随机系统的时滞镇定和鲁棒控制
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摘要
随机控制理论广泛地应用于经济、人口系统等社会领域以及航空航天、导航与控制、制造工程等工程领域。随机系统的研究已成为现代控制理论研究中的一个热点问题。在实际系统中,非线性、时滞是普遍存在的,通常时滞是引起系统不稳定或产生振荡的根源。另一方面,被控系统往往受到一些参数误差、未建模动态以及不确定的外界干扰等不确定因素的影响,系统模型具有某种不确定性。控制界针对不确定性对系统性能影响的研究产生了鲁棒控制理论。因此,随机系统的时滞镇定和鲁棒控制研究就具有重要的理论研究意义和实际应用价值。
本文首先构造一个Lyapunov‐Krasovskii泛函,利用Ito微分公式结合线性矩阵不等式(LMI)技术,研究了一类随机系统的时滞反馈鲁棒镇定和鲁棒H_∞控制问题,给出了系统随机稳定和满足随机H_∞性能指标的充分条件,得到了随机系统时滞反馈镇定和鲁棒H_∞控制器的设计方法,并通过仿真算例说明了方法的有效性。
进而,对该类系统考虑了当系统状态不能直接得到时,利用Ito微分结合非线性矩阵不等式(NMI)和半线性矩阵不等式研究了随机动态输出时滞反馈镇定和H_∞控制问题并给出了随机动态输出时滞反馈控制器的设计方法。这类控制器不同于传统的状态反馈控制器和状态输出反馈控制器。并且通过数值仿真说明了方法的有效性和结论的正确性。
最后对全文进行了总结,并对未来的研究方向进行了展望。
Stochastic control theory is widely used in economy, population system, social fields and aerospace, navigation and control, manufacturing engineering and engineering field. Stochastic system research has become one of a hot issue of modern control theory research. In the actual system, nonlinear, delay is universal exist, delay usually causing the instability or produce the root of oscillation. On the other hand, control system often affected by uncertainty factors like Parameter errors,Unmodeled dynamics and uncertainty outside interference. System model has some kind of uncertainty .Field of control produces robust control theory bases on the uncertainties on system performance impact study. Therefore, the delay stabilization and robust control problems for stochastic systems have important theoretical research significance and actual application value.
In this paper, we construct a Lyapunov-Krasovskii functional firstly, the delay feedback robust stabilization and H_∞control problems for a class of stochastic system are reserched by Ito? differential formula and linear matrix inequality(LMI) techniques, the delay-dependent sufficient conditions which provide the feasibility of the H_∞controller are given. The numerical example with simulation also has given to show the feasibility and effectiveness of the proposed method.
Secondly,the states of the stochastic systems under consideration in chapter 4 are not available directly, the stochastic dynamic output feedback stabilization and delay control problems are studied by Ito? differential formula, linear matrix inequality(LMI) and bilinear matrix inequality (BMI)techniques, and given the design method of stochastic dynamic output feedback controller. This kind of controller is different from traditional state feedback controller and state output feedback controller. The feasibility and effectiveness of the proposed method and the correctness of the conclusion are shown by the illustrate numerical example .
At last,we summarize the full text and gives the future research direction.
本文首先构造一个Lyapunov‐Krasovskii泛函,利用Ito微分公式结合线性矩阵不等式(LMI)技术,研究了一类随机系统的时滞反馈鲁棒镇定和鲁棒H_∞控制问题,给出了系统随机稳定和满足随机H_∞性能指标的充分条件,得到了随机系统时滞反馈镇定和鲁棒H_∞控制器的设计方法,并通过仿真算例说明了方法的有效性。
进而,对该类系统考虑了当系统状态不能直接得到时,利用Ito微分结合非线性矩阵不等式(NMI)和半线性矩阵不等式研究了随机动态输出时滞反馈镇定和H_∞控制问题并给出了随机动态输出时滞反馈控制器的设计方法。这类控制器不同于传统的状态反馈控制器和状态输出反馈控制器。并且通过数值仿真说明了方法的有效性和结论的正确性。
最后对全文进行了总结,并对未来的研究方向进行了展望。
Stochastic control theory is widely used in economy, population system, social fields and aerospace, navigation and control, manufacturing engineering and engineering field. Stochastic system research has become one of a hot issue of modern control theory research. In the actual system, nonlinear, delay is universal exist, delay usually causing the instability or produce the root of oscillation. On the other hand, control system often affected by uncertainty factors like Parameter errors,Unmodeled dynamics and uncertainty outside interference. System model has some kind of uncertainty .Field of control produces robust control theory bases on the uncertainties on system performance impact study. Therefore, the delay stabilization and robust control problems for stochastic systems have important theoretical research significance and actual application value.
