几类离散系统的稳定性分析
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摘要
自20世纪50年代以来,离散系统的理论研究与实际应用工作,逐渐受到控制理论界的广泛重视,并取得了很大成就,使离散控制系统的分析与设计成为控制理论的重要组成部分.
本文针对几类离散系统稳定性进行分析.
在第二章中,研究了一类离散脉冲切换系统的鲁棒H∞控制问题,在满足能稳能检测的条件下,给出了系统鲁棒稳定且具有γ性能的充分条件.
在第三章中,对两类离散系统的稳定性进行了分析,分别给出了这两类离散系统的有限时间稳定性的充分条件,并对第二类离散系统设计了控制器.
在第四章中,考虑了一类广义离散时滞系统,利用Lyapunov理论和LMI处理方法给出了系统Luenberger观测器的充分条件.
最后,对一类离散双线性系统的反馈问题进行分析,利用Lvapunov理论和LaSalle不变原理以LMIs的形式给出了系统镇定的条件.
Since 1950s,the discrete-time systems theoretical research and practical application of the discrete-time systems,have been considered and made great achievements, so that discrete control systems analysis and design of control theory are important component.
In this paper,we consider several types of discrete-time systems'stability.
In the second chapter, the problem of robust H∞control for a class of impulsive switched systems is discussed.On the conditions of stabilization and detected,sufficient conditions of the robust stability andγperformance are given.
In the third chapter,the stability of two kinds of discrete systems is analyzed.The finite-time stability of the discrete systems sufficient conditions,and sufficient conditions for the second discrete system design of the controller are presented.
In the fourth chapter,for the generalized discrete time-delay systems, using Lya-punov theory and LMI approach the sufficient conditions for Luenberger observer of this type of systems are derived.
Finally, a class of discrete bilinear systems of the feedback problems are ana-lyzed. Using Lyapunov theory and LaSalle invariance principle the sufficient conditions in the form of LMIs are given.
本文针对几类离散系统稳定性进行分析.
在第二章中,研究了一类离散脉冲切换系统的鲁棒H∞控制问题,在满足能稳能检测的条件下,给出了系统鲁棒稳定且具有γ性能的充分条件.
在第三章中,对两类离散系统的稳定性进行了分析,分别给出了这两类离散系统的有限时间稳定性的充分条件,并对第二类离散系统设计了控制器.
在第四章中,考虑了一类广义离散时滞系统,利用Lyapunov理论和LMI处理方法给出了系统Luenberger观测器的充分条件.
最后,对一类离散双线性系统的反馈问题进行分析,利用Lvapunov理论和LaSalle不变原理以LMIs的形式给出了系统镇定的条件.
Since 1950s,the discrete-time systems theoretical research and practical application of the discrete-time systems,have been considered and made great achievements, so that discrete control systems analysis and design of control theory are important component.
In this paper,we consider several types of discrete-time systems'stability.
In the second chapter, the problem of robust H∞control for a class of impulsive switched systems is discussed.On the conditions of stabilization and detected,sufficient conditions of the robust stability andγperformance are given.
In the third chapter,the stability of two kinds of discrete systems is analyzed.The finite-time stability of the discrete systems sufficient conditions,and sufficient conditions for the second discrete system design of the controller are presented.
In the fourth chapter,for the generalized discrete time-delay systems, using Lya-punov theory and LMI approach the sufficient conditions for Luenberger observer of this type of systems are derived.
Finally, a class of discrete bilinear systems of the feedback problems are ana-lyzed. Using Lyapunov theory and LaSalle invariance principle the sufficient conditions in the form of LMIs are given.
引文
[1]D.Liberzon,A.S.Morse.Basic problems in Stability and Design of Switched Systems [J]. IEEE Control System Magazine,1999,19(5):59-70.
[2]J.P.Hespanh,A.S.Morse.Switching between stabilizing controllers[J].Automatic,2002, 38(11):1905-1917.
[3]M.S.Branicky. Stability of Switched an Hybrid Systems[C]. In Proceedings of the IEEE Conference Decision and Control, Lake Buena Vista,1994:3498-3503.
[4]Zhang Xiaoli,Liu Yuzhong,Zhao Jun.Asymototic stability of a class of discrete switched systems [J]. Control Theory and Applications,2002; 19(5):774-776.
[5]P.Peleties, R.Decarlo.Asymptotic Stability of m-Switched Systems using Lyapunov-Like Functions[C]. In Proceedings of the American Control Conference,Boston,1991:1679-1684.
