不确定时滞系统的鲁棒稳定性分析
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摘要
不确定时滞系统的鲁棒稳定性分析是目前自动控制领域研究的一个热点.由于系统环境的变化、元器件的老化以及某些物理参数随时间的未知变化等因素所带来的系统行为的改变都会导致系统模型不确定性和时滞的产生.不确定性和时滞的存在使得系统的分析与综合变得更加复杂和困难,同时也是系统不稳定和系统性能变差的根源.本文从理论分析的角度,考虑不确定性、时滞、线性、非线性、干扰和自适应跟踪等动态特性,利用二次稳定性、Lyapunov稳定性以及线性矩阵不等式等重要理论与方法,分别研究了系统的状态反馈二次稳定控制器、在H∞范数界γ约束下稳定性控制器、鲁棒H∞保性能控制器、自适应输出反馈控制器以及系统的自适应跟踪控制器的综合问题,提出了相对统一的处理框架.研究结果表明,在一定的条件下,各类系统稳定化控制器的综合问题均可以转化为相应的Riccati方程(不等式)或线性矩阵不等式的可行性解问题,而Riccati方程(不等式)或线性矩阵不等式的求解已有成熟的算法,从而使研究结果具有可操作性,为进一步研究不确定时滞系统的鲁棒控制问题奠定了基础.
At present the robust stability analysis of systems with uncertain time-delay is focused on automatic control field. Many factors, such as changing of systems environment, the age of components, and some physical param-eters and so on, can bring about changes of the systems behaviors. Those will cause uncertainty and time-delay of the system models. The existence of uncertainty and time-delay makes the system analysis and synthesis become more complex and more difficult. At the same time, it is also the source of the systems instability and systems performance deterioration. Uncer-tainty, time-delay, linear, nonlinear, disturbances and adaptive tracking are considered from the perspective of theoretical analysis. Quadratic stabiliza-tion, Lyapunov stabilization, linear matrix inequalities and other important theories and methods are used in this paper. The paper studys the synthe-sis problems of the systems state feedback quadratic stabilization controller, stability controller constrained under the H∞bounded normγ, robust H∞guaranteed cost controller, adaptive output feedback controller and adaptive tracking controller, respectively. A relatively unified processing framework is proposed. The results show that the integrated problems of various systems stabilizing controller can be transformed into the feasibility of solutions of the corresponding Riccati equations (inequalities) or linear matrix inequali-ties under the certain conditions, however, the solution of Riccati equations (inequalities) and linear matrix inequalities have the sophisticated algorithms. So the research results are operable which establish the foundation for further study robust control problem of uncertain time-delay systems.
At present the robust stability analysis of systems with uncertain time-delay is focused on automatic control field. Many factors, such as changing of systems environment, the age of components, and some physical param-eters and so on, can bring about changes of the systems behaviors. Those will cause uncertainty and time-delay of the system models. The existence of uncertainty and time-delay makes the system analysis and synthesis become more complex and more difficult. At the same time, it is also the source of the systems instability and systems performance deterioration. Uncer-tainty, time-delay, linear, nonlinear, disturbances and adaptive tracking are considered from the perspective of theoretical analysis. Quadratic stabiliza-tion, Lyapunov stabilization, linear matrix inequalities and other important theories and methods are used in this paper. The paper studys the synthe-sis problems of the systems state feedback quadratic stabilization controller, stability controller constrained under the H∞bounded normγ, robust H∞guaranteed cost controller, adaptive output feedback controller and adaptive tracking controller, respectively. A relatively unified processing framework is proposed. The results show that the integrated problems of various systems stabilizing controller can be transformed into the feasibility of solutions of the corresponding Riccati equations (inequalities) or linear matrix inequali-ties under the certain conditions, however, the solution of Riccati equations (inequalities) and linear matrix inequalities have the sophisticated algorithms. So the research results are operable which establish the foundation for further study robust control problem of uncertain time-delay systems.
引文
[1]姜长生,吴庆宪,陈文华.现代鲁棒控制基础[M].哈尔滨：哈尔滨工业大学出版社,2005.
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[6]郑大钟.线性系统理论(第2版)[M].北京：清华大学出版社,2007.
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[11]SHIEH CHENG-SHION. Robust control of uncertain systems with time-varying state and control input delays [J]. Journal of Mathematical Con-trol and Information,2002,19:353-361.
[12]R.M. Palhares, C.D. Campos, P.Y. Ekel. Delay-dependent robust H∞ control of uncertain linear systems with time-varying delays [J]. Com-puters and Mathematics with Applications,2005,50:13-32.
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[17]Dong Yue. Robust stabilization of uncertain systems with unknow input delay [J]. Automatica,2004,40:331-336.
[18]X.M. Zhang, M. Wu, J.H. She, Y. He. Delay-dependent stabilization of linear systems with time-varying state and input delays [J]. Automatica, 2005,41:1405-1412.
[19]X.M. Zhang, M. Wu, Q.L. Han, J.H. She. A new integral inequality approach to delay-dependent robust H∞ control [J]. Asian Journal of Control,2006,8:153-160.
[20]W.H. Chen, Z.H. Guan, X.M. Lu. Delay-dependent output feedback guaranteed cost control for uncertain time-delay systems [J]. Automat-ica,2004,40:1263-1268.
[21]N. Xie, G.Y. Tang. Delay-dependent nonfragile guaranteed cost control for nonlinear time-delay systems [J]. Nonlinear Analysis,2006,64:2084-2097.
[22]H.Y. Zhao, Q.W. Chen, S.Y. Xu. H∞ guaranteed cost control for uncer-tain Markovian jump systems with mode-dependent distributed delays and input delays [J]. Journal of the Franklin Institute,2009,346:945-957.
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[25]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.
[26]X.F. Zhang, Z.L. Cheng. Output feedback stabilization of nonlinear sys-tems with delays in the input [J]. Applied Mathematics and Computa-tion,2005,167:1026-1040.
[27]O.M. Kwon, J.H. Park, S.M. Lee. On robust stability criterion for dy-namic systems with time-varying delays and nonlinear perturbations [J]. Applied Mathematics and Computation,2008,203:937-942.
[28]C.C. Hua, X.P. Guan, P. Shi. Robust stabilization of a class of nonlinear time-delay systems [J]. Applied Mathematics and Computation,2004, 155:737-752.
[29]C.H. Lien, K.W. Yu, Y.F. Lin, Y.J. Chung, L.Y. