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

      2018-03-22 07:10:46劉興文
      關(guān)鍵詞:例子時(shí)滯線性

      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.

      [1]KOLMANOVSKII V,NOSOV V,EDS.Stability of Functional Differential Equations[M].Academic Press,1986.

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      [3]LIU X.Stability criterion of 2-D positive systems with unbounded delays described by Roesser model[J].Asian Journal of Control,2015,17(2):544-553.

      [4]OOBA T,F(xiàn)UNAHASHI Y.Two conditions concerning common quadratic Lyapunov functions for linear systems[J].IEEE Trans.on Automatic Control,1997,42(5):719-722.

      [5]JOHANSSON M,RANTZER A.Computation of piecewise quadratic Lyapunov functions for hybrid systems[J].IEEE Trans on Automatic Control,vol,1998(4):555-559.

      [6]SUN Y G,WANG L.Stabilization of planar discrete-time switched systems:Switched Lyapunov functional approach[J].Nonlinear Analysis:Hybrid Systems,2008,2(4):1062-1068.

      [7]LIU Y,F(xiàn)ENG W.Razumikhin-Lyapunov functional method for the stability of impulsive switched systems with time delay[J].Mathematical& Computer Modelling,2009,49(1):249-264.

      [8]MAZENC F,MALISOFF M.Stability analysis for time-varying systems with delay using linear Lyapunov functionals and a positive systems approach[J].IEEE Trans.on Automatic Control,2016,61(3):771-776.

      [9]BOYD S,GHAOUI E,F(xiàn)ERON E,et al.Linear Matrix Inequalities in System and Control Theory[J].Philadelphia:SIAM,1994.

      [10]WU H N.Delay-dependent stability analysis and stabilization for discrete-time fuzzy systems with state delay:a fuzzy Lyapunov-Krasovskii functional approach[J].IEEE Trans on Systems,Man,& Cybernetics-Part B,2006,36(4):954-962.

      [11]HE Y,WANG Q G,LIN C,et al.Delay-range-dependent stability for systems with time-varying delay[J].Automatica,2007,43(2):371-376.

      [12]SONG Y,F(xiàn)AN J,F(xiàn)EI M,et al.Robust H∞control of discrete switched system with time delay[J].Applied Mathematics& Computation,2008,205(1):159-169.

      [13]XU L,XU D.Mean square exponential stability of impulsive control stochastic systems with time-varying delay[J].Physics Letters A,2009,373(3):328-333.

      [14]DAAFOUZ J,RIEDINGER P,IUNG C.Stability analysis and control synthesis for switched systems:A switched Lyapunov function approach[J].IEEE Trans.on Automatic Control,2002,47(11):1883-1887.

      [15]GEROMEL J C,COLANERI P.Stability and stabilization of discrete time switched systems[J].International Journal of Control,2006,79(7):719-728.

      [16]ZHAO X,LIU X,YIN S,et al.Improved results on stability of continuous-time switched positive linear systems[J].Automatica,2014,50(2):614-621.

      [17]CHESI G,COLANERI P,GEROMEL J C,et al.A nonconservative LMI condition for stability of switched systems with guaranteed dwell time[J].IEEE Trans on Automatic Control,2012,57(5):1297-1302.

      [18]LIU X,ZHAO X.Stability analysis of discrete-time switched systems:A switched homogeneous Lyapunov function method[J].International Journal of Control,2016,89(2):297-305.

      [19]CHESI G.Sufficient and necessary LMI conditions for robust stability of rationally time-varying uncertain systems[J].IEEE Trans on Automatic Control,2013,58(6):1546-1551.

      [20]CHESI G,MIDDLETON R H.H∞and H2norms of 2-D mixed continuous-discrete-time systems via rationally-dependent complex Lyapunov functions[J].IEEE Trans on Automatic Control,2015,60(10):2614-2625.

      [21]ZHANG H,XIA J,ZHUANG G.Improved delay-dependent stability analysis for linear time-delay systems:Based on homogeneous polynomial Lyapunov-Krasovskii functional method[J].Neurocomputing,2016,193:176-180.

      [22]PANG G C,ZHANG K J.Stability of time-delay system with time-varying uncertainties via homogeneous polynomial lyapunov-krasovskii functions[J].International Journal of Automation and Computing,2015,12(6):657-663.

      [23]CHESI G,GARULLI A,TESI A,et al.Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems[M].New York:Springer,2009.

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      [27]LIU X,ZHANG H.New stability criterion of uncertain systems with time-varying delay[J].Chaos,Solitons &Fractals,2005,26(5):1343-1348.

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