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Controller design for switched systems with nonlinear sub-systems using common Lyapunov functions

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Abstract

Hybrid systems are common in real life and have been studied in many different fields. However, due to the interaction of different sub-systems, a hybrid system is much more complex than a mono-dynamic one, and great challenges are confronted when seeking stable controllers either for linear or nonlinear hybrid systems. The traditional design scheme for such a system, namely, to design a controller for each sub-system separately, cannot yield satisfactory performance, since the switches between sub-systems are not specifically considered. In fact, the controllers constructed in this way are usually of poor performance, or even unstable, as a disastrous effect caused by the alternation of sub-systems. In this paper, considering the aforementioned problem, a control scheme is proposed to design suitable controllers for hybrid systems to achieve overall stability by taking account of the switching behaviors of the system, as well as its sub-systems carefully. The design scheme consists of two steps, wherein the first step aims to design sub-controllers for different sub-systems, usually with different sub-Lyapunov functions, while a common Lyapunov function candidate is composed in the second step to modify the previously designed sub-controllers correspondingly. Following this design scheme, not only the stability, but also the closed-loop performance is successfully guaranteed. Some simulation results are provided to show the satisfactory performance of the proposed design scheme.

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Funding

This work was supported by the National Natural Science Foundation of China under Grant 61633012 and 62003172.

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Authors and Affiliations

  1. Institute of Robotics and Automatic Information System, College of Artificial Intelligence, Nankai University, Tianjin, China

    Zhi Fan, Yongchun Fang, Yinan Wu & Cunhuan Liu

  2. Tianjin Key Laboratory of Intelligent Robotics, Nankai University, Tianjin, China

    Zhi Fan, Yongchun Fang, Yinan Wu & Cunhuan Liu

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  1. Zhi Fan
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Correspondence to Yongchun Fang.

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Fan, Z., Fang, Y., Wu, Y. et al. Controller design for switched systems with nonlinear sub-systems using common Lyapunov functions. Nonlinear Dyn 107, 2275–2289 (2022). https://doi.org/10.1007/s11071-021-07012-y

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