广义时滞系统的鲁棒同时H_∞控制
|
摘要
讨论不确定广义系统时滞相关的鲁棒同时H_∞控制问题。首先,通过引入Lyqpunov-Krasovskii泛函以线性矩阵不等式(LMI)形式给出时滞相关的新有界实引理,使得系统正则、无脉冲、稳定且具有H_∞性能。其次,在有界实引理基础上得到广义闭环系统的同时H_∞控制器存在的时滞相关的充分条件。进一步推广到不确定广义时滞闭环系统中,并给出鲁棒同时H_∞控制器的的设计方法。控制器可由一组线性矩阵不等式(LMI)求解得到,避免了系统中矩阵的分解。最后,由仿真算例说明该设计方法的有效性。
This paper considers the problems of time delay dependent robust simultaneous H-infinity control for uncertain singular systems.Firstly,the new bounded real lemma is given in the terms of linear matrix inequalities(LMI) by introducing Lyapunov-Krasovskii functional,which makes the uncertain singular system regular,impulse free,stable and have H-infinity performance.Secondly,delay dependent sufficient condition of the simultaneous H-infinity controller for singular closed-loop systems is derived based on the bounded real lemma.Moreover,the approach is generalized to regular closed-loop systems with time delay and admissible uncertainties,and the design approach for robust simultaneous H-infinity controller is gained.All the obtained results are formulated by strict LMI involving no decomposition of system matrixes.Finally,a numerical example is provided to demonstrate the availability of the proposed method.
This paper considers the problems of time delay dependent robust simultaneous H-infinity control for uncertain singular systems.Firstly,the new bounded real lemma is given in the terms of linear matrix inequalities(LMI) by introducing Lyapunov-Krasovskii functional,which makes the uncertain singular system regular,impulse free,stable and have H-infinity performance.Secondly,delay dependent sufficient condition of the simultaneous H-infinity controller for singular closed-loop systems is derived based on the bounded real lemma.Moreover,the approach is generalized to regular closed-loop systems with time delay and admissible uncertainties,and the design approach for robust simultaneous H-infinity controller is gained.All the obtained results are formulated by strict LMI involving no decomposition of system matrixes.Finally,a numerical example is provided to demonstrate the availability of the proposed method.
引文
[1]geometry,and the simultaneous stabilization problem[J],IEEE Trans.Automat.Control,1982,27(1):895-903.
[2]曹永岩,孙优贤.同时镇定问题研究[J].信息与控制,1998,27(3):161-166 Saeks R and Murray J,Fractional representation,algebraic.Cao Y Y.Sun YX.The study of simultaneous stabilization[J].Information and control,1998,27(3):161-166
[3]Cao YY,Sun YX,Lam J.Simultaneous stabilization via static output feedback and state feedback[J].IEEE Trans.Aut.Contr.,1999,44(5):1277-1282
[4]关强,何冠生,王龙,郁文生,等.线性系统的同时镇定问题[J].控制理论与应用,2011,28(1):1-10.Guan Q,He G S,Wang L,Yu W S,et al.Simultanous stabilization of linear systems[J].Control Theroy&Application,2011,28(1):1-10.
[5]ZamesG Feedback and optimal sensitivity:modle reference transformations,multiplicative seminorms and approximate invers[J].IEEE Transaction on Automatic Control,1981,26(1):831-847.
[6]申铁龙.H_∞岫控制理论与应用[M].清华大学出版社,1996.Shen T L.H_∞Control Theroy and Applications[M].TsingHua University Press,1996.
[7]王德进.H_2H_∞优化控制理论[MI.哈尔滨工业大学出版社,2001.Wang D J.H_2H_∞Optimal Control Theroy[M].Harbin Institute of Technology Press,2001.
[8]徐胜元,牛玉刚.参数不确定性奇异系统的鲁棒控制[J].自动化学报,2001,3(7):397-400.Xu S Y,Niu Y G Robust H_∞control for singular systems with parameter uncertainty[J].ActaAutomaticaSinica,2001,3(7):397-400.
[9]王娟,张秀华.基于LMI的一类不确定奇异系统的鲁棒控制[J].控制工程,2013,20(4):691-693.Wang J,Zhang X H.Robust control of a uncertain generalized systems based on the LMI[J].Control Engineering of China,2013,20(4):691-693.
