Reset control and ℒ₂-gain analysis of piecewise-affine systems

Abstract

This paper is concerned with reset controller design and ${\mathcal{L}}_2$-gain stability of piecewise-affine systems under the framework of hybrid systems. Firstly, stability conditions of piecewise-affine systems under dynamic state-feedback control are established through bilinear matrix inequality conditions. Secondly, a reset controller with reset rules under the hybrid systems framework is proposed in the sense of Lyapunov and sufficient conditions for exponential and ${\mathcal{L}}_2$-gain stability of the closed-loop systems are provided. Different from the piecewise-affine systems with the dynamic state-feedback controller, the reset controller is designed such that the ${\mathcal{L}}_2$-gain performance of piecewise-affine systems can be enhanced. Thirdly, an LMI approach is proposed to avoid the difficulty of solving the bilinear matrix inequalities. Furthermore, robustness to inflations of the flow and jump sets is established, and robustness to the norm-bounded uncertainties is proposed. Finally, numerical simulations including a robot arm system, an inverted pendulum system and the Chua's circuit system are provided to illustrate the results.

Keywords:

• Reset control
• ${\mathcal{L}}_2$-gain stability
• piecewise-affine system
• hybrid system

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Natural Science Foundation of Jiangsu Province [grant number BK20201340] and China Postdoctoral Science Foundation [grant number 2018M642160].