切换时滞系统的稳定性研究
1. 大连交通大学, 自动化与电气工程学院, 辽宁大连 116028
2. 大连交通大学, 理学院, 辽宁大连 116028
3. 大连交通大学, 软件学院, 辽宁大连 116028
| 摘 要： | 切换时滞系统是一类具有多模态特性的系统模型，具有重要的理论研究价值。本文使用李亚普诺夫函数法和模型依赖平均驻留时间方法（MDADT）研究了非线性和线性切换时滞系统的指数稳定性问题。将模型依赖平均驻留时间方案引入切换时滞系统，基于模型依赖平均驻留时间设计，以线性矩阵不等式（LMIs）的形式给出了指数稳定的充分条件。切换系统中可以含有不稳定子系统。利用稳定的子系统和不稳定的子系统被激活总时间比，给出了保证系统指数稳定的充分条件。随后对于一类特殊的切换时滞系统，通过求解线性矩阵不等式（LMIs）和某些方程，可以分别得到使系统保持指数稳定所需要的模型依赖平均驻留时间（MDADT）的下界及子系统之间被激活的时间比率。并给出了系统的状态指数衰减的具体形式。证明了在任意切换的条件下，切换闭环系统中的所有信号最终有界，同时每个子系统输出都能在一个小的有界误差范围内跟踪上相应的目标轨迹。最后通过数值算例和MATLAB仿真说明了该方法的有效性。 |
| 关 键 词： | 模型依赖平均驻留时间; 切换时滞系统; 指数稳定性; 李亚普诺夫函数; 线性矩阵不等式（LMI） |
| DOI： | 10.57237/j.se.2023.04.001 |
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收稿日期: 2023-05-05
接受日期: 2023-06-25
在线出版日期: 2023-08-29
Stability of Switching Time Delay Systems
1. School of Electric Engineering and Automation, Dalian Jiaotong University, Dalian 116028, China
2. School of Mathematics, Dalian Jiaotong University, Dalian 116028, China
3. School of Software, Dalian Jiaotong University, Dalian 116028, China
| Abstract: | Switched systems, as a kind of system models with multi-modal characteristics, Therefore, switched systems have important theoretical research value. In this paper, Lyapunov function and model dependent average dwell time method (MDADT) is used to study the exponential stability of nonlinear and linear switched time delay systems. The mode dependent average dwell time scheme is introduced into the system with switched time delay. Based on the mode dependent average dwell time design, the sufficient conditions for exponential stability are given in the form of linear matrix inequalities (LMIs). Exponential stability for switched delay systems consisting of stable and unstable subsystems is addressed. Under the condition that activation time ratio between stable subsystems and unstable ones is not less than a specified constant, sufficient conditions are given to guarantee exponential stability of the switched delay systems. Subsequently, for a special class of switched delay systems, a lower bound on the mode dependent average dwell time and the activation time ratio of subsystems are calculated respectively by solving a set of linear matrix inequalities (LMIs) and certain equations. The concrete form of state exponential decay of the system is given. It is proved that under the condition of arbitrary switching, all the signals in the switched closed loop system are finally bounded, each subsystem output can track the corresponding target trajectory within a small bounded error range. Finally, the effectiveness of the proposed method is demonstrated by numerical examples and MATLAB simulation. |
| Keywords: | Mode Dependent Mean Dwell Time; Switched Delay Systems; Exponential Stability; Lyapunov Function; Linear Matrix Inequality (LMI) |
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