[10]CampbellL.Singular systems of differential equations[M].San Francisco:Pitman,1980.
[11]DaiL.Singularcontrol systems[M].Berlin,Germany:Springer-Verlag,1989.Gu K,KharitonvV L,Chen J.Stability of Time-Delay.
[12]Gu K,KharitonvV L,Chen J.Stability of Time-Delay Systems[M].Boston:Birhauser,2003.
[13]马跃超,张娜,梁建云.一类时滞广义系统的非脆弱H_∞控制[J].控制工程,2013,20(4):703-706.Ma Y C,Zhang N,Liang J Y.Non-fragile H_∞control for a class of singular time-delay systems[J].Control Engineering of China,2013,20(4):703-706.
[14]Xu S Y,LamJand Yang C W.Robust H_∞control for uncertain singular systems with state delay[J].Intemational Journal of Robust and Nonlinear Control,2003,13(5):1213-1223.
[15]XuSY,Dooren PV,Stefan R,et al.Robust stability and stabilization for singular systems with state delay and parameter uncertainty[J].IEEETransactiononAutomaticControl,2002,47(7):1122-1128.
[16]Ji X F,Su H Y,Chu J.An LMI approach to robust H-infinity control for uncertain singular time-delay systems[J].Journal of Control Theory andApplication,2006,4(3):361-366.
[17]FridmanE.Effects of small delays on stabilityof singular perturbed systems[J].Automatica,2002,38(5):897-902.
[18]Lu RQ,Su H Y,ChuJ.A memoryless robust stabilizing controller for a class of uncertain singular time delay systems[J].Int.Journal of Systems Science,2004,36(10):973-981.
[19]张先明.基于积分不等式方法的时滞相关鲁棒控制研究[D].湖南:中南大学.2006.ZhangXM.Study on dely-dependent robust control based on an integral inequality approach[D].Hunan:Central South University,2006.
[20]张先明,吴敏,何勇.线性时滞广义系统的时滞相关稳定性[J].电路与系统学报,2003,8(4):3-7.Zhang X M,Wu M,He Y.On delay-dependent stability for linear singular systems with delay[J].Journal of circuits and systems,2003,8(4):3-7.
[21]高焕丽,胥布工,张友刚.两类不确定奇异时滞系统鲁棒稳定的相关型判据[J].系统工程与电子技术,2008,30(1):135-140.Gao H L,Xu B Q Zhang Y G Delay-dependent robust stability cirteria for two classes of uncertain singular time-delay systems[J].Systems Engineering and electronics,2008,30(1):135-140.
[22]刘雅妹,兰奇逊,梁家荣.时滞相关广义系统的H_∞控制改进方法[J].河南大学学报(自然科学版)'2011,41(1):13-16.Liu Y M,Lan Q X,Liang J R.An improved method of delay-dependent H_∞control for singular systems with time-dely[J].Journal of Henan University,2011,41(l):13-16.
[23]李中彬.不确定时滞相关广义系统的H_∞鲁棒控制[J].科技通报,2012,28(2):42-46.LI Z B.Dely-dependent robust H_∞control and uncertain Singular systems with time-dely[J].Bulletin of science and technology,2012,28(2):42-46.
[24]Wu,Z.G,Park,J.H.,Su,H.,Chu,J.Reliable passive control for singular systems with time-varying delays[J].,Journal of Process Control,2013,23(8):1217-1228.
[25]Xu SY,Lam J,Zou Y,Li JZ.Robust admissibility of time-varying singular systems with commensurate time delays[J].Automatica2009,45(11):2714-2717.
[26]Liu Z Y,Lin C,Chen B.A neutral system approach to stability of singular timedelay systems[J].Jurnal Franklin Institute,2014,351(10):4939-4948.
[27]Fridman E.Stability of linear descriptor systems with delay:a Lyapunov-based approach[J].Journal of Mathematical Analysis and Applications,2002,273(1):24-44.
[28]杨冬梅,孟新全.不确定时滞离散切换广义系统的鲁棒H∞控制[J].东北大学学报(自然科学版).2015,35(3):309-313.Yang D M,Meng X Q.Robust H∞Control for Discrete-Time Switched Singular Systemswith Norm-Bounded Uncertainties and Time-Delay[J],Journal of Northeastern University(Natural Science),2015,35(3):309-313.