In this paper, we construct a Lyapunov-Krasovskii functional firstly, the delay feedback robust stabilization and H_∞control problems for a class of stochastic system are reserched by Ito? differential formula and linear matrix inequality(LMI) techniques, the delay-dependent sufficient conditions which provide the feasibility of the H_∞controller are given. The numerical example with simulation also has given to show the feasibility and effectiveness of the proposed method.
Secondly,the states of the stochastic systems under consideration in chapter 4 are not available directly, the stochastic dynamic output feedback stabilization and delay control problems are studied by Ito? differential formula, linear matrix inequality(LMI) and bilinear matrix inequality (BMI)techniques, and given the design method of stochastic dynamic output feedback controller. This kind of controller is different from traditional state feedback controller and state output feedback controller. The feasibility and effectiveness of the proposed method and the correctness of the conclusion are shown by the illustrate numerical example .
At last,we summarize the full text and gives the future research direction.
引文
[1] K.J.Astrom,Introduction to stochastic control theory,New York:Dover Publications,1970
[2] K. Ito? ,On stochastic differential equations,Mem.Amer.Math.Soc.,1951.
[3] N. Kolmanovskii,Stability of Functional Differential equations,London : Academic Press,1986
[4] J.C.Doyle, K.Glover,P.P.Khargonekar et al.State-Space solutions to standard H 2and H_∞control problems.IEEE Trans.on Automatic Control,1989,34(8):831-847
[5] D. Hinriechsen, A.J. Pritchard,Stochastic H_∞,SIAM J.Control Optim,1998,36(5):1504- 1538
[6] A.E. Bouhtouri, D. Hinrichsen, A.J.Pritchard, H_∞-type control for discrete-time sto- chastic systems,International Journal of Robust and Nonlinear Control,1999,9(13):923-948
[7]盛梅,王为群,邹云,多细胞不确定性随机时滞系统鲁棒H_∞控制,信息与控制,2006,35(4):532-536
[8]L.N.ZHOU, X.H.Liu,Robust H_∞control for uncertain eutral stochastic time-Delay systems,控制工程,2007,14(2):125-128
[9]华民刚,邓飞其,不确定中立随机分布时滞系统的鲁棒H_∞控制,华南理工大学学报(自然科学版),2008,36(5):106-112
[10]B.S.Chen, W.H.Zhang,Stochastic H 2/ H_∞control with state-dependent nosie,IEEE Trans.on Automatic Control,2004,49(1):45-57
[11] Y.L. Huang, W.H. Zhang, G. Feng,Infinte horizon stochastic H 2/ H_∞control for stochastic systems with Markovian jumps, Automatica,2008,44(3):857-863
[12] W.H. Zhang, Y.L. Huang, L.H. Xia, Infinte horizon stochastic H 2/ H_∞control for discrete-time systems with state and disturbance dependent noise, Automatica,2008,44(9): 2306-2316
[13]S. Xu, T. Chen,Reduced-order H_∞filtering for stochastic systems,IEEE Trans.On Signal Process,2002,50(12):2998-3007
[14] S.Y. Xu, P. Shi, Y.M. Chu et al.Robust stochastic stalility and H_∞control of Uncertain neuteal stochastic time-delay systems,Journal of Mathematical Analysis and Applications, 2006,314(1):1-16
[15] G.C. Chen, Y. Shen, Robust H_∞Control for stochastic Markovian jump systems, International Journal of Nonlinear Sciences and Numerical Simu-lation,2009,10(9):1201- 1210.
[16]S.Y.Xu, T.W.Chen, Output feedback control for uncertain stochastic systems with time-varying delays,automatica,2004,40(12):2091-2098
[17]康宇,奚宏生,季海波等,不确定状态滞后线性跳跃系统的时滞相关非脆弱H_∞控制,控制理论与应用,2005,22(6):939-946
[18]吴敏,肖伸平,张先明等,中立型系统的时滞相关非常脆弱H_∞控制,系统工程与电子技术,2008,30(9):1768-1773.
