Finite-time stability analysis of a class of nonlinear time-varying systems: a numerical algorithm

ABSTRACT

The paper investigates the finite-time stability (FTS) analysis of a very general class of nonlinear time-varying systems. The FTS of the considered system, whose vector field consists of a nonlinear part which can be sublinear or superlinear, and a linear part which can be time-varying, has not been fully studied before. By estimating the bound of the norm of the considered system's states with the generalised Gronwall–Bellman inequality, a sufficient criterion is established to guarantee the FTS of the considered system. To facilitate checking the criterion in practice, a novel numerical algorithm is proposed by numerically solving certain differential equations. Therefore, the FTS of the considered class of nonlinear time-varying systems can be easily analysed by the numerical algorithm. Further considering the numerical errors in the practical numerical computation, we strictly prove the credibility and programmability of the numerical algorithm in theory. Finally, three numerical examples are provided to illustrate the effectiveness the proposed results.

KEYWORDS:

• Finite-time stability
• nonlinear system
• time-varying system
• generalised Gronwall–Bellman inequality
• numerical algorithm

Acknowledgments

The authors would like to thank the editors and the reviewers for their very helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Basic Research Program of China [grant number 2013CB733100] and the National Natural Science Foundation of China [grant number 61333008].

Notes on contributors

Zhihua Chen

Zhihua Chen received the B.S. and M.S. degrees in Applied Mathematics and Control Theory and Control Engineering, respectively, both from Harbin Institute of Technology in 2012, 2014. Since 2014, he is a Ph.D. candidate in Control Theory and Control Engineering from China Academy of Space Technology (CAST), China. His research interests include analysis and design of control systems based on the computer platform, hybrid control and spacecraft control.

Yongchun Xie

Yongchun Xie received the B.S. degree in Electronic Engineering from Tsinghua University in 1989, and received her M.S. and Ph.D. degrees in automatic control theory and its applications from CAST in 1991 and 1994, respectively. Since 1994 she has been working at Beijing Institute of Control Engineering (BICE), CAST, China. From 1998 to 1999 she worked at the Institute of Space and Astronautical Science in Japan as a foreign researcher of Center of Excellence. She is currently the director of Science and Technology Committee of BICE and senior chief professor of CAST. She has co-authored 1 book and authored/co-authored over 100 papers. She has won two ministerial First Prize of Science and Technology Progress. In 2011 and 2012 she was named as ‘the Prominent Contributor for China's Manned Space Engineering’, ‘2011 Annual Figure in Chinese Automation Field’ and ‘2011 China Economic Annual Woman’. In 2013, she was awarded the ‘2012 Annual China Space Foundation Award’ and the First Prize of Jiachi Yang Science and Technology. Her interests include spacecraft intelligent and autonomous control, especially autonomous rendezvous and docking. Professor Xie is a member of the council of Chinese Association of Automation, and editorial board member for Journal of Astronautics, and Aerospace Control and Applications.