時(shí)滯系統(tǒng)穩(wěn)定性分析
——齊次多項(xiàng)式Lyapunov泛函方法

劉興文

(西南民族大學(xué)電氣信息工程學(xué)院，四川 成都 610041)

穩(wěn)定性是動(dòng)力學(xué)系統(tǒng)最重要的性質(zhì)之一，在數(shù)學(xué)和工程領(lǐng)域得到廣泛的研究[1-3].眾所周知，Lyapunov理論是分析穩(wěn)定性最有效和最流行的工具，其核心是構(gòu)造合適的Lyapunov函數(shù)(無時(shí)滯系統(tǒng))[4-5]或Lyapunov泛函(時(shí)滯系統(tǒng))[6-8].

由于計(jì)算技術(shù)不斷發(fā)展，二次Lyapunov函數(shù)得到廣泛應(yīng)用，所得的穩(wěn)定性判據(jù)一般用線性矩陣不等式來描述[9].對(duì)時(shí)滯系統(tǒng)，二次Lyapunov泛函是分析穩(wěn)定性的有力工具[10-13].然而在很多情況下，用二次Lyapunov函數(shù)或二次Lyapunov泛函獲得低保守性的穩(wěn)定性判據(jù)相當(dāng)不易[14-15].因此，需要尋求穩(wěn)定性分析的新方法.最近，協(xié)正多項(xiàng)式Lyapunov函數(shù)(齊次多項(xiàng)式Lyapunov函數(shù)的一種特殊形式)被用于任意切換信號(hào)的切換系統(tǒng)[16].Chesi等人提出一種無保守性線性矩陣不等式條件驗(yàn)證滿足滯留時(shí)間的切換系統(tǒng)的指數(shù)穩(wěn)定性[17]，這啟發(fā)了廣大學(xué)者構(gòu)造高次多項(xiàng)式Lyapunov函數(shù)，而不是二次Lyapunov函數(shù)，分析動(dòng)力學(xué)系統(tǒng)的穩(wěn)定性[17-20].

在此背景下，人們開始用多項(xiàng)式Lyapunov泛函研究時(shí)滯系統(tǒng).然而，多項(xiàng)式Lyapunov泛函方法尚未得到深入研究.文獻(xiàn)[21-22]嘗試采用該方法建立時(shí)滯系統(tǒng)的穩(wěn)定性條件.需要注意的是，這兩篇文獻(xiàn)的主要推證有誤.因此，本文將進(jìn)一步探索多項(xiàng)式Lyapunov泛函.

本文結(jié)構(gòu)安排如下:第1節(jié)介紹了預(yù)備知識(shí)，第2節(jié)給出主要結(jié)果，第3節(jié)給出一個(gè)數(shù)值例子，第4節(jié)總結(jié)全文.

1 問題陳述及預(yù)備知識(shí)

2 主要結(jié)果

3 數(shù)值例子

本節(jié)給出一個(gè)數(shù)值例子驗(yàn)證所得的理論結(jié)果.

考慮下面的系統(tǒng)方程:

表1 時(shí)滯的上界:時(shí)變時(shí)滯(q＝2)Table 1 Upper bound of delays:Time-varying delays(q＝2)

4 結(jié)論

本文針對(duì)時(shí)滯系統(tǒng)提出一種齊次多項(xiàng)式Lyapunov泛函方法，建立了系統(tǒng)的穩(wěn)定性條件.數(shù)值例子表明本文給出的方法對(duì)快速變化的時(shí)滯顯著效果.

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