[19]宋博,徐胜元,夏建伟,具有分布时滞的中立型随即系统的非易碎鲁棒H_∞控制,南京理工大学学报(自然科学版),2008,32(3):261-264.
[20] S.Y. Xu, J.Lam, G.H.Yang et al.,Stabilization and H_∞control for uncertain stochastic time-delay systems via non-fragile controllers, Asian Journal of Control,2006,8(2):197-200.
[21]J.W.Xia, S.Y.Xu,B. Song, Delay-dependent L2 /L∞filter design for stochastic time-delay systems, Systems & Control Letters,2007,56(9-10):579-587
[22] G.C. Chen, Y.Shen, Robust H_∞filter design for neutral stochastic uncertain systems with time-varying delay,Journal of Mathematical Analysis and Applica-tions,2009,353(1): 196-204.
[23]S.Y. Xu, T.W.Chen,Robust H_∞filtering for uncertain impulsive stochastic systems under sampled measurements,Automatica,2003,39(3):509-516.
[24]H.D. Tuan, P. Apkarian,Low nonconvexity-rank bilinear matrix inequalities: Algorithms and applications in robust controller and structure designs.IEEE Trans.on Automatic Control,2000,45(11):2111-2117.
[25] M. Fukuda, M. Kojuma,Branch-and-cut algorithms for the bilinear matrix inequality eigenvalue problem,Computational Optimization and Applications,2001,19(1):79-105.
[26] E.Gershon, D.J.N. Limebeer, U. Shaked et al. Robust H_∞filtering of statio-nary continuous time linear systems with stochastic uncertainties IEEE Trans. on Automatic Control,2001,46(11):1788-1793.
[27]魏波,季海波,模型不确定非线性随机系统的鲁棒性能准则设计,系统科学与数学,2007,27(3):422-430.
[28]蔺香运,张维海,王向荣,一类非线性随机系统的状态反馈H_∞控制,应用数学报,2009,32(2):476-484.
[29]W.Zhang, B.S.Chen,C.S.Tseg,Robust H_∞filtering for nonlinear stochastic systems,IEEE Trans.on Signal Processing,2005,53(2):589-598.
[30] Wei G L.,Wang Z D.,Shu H S.et al.,Robust H_∞control of stochastic time-delay jumping systems with nonlinear disturbances,Optimal Control Application and Methods, 2006, 27(5):255-271.
[31]林丹凤,刘立山,一类具有状态及控制滞后的非线性不确定随机系统的鲁棒H_∞控制曲阜师范大学学报,2009,35(1):19-28.
[32] N.Berman, U.Shaked, H_∞-like control for nonlinear stochastic Systems & Control Letters,2006,55(3):247-257.
[33] W.H. Zhang, B.S. Chen, Statet feedback H_∞control for a class of nonlinear stochastic systems,SIAM:Journal on Control Optimization,2006,42(11):2869-2875.
[34] H.S. Shu, G.L.Wei, H_∞analysis of nonlinear stochastic time-delay systems,Chaos,
Solitons & Fractals,2005,26(2):637-647.
[35]罗琦,随机反应扩散系统的稳定、镇定与控制,华南理工大学博士学位论文,2004:1
[36]胡适耕,黄乘明,吴付科,随机微分方程,北京:科学出版社,2008
[37]陈贵词,中立型随机时滞系统的鲁棒H_∞控制和滤波器设计,华中科技大学博士学位论文.2010:1
[38] E.K. Boukas,“Stabilization of stochastic nonlinear hybrid systems[J],International Journal of Innovative Computing”, Information and Control,vol.1, pp.131-141, 2005.
[39] H. Kushner, Stochastic stability and control, Academic Press, New York, 1967.
[40] V.B. Kolmanovskii, A.D. Myshkis, Applied theory of functional differential equations, Kluwer Academic Publishers. Dordrecht, 1992.
[41] G.C. Chen, Y. Shen,“Robust H1 filter design for neutral stochastic uncertain systems with time-varying delay”, Journal of Mathematical Analysis and Applications, vol. 353, pp.196-204, 2009.