[6]S.Pettersson, B.Lennartson.LMI for Stability and Robustness of Hybrid Systems[C].In Proceedings of the American Control Conference;Albuquerque;1997:1714-1718.
[7]S.Pettersson. B.Lennartson.Exponential Stability Analysis of Nonlinear Systems using LMIs[C].In Proceedings of 36th IEEE Conference Decision an Control,San Diego,CA,1997:199-204.
[8]S.Pettersson, B.Lennartson.Exponential Stability of Hybrid Systems using Piecewise Quadratic Lyapunov Functions Resulting in an LMI Problems [C]. In Proceedings of the 14th IFAC World Congress, Beijing,China,1999.
[9]Xu Honglei,Liu Xinzhi.Robust exponential stabilization of a class of impulisive switched systems [J]. Techniques of Automatic and Applications,2004,23(11):14-16.
[10]Li jiao,Liu yuzhong.Roust H-infinity dynamic output feedback control for impulsive switched systems [J]. Control Theory and Applications,2008,25(3).
[11]Zhang Hongtao,Liu Xinzhi.Roust H-infinity control on impulsive switched systems with disturbance[J].Control Theory and Applications,2004,21(2):261-266.
[12]P.Dorato.Short time stability in linear time-varying systems.In Proc.IRE Intern.Convention Record Pt.4,1961,pp.83-87.
[13]F.Amato.M.Ariola,and C.Cosentino.Finite-time stabilization via dynamic output feed-back. Automatica.vol.42.pp.337-342,2006.
[14]F.Amato,M.Ariola,M.Carbone and C.Cosentino.Finite-time output feedback control of linear systems via differential linear matrix conditions.In Proc.Conf.Decision Control.San Diego,CA,2006,pp.5371-5375.
[15]F.Amato.Finite-time control of discrete-time linear system.IEEE Transaction on auto-matic control,2005,50(5):724-729.
[16]R.Ambrosino,F.Calabrese.C.Cosentino and Gianmaria De Tommasi.Sufficient conditions for finite-time Stability of impulsive Dynamical Systems.IEEE Transaction on automatic control,vol.54,No.4.
[17]F.Amato,R.Ambrosino.M Ariola,C.Cosentino.Finite-time stability of linear time-varying systems with jumps.Automia,2009.
[18]S.P.Bhat and D.S.Bernstein,Finite-time stability of continuous autonomous systems.SIAM J.Control Optim.,vol.38,no.3,pp.751-766,2000.
[19]S.G.Nersesov and W.M.Haddad.Finite-time stabilization of nonlinear impulsive dynam-ical systems.In Proc.Control Conf.,Kos,Greece,Jul.2007,pp.91-98.
[20]S.Pettersson and B.Lennartson.Hybird System Stability and Robustness verication using linear matrix inequalities.Int. J.Control,vol.75.no.16-17.pp1335-1355,2002.
[21]F.Amato,M.Ariola,and P.Dorato. Finite-Time control of linear time-varying systems subject to parametric uncertainties and disturbances. Automatica,vol.37,no.9.pp.1459-1463,2001.
[22]L.Weiss and E.Infante.Finite time stability ander perturbing forces and on product spaces.IEEE Trans.Automat.Control,vol.AC-12,no.1,pp.54-59,feb.1967.
[23]P.O.Gutman.Stabilizing Controllers for Bilinear Systems.IEEE Transaction on Auto-matic Control vol.25,pp.302-306,1980.
[24]C.Bruni,G.Di Pliio,and G.Koch.Bilinear Systems:An Appealing Class of "Nearly Linear" Systems in Theory and Applications.IEEE Transactions on Automatic Control, vol.19,pp.334-348,1974.
[25]R.Longchamp. Stable Feedback Control of Bilinear Systems.IEEE Transactions on Auto-matic Control vol.25,pp.302-302,1980.
[26]R.R.Mohler and A.Y.Khapalov.Bilinear Control and Applications.vol.105,pp.621-637,2000.
[27]T.Yang.Impulsive Control Theory.Springer.2001.
[28]R.Boel,G.Stremersch.Discrets event systems:Analysis and Control.Springer.2000.
[29]M.Darouach.Linear functional observers for systems with Delays in State Variables:The Discrete-Time Case.IEEE Trans. Automat.Contr.,AC-50,no.2,pp.228-223,2005 Control, AC-41,no.7,pp.1068-1072,1996.
