Necessary exponential stability conditions for linear discrete time-delay systems and application

ABSTRACT

Necessary conditions for the exponential stability of the linear discrete time-delay systems are presented by employing the so-called Lyapunov–Krasovskii functional approach. These conditions not only provide a new tool for stability analysis of the linear discrete time-delay system by characterising instability domains, but also extend the existing results of the linear discrete time-delay system. Simultaneously, we investigate several crucial properties that connect the Lyapunov matrix and the fundamental matrix of the system. Finally, the robust stability analysis of the linear discrete time-delay systems with norm-bounded uncertainties is presented. Numerical examples illustrate the validity of the obtained results.

KEYWORDS:

• Linear discrete time-delay system
• exponential stability
• fundamental matrix
• Lyapunov matrix
• Lyapunov–Krasovskii functional

Acknowledgments

The authors would like to thank the associate editor and anonymous reviewers for their insightful comments, which have improved the quality of this article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (11371006 and 61703148), the Basic Research Operating Expenses Program of Colleges and Universities in Heilongjiang Province (HDJCCX-2016212 and RCCX201717), the Natural Science Foundation of Heilongjiang Province (QC2018083) and the Heilongjiang University Innovation Fund for Graduates (YJSCX2018-057HLJU).

Notes on contributors

Haifang Li

Haifang Li received the B.S. degree in School of Mathematics and Statistics from Hulunbeir College, Hulunbeir, China, in 2016. She is an M.S. student in Heilongjiang University, Harbin, China. Her research interest is stability analysis of delayed dynamic systems.

Ning Zhao

Ning Zhao received the B.S. degree in School of Mathematical Science from Hulunbeir College, Hulunbeir, China, in 2015. He is an M.S. student in Heilongjiang University, Harbin, China. His research interest is stability analysis of delayed dynamic systems.

Xian Zhang

Xian Zhang received Ph.D. degree in control theory from Queen 's University of Belfast in UK in 2004. Since 2004 he has been at Heilongjiang University, where he is currently a Professor in the School of Mathematical Science. His current research interests include neural networks, genetic regulatory networks, mathematical biology and stability analysis of delayed dynamic systems. He has received the Second Class of Science and Technology Awards of Heilongjiang Province in 2005 and the Three Class of Science and Technology Awards of Heilongjiang Province in 2015. He is a member of the IEEE, and a Vice President of Mathematical Society of Heilongjiang Province. Since 2006, he served as an Editor of the Journal of Natural Science of Heilongjiang University. He has authored more than 100 research papers.

Xin Wang

Xin Wang received the B.S. and M.S. degrees in School of Mathematical Science from Heilongjiang University, Harbin, China, in 2008 and 2011, respectively, and the Ph.D. degree in navigation guidance and control from Northeastern University, Shenyang, China, in 2016. He is currently a Lecturer with the School of Mathematical Science, Heilongjiang University, Harbin, China. Dr. Xin Wang was a Visiting Professor at the University of Victoria from November 2017 to October 2018. His research interests include fault diagnosis, fault-tolerant control, multiagent coordination, and time-delay systems.