关于系统有限时间稳定的若干问题的讨论
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摘要
本文主要讨论线性系统的有限时间稳定性。首先讨论离散线性系统在动态观测器下的有限时间稳定问题,利用线性矩阵不等式,给出三个相关的判定定理。而后对离散线性时滞系统进行讨论,同样利用线性矩阵不等式,得到该系统不是有限时间稳定的一个充要条件。最后,本文将有限时间稳定问题和传统的H~∞控制问题相结合,研究有限时间H~∞控制问题,并给出连续系统满足有限时间H~∞控制的充分条件。
In this paper,the finite-time stability of linear systems is discussed. First of all,the problem of the finite-time stability of a discrete system is studied and three relative theorems are given based on LMI. Then linear time-delay system is discussed and a sufficient and necessary condition is given based on LMI also. At last, combined the finite-time stability with traditional H~∞control problem, a sufficient condition which solves the finite-time H~∞control problem of a class of continuous systems is obtained.
In this paper,the finite-time stability of linear systems is discussed. First of all,the problem of the finite-time stability of a discrete system is studied and three relative theorems are given based on LMI. Then linear time-delay system is discussed and a sufficient and necessary condition is given based on LMI also. At last, combined the finite-time stability with traditional H~∞control problem, a sufficient condition which solves the finite-time H~∞control problem of a class of continuous systems is obtained.
引文
[1] P. Dorato, Short time stability in linear time-varying systems, in Proc. IRE International Convention Record Part 4, 1961, pp. 83-87.
[2] L.Weiss and E. F. Infante, Finite time stability under perturbingforces and on product spaces, IEEE Trans. Auto.Contr., vol.12,1967,pp. 54-59.
[3] R.P. Agarwal, V. Lakshmikantham, Uniqueness and nonuniqueness criteria for ordinary differential equations, World Scientific, Singapore, 1993.
[4] A.F. Filippov, Differential equations with discontinuous right-hand sides. Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers,Dordrecht,1988.
[5] Kawski.M, Stabilization of nonlinear systems in the plane,Syst. Control Lett. 12,1989 pp. 169-175.
[6] T. Yoshizawa, Stability theory by Lyapunov's second method, The Mathematical Society of Japan, 1966.
[7] E.A. Coddington, N. Levinson, Theory of ordinary differentia] equations, McGraw-Hill, New York, 1955.
[8] S.P. Bhat,D.S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim. 38 (3) ,2000. pp. 751-766.
[9] S.P.Bhat, D.S. Bernstein, Geometric homogeneity with applications to finite-time stability, Math. Control Signals Syst. 17 ,2005, pp. 101-127.
[10] V.T. Haimo, Finite-time controllers, SIAM J. Control Optim. 24 ,1986,pp. 760-770.
[11] S.P. Bhat, D.S. Bernstein,Continuous finite-time stabilization of the translational and rotational double integrators, IEEE Trans. Autom. Control 43 ,1998,pp.678-682.
[12] Y. Hong, Finite-time stabilization and stabilizability of a class of controllable systems, Syst. Control Lett. 46 ,2002,pp.231-236.
[13] Y. Hong, J. Huang, Y. Xu, On an output feedback finite-time stabilization problem, IEEE Trans. Autom. Control 46,2001,pp. 305-309.
[14] C. Qian, W. Lin, A continuous feedback approach to global strong stabilization of nonlinear systems, IEEE Trans. Autom. Control 46 ,2001,pp. 1061-1079.
[15] A.T.Fuller, Optimization of some nonlinear control systems by means of Bellman's equation and dimensional analysis, Int. J. Control 3,1966,pp. 359-394.
[16] E.P. Ryan, Singular optimal controls for second-order saturating systems, Int. J. Control 30 ,1979,pp. 549-564.
[17] E.P. Ryan, Finite-time stabilization of uncertain nonlinear planar systems, Dyn. Control 1 ,1991,pp. 83-94.
[18] S.G. Nersesov, W.M. Haddad,Finite-time stabilization of nonlinear impulsive dynamical systems,Elsevier,Nonlinear Analysis: Hybrid Systems 2,2008,832 - 845.
[19] E. Moulay et al.,Finite-time stability and stabilization of time-delay systems,Elsevier,Systems and Control Letters 57,2008pp.561 - 566.
[20] Y. Hong et al.,Finite-time control for robot manipulators,Systems and Control Letters 46 ,2002,pp. 243-253.