[30]D.Koening and B.Marx.Observer Design with H∞ Performance for Delay Descriptor Systems.Proc. IEEE American Control Conf. ACC 2007.
[31]J.H.Kim.Delay and its time-derivative dependent robust stability of time-delayed linear systems with uncertainty.IEEE Transaction on Automatic Control,AC-46,no.5,pp.789-792,2001.
[32]P.Gahinet.A Nemirovskii,A. J.Laub and M.Chilali.The LMI control toolbox.In Proc.33th IEEE Conf.Decision Control,1994,pp.2038-2-41.
[33]F.Garofalo,G.Celentano and L.Glielmo.Stability robustness of interval matrices via quadratic Lyapunov forms.IEEE Transactions on Automatic Control,vol.38,pp.281-284,1993.
[34]S.Boyd,L.El Ghaoui,E.Feron and V.Balakrishnan.Linear Matrix Inequalities in System and Control Theory.SIAM Press,1994.
[35]Verghese,et al.A generalized state-space for sigular systems.IEEE Trans.Automat.Contr.,1981,26:811-830.
[36]Xu S,Dooren P V,Lam J.Robust stability and stabilization for singular systems with state delay and parameter uncertainty.IEEE Trans.Automa.Control,2002,47:1122-1128.
[37]H.K.Khalil,Nonlinear Systems.MacMillan,1992.
[38]张卫,鞠培军.一类不确定离散线性系统有限时间控制器设计.泰山学院学报,2007.
[39]胡跃明,非线性控制系统理论与应用(第二版),国防工业出版社,2005.
[40]廖晓昕,动力系统的稳定性理论和应用,国防工业出版社,2000.
[41]郑大钟,线性系统理论(第二版),清华大学出版社,Springer,2002.
[42]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.
[43]段广仁,线性系统理论(第二版),哈尔滨工业大学出版社,2004.
[2]J.P.Hespanh,A.S.Morse.Switching between stabilizing controllers[J].Automatic,2002, 38(11):1905-1917.
[3]M.S.Branicky. Stability of Switched an Hybrid Systems[C]. In Proceedings of the IEEE Conference Decision and Control, Lake Buena Vista,1994:3498-3503.
[4]Zhang Xiaoli,Liu Yuzhong,Zhao Jun.Asymototic stability of a class of discrete switched systems [J]. Control Theory and Applications,2002; 19(5):774-776.
[5]P.Peleties, R.Decarlo.Asymptotic Stability of m-Switched Systems using Lyapunov-Like Functions[C]. In Proceedings of the American Control Conference,Boston,1991:1679-1684.
[6]S.Pettersson, B.Lennartson.LMI for Stability and Robustness of Hybrid Systems[C].In Proceedings of the American Control Conference;Albuquerque;1997:1714-1718.
[7]S.Pettersson. B.Lennartson.Exponential Stability Analysis of Nonlinear Systems using LMIs[C].In Proceedings of 36th IEEE Conference Decision an Control,San Diego,CA,1997:199-204.
[8]S.Pettersson, B.Lennartson.Exponential Stability of Hybrid Systems using Piecewise Quadratic Lyapunov Functions Resulting in an LMI Problems [C]. In Proceedings of the 14th IFAC World Congress, Beijing,China,1999.
[9]Xu Honglei,Liu Xinzhi.Robust exponential stabilization of a class of impulisive switched systems [J]. Techniques of Automatic and Applications,2004,23(11):14-16.
[10]Li jiao,Liu yuzhong.Roust H-infinity dynamic output feedback control for impulsive switched systems [J]. Control Theory and Applications,2008,25(3).
[11]Zhang Hongtao,Liu Xinzhi.Roust H-infinity control on impulsive switched systems with disturbance[J].Control Theory and Applications,2004,21(2):261-266.
[12]P.Dorato.Short time stability in linear time-varying systems.In Proc.IRE Intern.Convention Record Pt.4,1961,pp.83-87.
[13]F.Amato.M.Ariola,and C.Cosentino.Finite-time stabilization via dynamic output feed-back. Automatica.vol.42.pp.337-342,2006.
[14]F.Amato,M.Ariola,M.Carbone and C.Cosentino.Finite-time output feedback control of linear systems via differential linear matrix conditions.In Proc.Conf.Decision Control.San Diego,CA,2006,pp.5371-5375.
[15]F.Amato.Finite-time control of discrete-time linear system.IEEE Transaction on auto-matic control,2005,50(5):724-729.
