切换系统的鲁棒二次公共Lyapunov函数矩阵寻找算法
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摘要
为了获得不确定线性切换系统稳定性判别的公共二次Lyapunov函数寻找方法,提出了鲁棒公共二次Lyapunov函数的概念,运用矩阵不等式分析,得到了在鲁棒稳定矩阵集对合和不对合的情况下,鲁棒公共二次Lyapunov函数存在的充分性条件以及LMI形式的递推搜寻算法。获得的结果便于计算机实现,对不确定切换系统鲁棒稳定性判别具有一定价值。应用仿真测试验证了其正确性。
To obtain a searching algorithm for the common quadratic Lyapunov function(CQLF) of an uncertain switched system(SS), the concept of the common robust quadratic Lyapunov function(CRQLF) is proposed. In addition, sufficient conditions for the CRQLF, and its corresponding recurrence search algorithm method in LMI forms, are obtained using a matrix inequality analysis when the stable matrix set is both involuntary and voluntary. These results enable easy computer implementation, and are valuable for making robust stability judgments of uncertain SSs. Furthermore, the application simulation test certificates their validity.
To obtain a searching algorithm for the common quadratic Lyapunov function(CQLF) of an uncertain switched system(SS), the concept of the common robust quadratic Lyapunov function(CRQLF) is proposed. In addition, sufficient conditions for the CRQLF, and its corresponding recurrence search algorithm method in LMI forms, are obtained using a matrix inequality analysis when the stable matrix set is both involuntary and voluntary. These results enable easy computer implementation, and are valuable for making robust stability judgments of uncertain SSs. Furthermore, the application simulation test certificates their validity.
引文
[1]LIBERZON D,MORSE A S.Basic problems in stability and design of switched systems[J].IEEE transaction on control systems,1999,19(5):59-70.
[2]DOGRUEL M,OZGUNER U.Stability of a set of matrices:an application to hybrid systems[C]//Proc of IEEE International Symposium on Intelligent Control.[S.l.],1995:59-64.
[3]BELDIMAN O,BUSHNELL L.Stability,linearization and control of switched systems[C]//Proc of American Control Conference.San Diego,CA,1999:2950-2955.
[4]OOBA T,FUNAHASHI Y.Two conditions concerning common quadratic Lyapunov functions for linear systems[J].IEEE trasactions on automatic control,1997,42(5):719-721.
[5]NARENDRA K S,BALAKRISHNAN J A.A common Lyapimov function for stable LTI systems with commuting A-matrices[J].IEEE transactions on automatic control,1994,39(12):2469-2471.
[6]LIBERZON D,MORSE J P.Stability of switched systems:a Lie algebraic condition[J].Systems and control letters,1999,37(3):117-122.
[7]MARGALIOTA M,LIBERZON D.Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions[J].Systems and control letters,2006,55(1):8-16.
[8]CHENG D Z,ZHU Y H,HU Q X.Stabilization of switched systems via common Lyapunov function[C]//Proc of the World Congress on Intelligent Control and Automation(WCICA).[S.l.],2006:183-187.
[9]DAYAWANSA W P,MARTIN C F.A converse Lyapunov theorem for a class of dynamical systems which undergo switching[J].IEEE transactions on automatic control,1999,44(4):751-760.
[10]LIN H,ANTSAKLIS P J.Stability and stabilizability of switched linear systems:a survey of recent results[J].IEEEtransactions on automatic control,2009,54(2):308-322.
[11]CHENG D Z,GUO L,HUANG J.On quadratic Lyapunov functions[J].IEEE transactions on automatic control,2003,48(5):885-890.
[12]ZHU Y H,CHENG D Z,QIN H S.Constructing conmmon Lyapunov functions for a class of stable matrices[J].Acta automatica sinica,2007,33(2):202-204.
[13]BOYD S,GHAOUI L E,FERON E,et al.Linear Matrix Inequalities in system and control theory[M].Philadelphia:SIAM,1994:7-35.
[14]俞立.鲁棒控制-线性矩阵不等式处理方法[M].北京:清华大学出版社,2002.YU Li.Robust control-linear matrix inequality methods[M].Beijing:Tsinghua University Press,2002.
[15]王向东,何南思.温室大棚温湿度跟踪切换最优控制器[J].沈阳工业大学学报,2014,36(5):543-549.WANG Xiangdong,HE Nansi.Tracking switching optimal controller for temperature and humidity of greenhouse[J].Journal of Shenyang university of technology,2014,36(5):543-549.
[2]DOGRUEL M,OZGUNER U.Stability of a set of matrices:an application to hybrid systems[C]//Proc of IEEE International Symposium on Intelligent Control.[S.l.],1995:59-64.
[3]BELDIMAN O,BUSHNELL L.Stability,linearization and control of switched systems[C]//Proc of American Control Conference.San Diego,CA,1999:2950-2955.
[4]OOBA T,FUNAHASHI Y.Two conditions concerning common quadratic Lyapunov functions for linear systems[J].IEEE trasactions on automatic control,1997,42(5):719-721.
[5]NARENDRA K S,BALAKRISHNAN J A.A common Lyapimov function for stable LTI systems with commuting A-matrices[J].IEEE transactions on automatic control,1994,39(12):2469-2471.
[6]LIBERZON D,MORSE J P.Stability of switched systems:a Lie algebraic condition[J].Systems and control letters,1999,37(3):117-122.
[7]MARGALIOTA M,LIBERZON D.Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions[J].Systems and control letters,2006,55(1):8-16.
[8]CHENG D Z,ZHU Y H,HU Q X.Stabilization of switched systems via common Lyapunov function[C]//Proc of the World Congress on Intelligent Control and Automation(WCICA).[S.l.],2006:183-187.
[9]DAYAWANSA W P,MARTIN C F.A converse Lyapunov theorem for a class of dynamical systems which undergo switching[J].IEEE transactions on automatic control,1999,44(4):751-760.
[10]LIN H,ANTSAKLIS P J.Stability and stabilizability of switched linear systems:a survey of recent results[J].IEEEtransactions on automatic control,2009,54(2):308-322.
[11]CHENG D Z,GUO L,HUANG J.On quadratic Lyapunov functions[J].IEEE transactions on automatic control,2003,48(5):885-890.
[12]ZHU Y H,CHENG D Z,QIN H S.Constructing conmmon Lyapunov functions for a class of stable matrices[J].Acta automatica sinica,2007,33(2):202-204.
[13]BOYD S,GHAOUI L E,FERON E,et al.Linear Matrix Inequalities in system and control theory[M].Philadelphia:SIAM,1994:7-35.
[14]俞立.鲁棒控制-线性矩阵不等式处理方法[M].北京:清华大学出版社,2002.YU Li.Robust control-linear matrix inequality methods[M].Beijing:Tsinghua University Press,2002.
[15]王向东,何南思.温室大棚温湿度跟踪切换最优控制器[J].沈阳工业大学学报,2014,36(5):543-549.WANG Xiangdong,HE Nansi.Tracking switching optimal controller for temperature and humidity of greenhouse[J].Journal of Shenyang university of technology,2014,36(5):543-549.