[21] Y. Hong et al. ,Adaptive finite time stabilization for a class of nonlinear systems,43rd IEEE Conference on Decision and Control December 14-17,2004
[22] X Huang et al.,Finite-time stabilization in the large for uncertain nonlinear sys-tems,Proceeding of the 2004 American Control Conference Boston. Massachusetts June 30 -July 2,2004
[23] E. Moulay and W. Perruquetti, Finite time stability of non linear systems,Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003.
[24] X. Huang et al.,Global finite-time stabilization of a class of uncertain nonlinear systems,Automatica 41 ,2005,pp. 881-888.
[25] R. Ding, J. Li,Nonlinear finite-time Lyapunov exponent and predictability,Physics Letters A 364 ,2007,pp. 396 - 400.
[26] W. Perruquetti, S.drakunov,Finite time stability and stabilization ,Proceedings of the 39th IEEE Conference on Decision and Control,2000.
[27] S.G. Nersesov et al.,Finite-time stabilization of nonlinear dynamical systems via control vector Lyapunov functions,Journal of the Franklin Institute,2008.
[28] E. Moulay and W. Perruquetti, Lyapunov-based approach for finite time stability and stabilization,Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15,2005.
[29] S. Onori et al.,Finite time stability design via feedback linearization,Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville,Spain, December 12-15,2005.
[30] G. BartoliniOn et al.,the Finite-time stabilization of uncertain nonlinear systems with relative degree three,IEEE Tran.Aut.control, VOL. 52, NO. 11,2007.
[31]Sanjay P.Bhaty,Dennis S.Bernstein,Finite-time stability of continuous autonomous systems,SIAM J.control optim.Vol.38,No.3,2000,pp.751-766.
[32]李世华,丁世宏,田玉平,一类二阶非线性系统的有限时间状态反馈镇定方法,自动化学报vol.33,No.1,2007.
[33]F.Amato,M.Ariola,C.Cosentino,C.T.Abdallah,and P.Dorato,Necessary and sufficient conditions for finite-time stability of linear systems,in Proc.American Control Conference,Denver,CO,2003,pp.4452-4456.
[34]F.Amato,M.Ariola,and C.Cosentino,Finite-time control of linear time-varying systems via output feedback,in Proc.American Control Conference,Portland,MD,2005,pp.4723-4727.
[35]F.Amato,M.Ariola,and P.Dorato,Finite-time control of linear systems subject to parametric uncertanties and disturbances,Automatica,vol.37,2001,pp.1459-1463.
[36]F.Amato,M.Ariola,and C.Cosentino,Finite-time stabilization via dynamic output feedback,Automatica,vol.42,2006,pp.337-342.
[37]P.Gahinet,A.Nemirovski,A.J.Laub,and M.Chilali,LMI control toolbox.The Mathworks Inc,Natick,MA,1995.
[38]F.Amato et al.,Robust finite-time stabilization of linear systems depending on para-metric uncertainties,Proceedings of the 37th IEEE Conf.on Decision and Control 1998.
[39]F.Amato et al.,Finite-time output feedback control ot linear systems via differential linear matrix conditions,45th IEEE CDC,San Diego,USA,Dec.13-15,2006.
[40]F.Amato et al.,Control of linear discrete-time systems over a finite-time inter-val,43rd IEEE Conference on Decision and Control December 14-17,2004.
[41]F.Amato,M.Ariola,Finite-time control of discrete-time linear systems,IEEE Tran.on Auto.control,VOL.50,NO.5,MAY 2005.
[42]F.Amato et al.,Finite-time stability of discrete-time systems,Proceeding of the 2004American Control Conference Boston,Massachusetts June30-July 2.2004.
[43]Y.Shen,C.Li,LMI-based finite-time boundedness analysis of neural networks with parametric uncertainties,Neurocomputing 71,2008,pp.502-507.
[44]L.Zhu et al.,Finite-time control of discrete-time systems with time-varying exoge-nous disturbance,Communications in Nonlinear Science and Numerical Simulation 14,2009,pp.361-370.
[45]辛道义,刘允刚,非线性系统有限时间稳定性分析与控制设计,山东大学学报(工学版)Vol.37,No.3,2007
[46]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,第29卷第3期2007
[47]Y.Hong;et al.,Finite time convergent control using terminal sliding mode,Journal of Control Theory and Applications 2,2004,pp.69-74.