[42] S.Y.Xu, T.W.Chen,“H1output feedback control for uncertain stochastic systems with time-varying delays”, Automatica, vol.40, pp.2091-2098, 2004.
[43] S.Y.Xu, P.Shi, Y.M.Chu, et.al.,“Robust stochastic stabilization and H1 control of uncertain neutral stochastic time-delay systems”,Journal of Mathematical of Analysis and Applications, vol. 314, pp.1-16, 2006.
[44] S.Y. Xu, T.W. Chen,“Robust H1 filtering for uncertain impulsive stochastic systems under sampled measurements”, Automatica, vol.39, pp. 509-516, 2003.
[45] S.Y. Xu, J. Lam,“Exponential H1 filtering design for uncertain Takagi-Sugeno fuzzy systems with time delay”, Engineering Applications of Artificial Intelligence, vol. 17, pp. 645-659, 2004.
[46] J.W. Xia, S.Y. Xu, B. Song,“Delay-dependent L2by delay feedback control”, Systems & Control Letters, vol.57, pp.927-935, 2008.
[53] G.C. Chen, Y. Shen, S. Zhu,“H_∞control for neutral linear stochastic time-delay systems via delay feedback controller”, In: Proceedings 2010 IEEE fifth International Conference on Bio-Inspired Computing:Theory and Applicaitons, vol. 1, pp. 16-20. 2010.
[54] X. Mao. Stochastic Differential Equations and Their Applicaitons.Horwood Publishers, Chichester, 1997.
[55] W. Feng, J.F. Zhang. Stability analysis and stabilization control of multi-variable switched stochastic systems. Automatica,2006, 42(1):169-176.
[56] Y. Shen, J. Wang. An Improved Algebraic Criterion for Global Exponential Stability of Recurrent Neural Networks With Time-Varying Delays. IEEE Transaction Neural Networks,2008, 19(3):528-531.
[57] V.B.Kolmanovskii, A.D.Myshkis. Applied theory of functional differential equations, Kluwer Academic Publishers.Dordrecht, 1992.
[58] V.B.Kolmanovskii, A.D.Myshkis. Introduction to the theory and applications of functional differential equations, Kluwer Academic Publishers. Dordrecht, 1999.
[59] X. Mao. Exponential stability of large-scale stochastic differential equations. Systems & Control Letters, 1992, 19(1):71-82.
[60] X. Mao. Stochastic stabilization and destabilization. Systems & Control Letters, 1994, 23(4):279-290.
[61] X. Mao, N. Koroleva, A. Rodkina. Robust stability of uncertain stochastic differential delay equations. Systems & Control Letters, 1998, 35(5):325-336.
[62] S.Y. Xu, P. Shi, Y.M. Chu et.al. Robust stochastic stabilization and H_∞control of uncertain neutral stochastic timedelay systems. Journal Mathematical Analysis and Applications,2006, 341(1):1-16.
[63] D. Hinriechsen, A.J. Pritchard. Stochastic H_∞. SIAM J. Control Optim, 1998, 36(5):1504-1538.
[64] K. Youcef-Toumi, O. Ito? , A Time Delay Controller for Systems With Unknown Dynamics, J. Dyn. Sys., Meas., Control,1990, 112(1):133-142.
[65] V. Kapila, A. Tzes, Q.G. Yan. Closed-Loop Input Shaping for Flexible Structures Using Time-Delay Control, J. Dyn. Sys.,Meas., Control, 2000, 122(3)454-460.
[66] X. Mao, J. Lam, L.R. Huang. Stabilisation of hybrid stochastic differential equations by delay feedback control, Systems& Control Letters, 2008, 57(11):927-935.
[67] G.C. Chen,Y. Shen, S. Zhu. Delay Feedback H_∞Control for Neutral Nonlinear Stochastic Systems with Time-varying Delay,accepted.
[2] K. Ito? ,On stochastic differential equations,Mem.Amer.Math.Soc.,1951.
