时滞系统稳定性分析——齐次多项式Lyapunov泛函方法

刘兴文

(西南民族大学电气信息工程学院，四川 成都 610041)

稳定性是动力学系统最重要的性质之一，在数学和工程领域得到广泛的研究[1-3].众所周知，Lyapunov理论是分析稳定性最有效和最流行的工具，其核心是构造合适的Lyapunov函数(无时滞系统)[4-5]或Lyapunov泛函(时滞系统)[6-8].

由于计算技术不断发展，二次Lyapunov函数得到广泛应用，所得的稳定性判据一般用线性矩阵不等式来描述[9].对时滞系统，二次Lyapunov泛函是分析稳定性的有力工具[10-13].然而在很多情况下，用二次Lyapunov函数或二次Lyapunov泛函获得低保守性的稳定性判据相当不易[14-15].因此，需要寻求稳定性分析的新方法.最近，协正多项式Lyapunov函数(齐次多项式Lyapunov函数的一种特殊形式)被用于任意切换信号的切换系统[16].Chesi等人提出一种无保守性线性矩阵不等式条件验证满足滞留时间的切换系统的指数稳定性[17]，这启发了广大学者构造高次多项式Lyapunov函数，而不是二次Lyapunov函数，分析动力学系统的稳定性[17-20].

在此背景下，人们开始用多项式Lyapunov泛函研究时滞系统.然而，多项式Lyapunov泛函方法尚未得到深入研究.文献[21-22]尝试采用该方法建立时滞系统的稳定性条件.需要注意的是，这两篇文献的主要推证有误.因此，本文将进一步探索多项式Lyapunov泛函.

本文结构安排如下:第1节介绍了预备知识，第2节给出主要结果，第3节给出一个数值例子，第4节总结全文.

1 问题陈述及预备知识

2 主要结果

3 数值例子

本节给出一个数值例子验证所得的理论结果.

考虑下面的系统方程:

表1 时滞的上界:时变时滞(q＝2)Table 1 Upper bound of delays:Time-varying delays(q＝2)

4 结论

本文针对时滞系统提出一种齐次多项式Lyapunov泛函方法，建立了系统的稳定性条件.数值例子表明本文给出的方法对快速变化的时滞显著效果.

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