[16]R.Ambrosino,F.Calabrese.C.Cosentino and Gianmaria De Tommasi.Sufficient conditions for finite-time Stability of impulsive Dynamical Systems.IEEE Transaction on automatic control,vol.54,No.4.
[17]F.Amato,R.Ambrosino.M Ariola,C.Cosentino.Finite-time stability of linear time-varying systems with jumps.Automia,2009.
[18]S.P.Bhat and D.S.Bernstein,Finite-time stability of continuous autonomous systems.SIAM J.Control Optim.,vol.38,no.3,pp.751-766,2000.
[19]S.G.Nersesov and W.M.Haddad.Finite-time stabilization of nonlinear impulsive dynam-ical systems.In Proc.Control Conf.,Kos,Greece,Jul.2007,pp.91-98.
[20]S.Pettersson and B.Lennartson.Hybird System Stability and Robustness verication using linear matrix inequalities.Int. J.Control,vol.75.no.16-17.pp1335-1355,2002.
[21]F.Amato,M.Ariola,and P.Dorato. Finite-Time control of linear time-varying systems subject to parametric uncertainties and disturbances. Automatica,vol.37,no.9.pp.1459-1463,2001.
[22]L.Weiss and E.Infante.Finite time stability ander perturbing forces and on product spaces.IEEE Trans.Automat.Control,vol.AC-12,no.1,pp.54-59,feb.1967.
[23]P.O.Gutman.Stabilizing Controllers for Bilinear Systems.IEEE Transaction on Auto-matic Control vol.25,pp.302-306,1980.
[24]C.Bruni,G.Di Pliio,and G.Koch.Bilinear Systems:An Appealing Class of "Nearly Linear" Systems in Theory and Applications.IEEE Transactions on Automatic Control, vol.19,pp.334-348,1974.
[25]R.Longchamp. Stable Feedback Control of Bilinear Systems.IEEE Transactions on Auto-matic Control vol.25,pp.302-302,1980.
[26]R.R.Mohler and A.Y.Khapalov.Bilinear Control and Applications.vol.105,pp.621-637,2000.
[27]T.Yang.Impulsive Control Theory.Springer.2001.
[28]R.Boel,G.Stremersch.Discrets event systems:Analysis and Control.Springer.2000.
[29]M.Darouach.Linear functional observers for systems with Delays in State Variables:The Discrete-Time Case.IEEE Trans. Automat.Contr.,AC-50,no.2,pp.228-223,2005 Control, AC-41,no.7,pp.1068-1072,1996.
[30]D.Koening and B.Marx.Observer Design with H∞ Performance for Delay Descriptor Systems.Proc. IEEE American Control Conf. ACC 2007.
[31]J.H.Kim.Delay and its time-derivative dependent robust stability of time-delayed linear systems with uncertainty.IEEE Transaction on Automatic Control,AC-46,no.5,pp.789-792,2001.
[32]P.Gahinet.A Nemirovskii,A. J.Laub and M.Chilali.The LMI control toolbox.In Proc.33th IEEE Conf.Decision Control,1994,pp.2038-2-41.
[33]F.Garofalo,G.Celentano and L.Glielmo.Stability robustness of interval matrices via quadratic Lyapunov forms.IEEE Transactions on Automatic Control,vol.38,pp.281-284,1993.
[34]S.Boyd,L.El Ghaoui,E.Feron and V.Balakrishnan.Linear Matrix Inequalities in System and Control Theory.SIAM Press,1994.
[35]Verghese,et al.A generalized state-space for sigular systems.IEEE Trans.Automat.Contr.,1981,26:811-830.
[36]Xu S,Dooren P V,Lam J.Robust stability and stabilization for singular systems with state delay and parameter uncertainty.IEEE Trans.Automa.Control,2002,47:1122-1128.
[37]H.K.Khalil,Nonlinear Systems.MacMillan,1992.
[38]张卫,鞠培军.一类不确定离散线性系统有限时间控制器设计.泰山学院学报,2007.
[39]胡跃明,非线性控制系统理论与应用(第二版),国防工业出版社,2005.
[40]廖晓昕,动力系统的稳定性理论和应用,国防工业出版社,2000.
[41]郑大钟,线性系统理论(第二版),清华大学出版社,Springer,2002.
[42]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.
[43]段广仁,线性系统理论(第二版),哈尔滨工业大学出版社,2004.