[48]H.Wang,et al.,Finite-time chaos control via nonsingular terminal sliding mode control,Commun Nonlinear Sci Numer Simulat 14,2009,pp.2728-2733.
[49]Yuqiang Wu,et al.,Finite time sliding mode control for time delay sys-tems,Proceedings of the 46 World Congress on Intelligent Control and Automation June 1C-14,2002.
[50]Yu.S,Continuous finite-time control for robotic manipulators with terminal sliding mode,Automatica 41,2005,PP.1957-1964.
[51]A.Polyakov,A.Poznyak,Lyapunov function design for finite-time convergence anal-ysis:"Twisting," controller for second-order sliding mode realization,Automatica 45,2009,pp.444-448
[2] L.Weiss and E. F. Infante, Finite time stability under perturbingforces and on product spaces, IEEE Trans. Auto.Contr., vol.12,1967,pp. 54-59.
[3] R.P. Agarwal, V. Lakshmikantham, Uniqueness and nonuniqueness criteria for ordinary differential equations, World Scientific, Singapore, 1993.
[4] A.F. Filippov, Differential equations with discontinuous right-hand sides. Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers,Dordrecht,1988.
[5] Kawski.M, Stabilization of nonlinear systems in the plane,Syst. Control Lett. 12,1989 pp. 169-175.
[6] T. Yoshizawa, Stability theory by Lyapunov's second method, The Mathematical Society of Japan, 1966.
[7] E.A. Coddington, N. Levinson, Theory of ordinary differentia] equations, McGraw-Hill, New York, 1955.
[8] S.P. Bhat,D.S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim. 38 (3) ,2000. pp. 751-766.
[9] S.P.Bhat, D.S. Bernstein, Geometric homogeneity with applications to finite-time stability, Math. Control Signals Syst. 17 ,2005, pp. 101-127.
[10] V.T. Haimo, Finite-time controllers, SIAM J. Control Optim. 24 ,1986,pp. 760-770.
[11] S.P. Bhat, D.S. Bernstein,Continuous finite-time stabilization of the translational and rotational double integrators, IEEE Trans. Autom. Control 43 ,1998,pp.678-682.
[12] Y. Hong, Finite-time stabilization and stabilizability of a class of controllable systems, Syst. Control Lett. 46 ,2002,pp.231-236.
[13] Y. Hong, J. Huang, Y. Xu, On an output feedback finite-time stabilization problem, IEEE Trans. Autom. Control 46,2001,pp. 305-309.
[14] C. Qian, W. Lin, A continuous feedback approach to global strong stabilization of nonlinear systems, IEEE Trans. Autom. Control 46 ,2001,pp. 1061-1079.
[15] A.T.Fuller, Optimization of some nonlinear control systems by means of Bellman's equation and dimensional analysis, Int. J. Control 3,1966,pp. 359-394.
[16] E.P. Ryan, Singular optimal controls for second-order saturating systems, Int. J. Control 30 ,1979,pp. 549-564.
[17] E.P. Ryan, Finite-time stabilization of uncertain nonlinear planar systems, Dyn. Control 1 ,1991,pp. 83-94.
[18] S.G. Nersesov, W.M. Haddad,Finite-time stabilization of nonlinear impulsive dynamical systems,Elsevier,Nonlinear Analysis: Hybrid Systems 2,2008,832 - 845.
[19] E. Moulay et al.,Finite-time stability and stabilization of time-delay systems,Elsevier,Systems and Control Letters 57,2008pp.561 - 566.
[20] Y. Hong et al.,Finite-time control for robot manipulators,Systems and Control Letters 46 ,2002,pp. 243-253.
[21] Y. Hong et al. ,Adaptive finite time stabilization for a class of nonlinear systems,43rd IEEE Conference on Decision and Control December 14-17,2004
[22] X Huang et al.,Finite-time stabilization in the large for uncertain nonlinear sys-tems,Proceeding of the 2004 American Control Conference Boston. Massachusetts June 30 -July 2,2004
[23] E. Moulay and W. Perruquetti, Finite time stability of non linear systems,Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003.
[24] X. Huang et al.,Global finite-time stabilization of a class of uncertain nonlinear systems,Automatica 41 ,2005,pp. 881-888.
[25] R. Ding, J. Li,Nonlinear finite-time Lyapunov exponent and predictability,Physics Letters A 364 ,2007,pp. 396 - 400.