[3] N. Kolmanovskii,Stability of Functional Differential equations,London : Academic Press,1986
[4] J.C.Doyle, K.Glover,P.P.Khargonekar et al.State-Space solutions to standard H 2and H_∞control problems.IEEE Trans.on Automatic Control,1989,34(8):831-847
[5] D. Hinriechsen, A.J. Pritchard,Stochastic H_∞,SIAM J.Control Optim,1998,36(5):1504- 1538
[6] A.E. Bouhtouri, D. Hinrichsen, A.J.Pritchard, H_∞-type control for discrete-time sto- chastic systems,International Journal of Robust and Nonlinear Control,1999,9(13):923-948
[7]盛梅,王为群,邹云,多细胞不确定性随机时滞系统鲁棒H_∞控制,信息与控制,2006,35(4):532-536
[8]L.N.ZHOU, X.H.Liu,Robust H_∞control for uncertain eutral stochastic time-Delay systems,控制工程,2007,14(2):125-128
[9]华民刚,邓飞其,不确定中立随机分布时滞系统的鲁棒H_∞控制,华南理工大学学报(自然科学版),2008,36(5):106-112
[10]B.S.Chen, W.H.Zhang,Stochastic H 2/ H_∞control with state-dependent nosie,IEEE Trans.on Automatic Control,2004,49(1):45-57
[11] Y.L. Huang, W.H. Zhang, G. Feng,Infinte horizon stochastic H 2/ H_∞control for stochastic systems with Markovian jumps, Automatica,2008,44(3):857-863
[12] W.H. Zhang, Y.L. Huang, L.H. Xia, Infinte horizon stochastic H 2/ H_∞control for discrete-time systems with state and disturbance dependent noise, Automatica,2008,44(9): 2306-2316
[13]S. Xu, T. Chen,Reduced-order H_∞filtering for stochastic systems,IEEE Trans.On Signal Process,2002,50(12):2998-3007
[14] S.Y. Xu, P. Shi, Y.M. Chu et al.Robust stochastic stalility and H_∞control of Uncertain neuteal stochastic time-delay systems,Journal of Mathematical Analysis and Applications, 2006,314(1):1-16
[15] G.C. Chen, Y. Shen, Robust H_∞Control for stochastic Markovian jump systems, International Journal of Nonlinear Sciences and Numerical Simu-lation,2009,10(9):1201- 1210.
[16]S.Y.Xu, T.W.Chen, Output feedback control for uncertain stochastic systems with time-varying delays,automatica,2004,40(12):2091-2098
[17]康宇,奚宏生,季海波等,不确定状态滞后线性跳跃系统的时滞相关非脆弱H_∞控制,控制理论与应用,2005,22(6):939-946
[18]吴敏,肖伸平,张先明等,中立型系统的时滞相关非常脆弱H_∞控制,系统工程与电子技术,2008,30(9):1768-1773.
[19]宋博,徐胜元,夏建伟,具有分布时滞的中立型随即系统的非易碎鲁棒H_∞控制,南京理工大学学报(自然科学版),2008,32(3):261-264.
[20] S.Y. Xu, J.Lam, G.H.Yang et al.,Stabilization and H_∞control for uncertain stochastic time-delay systems via non-fragile controllers, Asian Journal of Control,2006,8(2):197-200.
[21]J.W.Xia, S.Y.Xu,B. Song, Delay-dependent L2 /L∞filter design for stochastic time-delay systems, Systems & Control Letters,2007,56(9-10):579-587
[22] G.C. Chen, Y.Shen, Robust H_∞filter design for neutral stochastic uncertain systems with time-varying delay,Journal of Mathematical Analysis and Applica-tions,2009,353(1): 196-204.
[23]S.Y. Xu, T.W.Chen,Robust H_∞filtering for uncertain impulsive stochastic systems under sampled measurements,Automatica,2003,39(3):509-516.
[24]H.D. Tuan, P. Apkarian,Low nonconvexity-rank bilinear matrix inequalities: Algorithms and applications in robust controller and structure designs.IEEE Trans.on Automatic Control,2000,45(11):2111-2117.
[25] M. Fukuda, M. Kojuma,Branch-and-cut algorithms for the bilinear matrix inequality eigenvalue problem,Computational Optimization and Applications,2001,19(1):79-105.
[26] E.Gershon, D.J.N. Limebeer, U. Shaked et al. Robust H_∞filtering of statio-nary continuous time linear systems with stochastic uncertainties IEEE Trans. on Automatic Control,2001,46(11):1788-1793.