[29]刘雅妹,兰奇逊,梁家荣.时滞相关广义系统的H∞控制的改进方法[J].河南大学学报(自然科学版),2011,41(1),13-17.Liu Y M,Lan Q X,Liang,JR.An improved method of delay dependent H∞control for singular systems with time delay[J].,Journal of Hennan University(Natural Science),2011,41(1):13-17.
[30]张伟,时宝,盖明久,广义时滞系统的时滞相关型H∞控制[J],控制理论与应用,2009,36(12):1430-1434.Zhang W,Shi B,Gai MJ.Delay-dependent H-infinity control for descriptor time-delay systems[J].Control Theory&Applications,2009,36(12):1430-1434.
[31]张先明,吴敏,何勇.线性时滞广义系统的时滞相关H∞控制[J].控制理论与应用,2005,22(4):649-652.ZHANG,X M,WU M,HE Y.Delay-dependent H∞control for linear descriptor systems with time delay[J].Control Theory&Applications,2005,22(4):649-652.
[32]杜昭平,岳东,胡松林,张美生.一类状态时滞广义串级控制系统的H∞镇定[J].控制理论与应用,2013,30(6):754-759.Du Z P,Yue D,Hu S L,Zhang M S.H-infinity stabilization for a class of singular cascade control systems with time-delay in state[J].,Control Theory&Applications,2013,30(6):754-759.
[33]李杰坤,穆春来.一类不确定时滞广义系统的鲁棒H∞控制[J].云南师范大学学报(自然科学版),2009,29(4):22-26.LI J K,Mu C L,non-fragile Robust control for a class of singular systems with time-delay[J].Journal of Yunnan Normal University(Natural Science),2009,29(4):22-26.
[34]王岩岩,刘伟,汪志鸣.不确定时滞奇异摄动系统的鲁棒H∞控制[J],北京工业大学学报,2016,42(2):217-222.Wang Y Y,Liu W,Wang Z M.Robust H∞control for uncertain singular perturbed systems with time delay[J],Journal of Beijing University of Technology,2016,42(2):217-222.
[35]张琴.一类不确定时变时滞系统的鲁棒镇定:积分不等式方法[J].,中国科技博览,2011,3(12):63-64.Zhang Q.Roubust stabilization for a class of uncertain time-varying systems:integral inequality method[J].,Chinese science andTechnology Expo,2011,3(12):63-64.
[36]马大中,王占山,冯健.区间时滞系统的非脆弱鲁棒H∞控制[J].东北大学学报(自然科学版),2011,32(4):476-480.Ma D Z,Wang Z S,Feng J.Non-fragile robust H∞control for linear systems with interval time-varying delay[J].School of Information Science and Engineering,Northeastem University,2011,32(4):476-480.
[37]李涛,张合新,孙鹏.含区间时变时滞的线性不确定系统统鲁棒稳定性新判据[J].控制与决策,2010,25(6):953-957Li T,Zhang H X,Sun P.New robust stability criteria for uncertain linear systems with interval time-varying delay[J],Control and Decision,2010,25(6):953-957.
[38]屈百达,张玉军,廖建庆.一类不确定线性时变时滞系统的鲁棒H∞控制[J].控制工程,2016,23(1):91-96.Qu,B D,Zhang Y J,Liao J Q.Robust H∞Controller Design for a Class of Uncertain Linear Systems withTime-varying and Control Delay
[39]苏宏业,褚健,鲁仁全等.不确定时滞系统的鲁棒控制理论[M].北京:科学出版社,2007.Su H Y,Zhu J,Lu R Q,et al.The robust control theory for uncertain time-delay systems[M].Beijing:Science Press,2007
[40]孙新柱,江明.不确定时滞系统的鲁棒稳定性研究[J].安徽工程科技学院学报,2008,23(2):39-41.Sun X Z,Jiang M.Study of robust stability for uncertain time-delay system[J]Journal of Anhui University of Technology and Science,2008,23(2):39-41.
[41]马喜成,李炜.不确定时滞系统的鲁棒容错控制[J].兰州理工大学学报,2007,33(3):79-82.Ma X C,Li W.Robust fault-tolerent control of uncertain time-delay system[J].Journal of Lanzhou University of Technology,2007,33(3):79-82.
