Xiong Jiandong, Wang Yidian, Huo Yanfang, Zhao Hongpeng
College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China.
Entropy (Basel). 2023 Sep 5;25(9):1300. doi: 10.3390/e25091300.
In this paper, stability is studied for a class of switched singular systems containing both stable and unstable modes. By introducing a time-varying piecewise Lyapunov function (TVPLF) and a mode-dependent average dwell time (ADT) switching rule, the computable sufficient conditions for system stability are derived. The time-varying piecewise Lyapunov functions are piecewise continuously differentiable on every mode (but may not be differentiable at the interpolating points of the dwell time). This Lyapunov function method is particularly advantageous in overcoming the limitations of traditional multiple Lyapunov function (MLF) methods, which may not have a feasible solution when dealing with switched systems containing only unstable modes. As such, the TVPLF offers greater flexibility in application. Compared with the conventional ADT switching rule, the mode-dependent ADT switching rule not only enables each mode to have its own ADT but also allows for its own switching strategy. Specifically, the stable mode adopts a slow switching strategy while the unstable mode adopts a fast one, thereby reducing the conservatism of the ADT switching rule. Furthermore, based on the stability analysis, the time-varying controllers are proposed to stabilize the switched singular system, which can be expressed as the sequential linear combination of a series of linear state feedback on each mode. The proposed controllers are continuous for each mode, which are different from the controllers designed through the traditional MLF and MDLF methods, where the controllers designed by traditional MLF are the time-invariant linear state feedback in each mode while the controllers designed by the MDLF are piecewise continuous for each mode.
本文研究了一类包含稳定和不稳定模式的切换奇异系统的稳定性。通过引入时变分段李雅普诺夫函数（TVPLF）和依赖模式的平均驻留时间（ADT）切换规则，推导出了系统稳定性的可计算充分条件。时变分段李雅普诺夫函数在每个模式上是分段连续可微的（但在驻留时间的插值点处可能不可微）。这种李雅普诺夫函数方法在克服传统多重李雅普诺夫函数（MLF）方法的局限性方面特别有利，传统MLF方法在处理仅包含不稳定模式的切换系统时可能没有可行解。因此，TVPLF在应用中具有更大的灵活性。与传统的ADT切换规则相比，依赖模式的ADT切换规则不仅使每个模式都有自己的ADT，还允许有自己的切换策略。具体来说，稳定模式采用慢切换策略，而不稳定模式采用快切换策略，从而降低了ADT切换规则的保守性。此外，基于稳定性分析，提出了时变控制器来稳定切换奇异系统，该控制器可以表示为每个模式上一系列线性状态反馈的顺序线性组合。所提出的控制器在每个模式上是连续的，这与通过传统MLF和MDLF方法设计的控制器不同，传统MLF设计的控制器在每个模式上是时不变线性状态反馈，而MDLF设计的控制器在每个模式上是分段连续的。