[27]魏波,季海波,模型不确定非线性随机系统的鲁棒性能准则设计,系统科学与数学,2007,27(3):422-430.
[28]蔺香运,张维海,王向荣,一类非线性随机系统的状态反馈H_∞控制,应用数学报,2009,32(2):476-484.
[29]W.Zhang, B.S.Chen,C.S.Tseg,Robust H_∞filtering for nonlinear stochastic systems,IEEE Trans.on Signal Processing,2005,53(2):589-598.
[30] Wei G L.,Wang Z D.,Shu H S.et al.,Robust H_∞control of stochastic time-delay jumping systems with nonlinear disturbances,Optimal Control Application and Methods, 2006, 27(5):255-271.
[31]林丹凤,刘立山,一类具有状态及控制滞后的非线性不确定随机系统的鲁棒H_∞控制曲阜师范大学学报,2009,35(1):19-28.
[32] N.Berman, U.Shaked, H_∞-like control for nonlinear stochastic Systems & Control Letters,2006,55(3):247-257.
[33] W.H. Zhang, B.S. Chen, Statet feedback H_∞control for a class of nonlinear stochastic systems,SIAM:Journal on Control Optimization,2006,42(11):2869-2875.
[34] H.S. Shu, G.L.Wei, H_∞analysis of nonlinear stochastic time-delay systems,Chaos,
Solitons & Fractals,2005,26(2):637-647.
[35]罗琦,随机反应扩散系统的稳定、镇定与控制,华南理工大学博士学位论文,2004:1
[36]胡适耕,黄乘明,吴付科,随机微分方程,北京:科学出版社,2008
[37]陈贵词,中立型随机时滞系统的鲁棒H_∞控制和滤波器设计,华中科技大学博士学位论文.2010:1
[38] E.K. Boukas,“Stabilization of stochastic nonlinear hybrid systems[J],International Journal of Innovative Computing”, Information and Control,vol.1, pp.131-141, 2005.
[39] H. Kushner, Stochastic stability and control, Academic Press, New York, 1967.
[40] V.B. Kolmanovskii, A.D. Myshkis, Applied theory of functional differential equations, Kluwer Academic Publishers. Dordrecht, 1992.
[41] G.C. Chen, Y. Shen,“Robust H1 filter design for neutral stochastic uncertain systems with time-varying delay”, Journal of Mathematical Analysis and Applications, vol. 353, pp.196-204, 2009.
[42] S.Y.Xu, T.W.Chen,“H1output feedback control for uncertain stochastic systems with time-varying delays”, Automatica, vol.40, pp.2091-2098, 2004.
[43] S.Y.Xu, P.Shi, Y.M.Chu, et.al.,“Robust stochastic stabilization and H1 control of uncertain neutral stochastic time-delay systems”,Journal of Mathematical of Analysis and Applications, vol. 314, pp.1-16, 2006.
[44] S.Y. Xu, T.W. Chen,“Robust H1 filtering for uncertain impulsive stochastic systems under sampled measurements”, Automatica, vol.39, pp. 509-516, 2003.
[45] S.Y. Xu, J. Lam,“Exponential H1 filtering design for uncertain Takagi-Sugeno fuzzy systems with time delay”, Engineering Applications of Artificial Intelligence, vol. 17, pp. 645-659, 2004.
[46] J.W. Xia, S.Y. Xu, B. Song,“Delay-dependent L2by delay feedback control”, Systems & Control Letters, vol.57, pp.927-935, 2008.
[53] G.C. Chen, Y. Shen, S. Zhu,“H_∞control for neutral linear stochastic time-delay systems via delay feedback controller”, In: Proceedings 2010 IEEE fifth International Conference on Bio-Inspired Computing:Theory and Applicaitons, vol. 1, pp. 16-20. 2010.
[54] X. Mao. Stochastic Differential Equations and Their Applicaitons.Horwood Publishers, Chichester, 1997.
[55] W. Feng, J.F. Zhang. Stability analysis and stabilization control of multi-variable switched stochastic systems. Automatica,2006, 42(1):169-176.
[56] Y. Shen, J. Wang. An Improved Algebraic Criterion for Global Exponential Stability of Recurrent Neural Networks With Time-Varying Delays. IEEE Transaction Neural Networks,2008, 19(3):528-531.
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