[42]Dong Y,Won S.An improvement on delay and its time-derivative dependent robust stability of time-delayed linear systems with uncertainty[J].IEEE Trans on Automatic Control,200247(2):407-408.
[43]张秀华,王娟,刘少廷.不确定时滞广义双线性系统的鲁棒镇定[J].控制工程,2010,17(5):584-586.
[44]Zhang X M,Wu M,She J H.Delay-dependent stabilization of linear systems with time-varying state and input delays[J].Automatica,2005,41(8):1405-1412.
[45]Zhang X H,Wang J,Liu S T.Robust Stabilization of Uncertain Time-delay Singular Bilinear Systems[J].Control Engineering of China.2010,17(5):584-586.
[46]王天成.不确定广义时滞系统的鲁棒稳方法研究[J].控制与决策,2007,22(9):1070-1072Wang T C.Improved robust stability criteria for uncertain descriptor system with time-delay[J].Control and Decision,2007,22(9):1070-1072.
[47]俞立.鲁棒控制·线性矩阵不等式方法[M].北京:清华大学出版社,2002.Yu L.Robust control-linear matrix inequality Method[M].Beijing:Tsinghua University press,2002.
[2]曹永岩,孙优贤.同时镇定问题研究[J].信息与控制,1998,27(3):161-166 Saeks R and Murray J,Fractional representation,algebraic.Cao Y Y.Sun YX.The study of simultaneous stabilization[J].Information and control,1998,27(3):161-166
[3]Cao YY,Sun YX,Lam J.Simultaneous stabilization via static output feedback and state feedback[J].IEEE Trans.Aut.Contr.,1999,44(5):1277-1282
[4]关强,何冠生,王龙,郁文生,等.线性系统的同时镇定问题[J].控制理论与应用,2011,28(1):1-10.Guan Q,He G S,Wang L,Yu W S,et al.Simultanous stabilization of linear systems[J].Control Theroy&Application,2011,28(1):1-10.
[5]ZamesG Feedback and optimal sensitivity:modle reference transformations,multiplicative seminorms and approximate invers[J].IEEE Transaction on Automatic Control,1981,26(1):831-847.
[6]申铁龙.H_∞岫控制理论与应用[M].清华大学出版社,1996.Shen T L.H_∞Control Theroy and Applications[M].TsingHua University Press,1996.
[7]王德进.H_2H_∞优化控制理论[MI.哈尔滨工业大学出版社,2001.Wang D J.H_2H_∞Optimal Control Theroy[M].Harbin Institute of Technology Press,2001.
[8]徐胜元,牛玉刚.参数不确定性奇异系统的鲁棒控制[J].自动化学报,2001,3(7):397-400.Xu S Y,Niu Y G Robust H_∞control for singular systems with parameter uncertainty[J].ActaAutomaticaSinica,2001,3(7):397-400.
[9]王娟,张秀华.基于LMI的一类不确定奇异系统的鲁棒控制[J].控制工程,2013,20(4):691-693.Wang J,Zhang X H.Robust control of a uncertain generalized systems based on the LMI[J].Control Engineering of China,2013,20(4):691-693.
[10]CampbellL.Singular systems of differential equations[M].San Francisco:Pitman,1980.
[11]DaiL.Singularcontrol systems[M].Berlin,Germany:Springer-Verlag,1989.Gu K,KharitonvV L,Chen J.Stability of Time-Delay.
[12]Gu K,KharitonvV L,Chen J.Stability of Time-Delay Systems[M].Boston:Birhauser,2003.
[13]马跃超,张娜,梁建云.一类时滞广义系统的非脆弱H_∞控制[J].控制工程,2013,20(4):703-706.Ma Y C,Zhang N,Liang J Y.Non-fragile H_∞control for a class of singular time-delay systems[J].Control Engineering of China,2013,20(4):703-706.
[14]Xu S Y,LamJand Yang C W.Robust H_∞control for uncertain singular systems with state delay[J].Intemational Journal of Robust and Nonlinear Control,2003,13(5):1213-1223.
[15]XuSY,Dooren PV,Stefan R,et al.Robust stability and stabilization for singular systems with state delay and parameter uncertainty[J].IEEETransactiononAutomaticControl,2002,47(7):1122-1128.
