Input-to-state contraction for impulsive systems

• Published in: International journal of control Vol. 95; no. 3; pp. 779 - 789
• Main Authors: Liu, Bin, Sun, Zhijie, Xu, Bo, Liu, Dong-Nan
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 04.03.2022; Taylor & Francis Ltd
• Subjects: contraction; Criteria; Discrete time systems; Dwell time; Fields (mathematics); impulse frequency (I.F.); impulsive systems; incremental stability; Input-to-state stability; Lipschitz condition; variation
• ISSN: 0020-7179; 1366-5820
• DOI: 10.1080/00207179.2020.1823018

This paper investigates the input-to-state contraction for impulsive systems. The notion of input-to-state contraction (ISC) is proposed via incremental input-to-state stability (δISS). By using the method of variation, the equivalence between global ISC and exponential δISS is established. Then by the technique of the average dwell-time (ADT), some criteria for global ISC and exponential δISS are derived for impulsive systems. Moreover, by using impulse frequency (I.F.) to replace ADT, the criteria are improved and less conservative criteria via I.F. are established for global ISC and exponential δISS. The criteria are used to derive global ISC and exponential δISS for continuous and discrete time systems. Discussions are given to relax the differentiable vector fields. And the criteria of exponential δISS via the Lyapunov-like function are derived for impulsive systems satisfying time-varying Lipschitz condition. Finally, three examples are worked out for illustrations, where impulsive control is designed for chaotic systems to achieve global ISC.