currently a PhD student in School of Mathematics and Systems Science at Beihang University, China. Her research interests include analysis and synthesis of switched systems.

Zhikun She

Zhikun She is a professor and vice-dean of the School of Mathematics and Systems Science at Beihang University, China. He received his BSc degree in 1999 and the PhD degree in 2005 both from Peking University, China and worked at MPI für Informatik as a post-doctor from January 2004 to December 2006. His research interests involve the theory, methodologies, and applications of safety verification and stability analysis of hybrid systems. He has been the principal investigator of more than 10 research projects and published more than 40 research papers. In particular, he has been awarded the first class of Natural Science Prize of the Ministry of Education of China in 2013 and the National Science Fund for Excellent Young Scholars of China in 2014.