In this paper, stability and stabilisation of switched time-varying delay systems are investigated. By developing, a multiple discontinuous Lyapunov function approach and using the mode-dependent average dwell time (MDADT) switching signal, sufficient conditions are first proposed to guarantee the exponential stability of the switched system with stable and unstable subsystems where the time-varying delay is considered in the states. Relaxed stability conditions are also obtained by using a form of the Bessel–Legendre inequality. Then stabilisation of the proposed system is presented by designing a sliding mode controller. Sufficient conditions and sufficient relaxed conditions of a reduced-order sliding mode dynamics are derived, and desired sliding surfaces in the form of linear matrix inequalities (LMIs) are also given. The obtained conditions are checked by using the combined genetic algorithm and convex programming techniques. Moreover, some numerical and practical examples are given to illustrate the effectiveness of the obtained theoretical results.

中文翻译：

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切换时变延迟系统的稳定性和稳定性：多重不连续李雅普诺夫函数方法

本文研究了切换时变延迟系统的稳定性和镇定性。通过开发多重不连续李雅普诺夫函数方法并使用依赖于模式的平均驻留时间 (MDADT) 切换信号，首先提出了充分条件来保证具有稳定和不稳定子系统的切换系统的指数稳定性，其中时变延迟为国家考虑。松弛稳定性条件也可以通过使用 Bessel-Legendre 不等式的形式获得。然后通过设计一个滑模控制器来稳定所提出的系统。推导了降阶滑模动力学的充分条件和充分松弛条件，并给出了线性矩阵不等式（LMI）形式的所需滑动面。使用组合遗传算法和凸编程技术检查获得的条件。此外，给出了一些数值和实际例子来说明所获得的理论结果的有效性。