[16]Ji X F,Su H Y,Chu J.An LMI approach to robust H-infinity control for uncertain singular time-delay systems[J].Journal of Control Theory andApplication,2006,4(3):361-366.
[17]FridmanE.Effects of small delays on stabilityof singular perturbed systems[J].Automatica,2002,38(5):897-902.
[18]Lu RQ,Su H Y,ChuJ.A memoryless robust stabilizing controller for a class of uncertain singular time delay systems[J].Int.Journal of Systems Science,2004,36(10):973-981.
[19]张先明.基于积分不等式方法的时滞相关鲁棒控制研究[D].湖南:中南大学.2006.ZhangXM.Study on dely-dependent robust control based on an integral inequality approach[D].Hunan:Central South University,2006.
[20]张先明,吴敏,何勇.线性时滞广义系统的时滞相关稳定性[J].电路与系统学报,2003,8(4):3-7.Zhang X M,Wu M,He Y.On delay-dependent stability for linear singular systems with delay[J].Journal of circuits and systems,2003,8(4):3-7.
[21]高焕丽,胥布工,张友刚.两类不确定奇异时滞系统鲁棒稳定的相关型判据[J].系统工程与电子技术,2008,30(1):135-140.Gao H L,Xu B Q Zhang Y G Delay-dependent robust stability cirteria for two classes of uncertain singular time-delay systems[J].Systems Engineering and electronics,2008,30(1):135-140.
[22]刘雅妹,兰奇逊,梁家荣.时滞相关广义系统的H_∞控制改进方法[J].河南大学学报(自然科学版)'2011,41(1):13-16.Liu Y M,Lan Q X,Liang J R.An improved method of delay-dependent H_∞control for singular systems with time-dely[J].Journal of Henan University,2011,41(l):13-16.
[23]李中彬.不确定时滞相关广义系统的H_∞鲁棒控制[J].科技通报,2012,28(2):42-46.LI Z B.Dely-dependent robust H_∞control and uncertain Singular systems with time-dely[J].Bulletin of science and technology,2012,28(2):42-46.
[24]Wu,Z.G,Park,J.H.,Su,H.,Chu,J.Reliable passive control for singular systems with time-varying delays[J].,Journal of Process Control,2013,23(8):1217-1228.
[25]Xu SY,Lam J,Zou Y,Li JZ.Robust admissibility of time-varying singular systems with commensurate time delays[J].Automatica2009,45(11):2714-2717.
[26]Liu Z Y,Lin C,Chen B.A neutral system approach to stability of singular timedelay systems[J].Jurnal Franklin Institute,2014,351(10):4939-4948.
[27]Fridman E.Stability of linear descriptor systems with delay:a Lyapunov-based approach[J].Journal of Mathematical Analysis and Applications,2002,273(1):24-44.
[28]杨冬梅,孟新全.不确定时滞离散切换广义系统的鲁棒H∞控制[J].东北大学学报(自然科学版).2015,35(3):309-313.Yang D M,Meng X Q.Robust H∞Control for Discrete-Time Switched Singular Systemswith Norm-Bounded Uncertainties and Time-Delay[J],Journal of Northeastern University(Natural Science),2015,35(3):309-313.
[29]刘雅妹,兰奇逊,梁家荣.时滞相关广义系统的H∞控制的改进方法[J].河南大学学报(自然科学版),2011,41(1),13-17.Liu Y M,Lan Q X,Liang,JR.An improved method of delay dependent H∞control for singular systems with time delay[J].,Journal of Hennan University(Natural Science),2011,41(1):13-17.
[30]张伟,时宝,盖明久,广义时滞系统的时滞相关型H∞控制[J],控制理论与应用,2009,36(12):1430-1434.Zhang W,Shi B,Gai MJ.Delay-dependent H-infinity control for descriptor time-delay systems[J].Control Theory&Applications,2009,36(12):1430-1434.
[31]张先明,吴敏,何勇.线性时滞广义系统的时滞相关H∞控制[J].控制理论与应用,2005,22(4):649-652.ZHANG,X M,WU M,HE Y.Delay-dependent H∞control for linear descriptor systems with time delay[J].Control Theory&Applications,2005,22(4):649-652.