[26] W. Perruquetti, S.drakunov,Finite time stability and stabilization ,Proceedings of the 39th IEEE Conference on Decision and Control,2000.
[27] S.G. Nersesov et al.,Finite-time stabilization of nonlinear dynamical systems via control vector Lyapunov functions,Journal of the Franklin Institute,2008.
[28] E. Moulay and W. Perruquetti, Lyapunov-based approach for finite time stability and stabilization,Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 12-15,2005.
[29] S. Onori et al.,Finite time stability design via feedback linearization,Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville,Spain, December 12-15,2005.
[30] G. BartoliniOn et al.,the Finite-time stabilization of uncertain nonlinear systems with relative degree three,IEEE Tran.Aut.control, VOL. 52, NO. 11,2007.
[31]Sanjay P.Bhaty,Dennis S.Bernstein,Finite-time stability of continuous autonomous systems,SIAM J.control optim.Vol.38,No.3,2000,pp.751-766.
[32]李世华,丁世宏,田玉平,一类二阶非线性系统的有限时间状态反馈镇定方法,自动化学报vol.33,No.1,2007.
[33]F.Amato,M.Ariola,C.Cosentino,C.T.Abdallah,and P.Dorato,Necessary and sufficient conditions for finite-time stability of linear systems,in Proc.American Control Conference,Denver,CO,2003,pp.4452-4456.
[34]F.Amato,M.Ariola,and C.Cosentino,Finite-time control of linear time-varying systems via output feedback,in Proc.American Control Conference,Portland,MD,2005,pp.4723-4727.
[35]F.Amato,M.Ariola,and P.Dorato,Finite-time control of linear systems subject to parametric uncertanties and disturbances,Automatica,vol.37,2001,pp.1459-1463.
[36]F.Amato,M.Ariola,and C.Cosentino,Finite-time stabilization via dynamic output feedback,Automatica,vol.42,2006,pp.337-342.
[37]P.Gahinet,A.Nemirovski,A.J.Laub,and M.Chilali,LMI control toolbox.The Mathworks Inc,Natick,MA,1995.
[38]F.Amato et al.,Robust finite-time stabilization of linear systems depending on para-metric uncertainties,Proceedings of the 37th IEEE Conf.on Decision and Control 1998.
[39]F.Amato et al.,Finite-time output feedback control ot linear systems via differential linear matrix conditions,45th IEEE CDC,San Diego,USA,Dec.13-15,2006.
[40]F.Amato et al.,Control of linear discrete-time systems over a finite-time inter-val,43rd IEEE Conference on Decision and Control December 14-17,2004.
[41]F.Amato,M.Ariola,Finite-time control of discrete-time linear systems,IEEE Tran.on Auto.control,VOL.50,NO.5,MAY 2005.
[42]F.Amato et al.,Finite-time stability of discrete-time systems,Proceeding of the 2004American Control Conference Boston,Massachusetts June30-July 2.2004.
[43]Y.Shen,C.Li,LMI-based finite-time boundedness analysis of neural networks with parametric uncertainties,Neurocomputing 71,2008,pp.502-507.
[44]L.Zhu et al.,Finite-time control of discrete-time systems with time-varying exoge-nous disturbance,Communications in Nonlinear Science and Numerical Simulation 14,2009,pp.361-370.
[45]辛道义,刘允刚,非线性系统有限时间稳定性分析与控制设计,山东大学学报(工学版)Vol.37,No.3,2007
[46]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,第29卷第3期2007
[47]Y.Hong;et al.,Finite time convergent control using terminal sliding mode,Journal of Control Theory and Applications 2,2004,pp.69-74.
[48]H.Wang,et al.,Finite-time chaos control via nonsingular terminal sliding mode control,Commun Nonlinear Sci Numer Simulat 14,2009,pp.2728-2733.
[49]Yuqiang Wu,et al.,Finite time sliding mode control for time delay sys-tems,Proceedings of the 46 World Congress on Intelligent Control and Automation June 1C-14,2002.
[50]Yu.S,Continuous finite-time control for robotic manipulators with terminal sliding mode,Automatica 41,2005,PP.1957-1964.
[51]A.Polyakov,A.Poznyak,Lyapunov function design for finite-time convergence anal-ysis:"Twisting," controller for second-order sliding mode realization,Automatica 45,2009,pp.444-448