Improved stability and stabilisation conditions of uncertain switched time-delay systems

Abstract

This article is concerned with the stability and stabilisation of switched time-delay systems (STDSs) with exponential uncertainty. Based on the Hurwitz convex combination and the energy attenuation principle, an improved state-dependent switching strategy is proposed, which switches to the next modes to obey the minimum energy. This approach fully considers the system dynamic of subsystems, which is more general. Considering the complex switching and delay dynamics, a mode-dependent Lyapunov–Krasovskii functional (LKF) that contains a triple integral term is constructed. The generalised free-matrix-based integral inequality (GFMBII) is used to estimate the integral terms in the derivative of the LKF, and an improved delay-dependent stability criterion is established in the form of linear matrix inequalities (LMIs). Further, to guarantee the stability of the STDSs with a large time-varying delay, a controller that considers the time delay and the exponential uncertainty is designed. Under this controller, a less conservative delay-dependent robust stabilisation criterion for STDSs with exponential uncertainty is established. The validity of the proposed methods is validated by two numerical examples and an application in river pollution control.

Keywords:

• Exponential uncertainty
• generalised free-matrix-based integral inequality (GFMBII)
• Lyapunov–Krasovskii functional (LKF)
• state-dependent switching strategy
• switched time-delay systems (STDSs)
• stability and stabilisation

Disclosure statement

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

Data availability statement

Data sharing is not applicable to this article as no new data were created or analysed in this study.

Additional information

Funding

This work was supported in part by the National Natural Science Foundation of China under Grant 62373373, and in part by the Science and Technology Innovation Program of Hunan Province under Grant 2022RC3051.