[32]杜昭平,岳东,胡松林,张美生.一类状态时滞广义串级控制系统的H∞镇定[J].控制理论与应用,2013,30(6):754-759.Du Z P,Yue D,Hu S L,Zhang M S.H-infinity stabilization for a class of singular cascade control systems with time-delay in state[J].,Control Theory&Applications,2013,30(6):754-759.
[33]李杰坤,穆春来.一类不确定时滞广义系统的鲁棒H∞控制[J].云南师范大学学报(自然科学版),2009,29(4):22-26.LI J K,Mu C L,non-fragile Robust control for a class of singular systems with time-delay[J].Journal of Yunnan Normal University(Natural Science),2009,29(4):22-26.
[34]王岩岩,刘伟,汪志鸣.不确定时滞奇异摄动系统的鲁棒H∞控制[J],北京工业大学学报,2016,42(2):217-222.Wang Y Y,Liu W,Wang Z M.Robust H∞control for uncertain singular perturbed systems with time delay[J],Journal of Beijing University of Technology,2016,42(2):217-222.
[35]张琴.一类不确定时变时滞系统的鲁棒镇定:积分不等式方法[J].,中国科技博览,2011,3(12):63-64.Zhang Q.Roubust stabilization for a class of uncertain time-varying systems:integral inequality method[J].,Chinese science andTechnology Expo,2011,3(12):63-64.
[36]马大中,王占山,冯健.区间时滞系统的非脆弱鲁棒H∞控制[J].东北大学学报(自然科学版),2011,32(4):476-480.Ma D Z,Wang Z S,Feng J.Non-fragile robust H∞control for linear systems with interval time-varying delay[J].School of Information Science and Engineering,Northeastem University,2011,32(4):476-480.
[37]李涛,张合新,孙鹏.含区间时变时滞的线性不确定系统统鲁棒稳定性新判据[J].控制与决策,2010,25(6):953-957Li T,Zhang H X,Sun P.New robust stability criteria for uncertain linear systems with interval time-varying delay[J],Control and Decision,2010,25(6):953-957.
[38]屈百达,张玉军,廖建庆.一类不确定线性时变时滞系统的鲁棒H∞控制[J].控制工程,2016,23(1):91-96.Qu,B D,Zhang Y J,Liao J Q.Robust H∞Controller Design for a Class of Uncertain Linear Systems withTime-varying and Control Delay
[39]苏宏业,褚健,鲁仁全等.不确定时滞系统的鲁棒控制理论[M].北京:科学出版社,2007.Su H Y,Zhu J,Lu R Q,et al.The robust control theory for uncertain time-delay systems[M].Beijing:Science Press,2007
[40]孙新柱,江明.不确定时滞系统的鲁棒稳定性研究[J].安徽工程科技学院学报,2008,23(2):39-41.Sun X Z,Jiang M.Study of robust stability for uncertain time-delay system[J]Journal of Anhui University of Technology and Science,2008,23(2):39-41.
[41]马喜成,李炜.不确定时滞系统的鲁棒容错控制[J].兰州理工大学学报,2007,33(3):79-82.Ma X C,Li W.Robust fault-tolerent control of uncertain time-delay system[J].Journal of Lanzhou University of Technology,2007,33(3):79-82.
[42]Dong Y,Won S.An improvement on delay and its time-derivative dependent robust stability of time-delayed linear systems with uncertainty[J].IEEE Trans on Automatic Control,200247(2):407-408.
[43]张秀华,王娟,刘少廷.不确定时滞广义双线性系统的鲁棒镇定[J].控制工程,2010,17(5):584-586.
[44]Zhang X M,Wu M,She J H.Delay-dependent stabilization of linear systems with time-varying state and input delays[J].Automatica,2005,41(8):1405-1412.
[45]Zhang X H,Wang J,Liu S T.Robust Stabilization of Uncertain Time-delay Singular Bilinear Systems[J].Control Engineering of China.2010,17(5):584-586.
[46]王天成.不确定广义时滞系统的鲁棒稳方法研究[J].控制与决策,2007,22(9):1070-1072Wang T C.Improved robust stability criteria for uncertain descriptor system with time-delay[J].Control and Decision,2007,22(9):1070-1072.
[47]俞立.鲁棒控制·线性矩阵不等式方法[M].北京:清华大学出版社,2002.Yu L.Robust control-linear matrix inequality Method[M].Beijing:Tsinghua University press,2002.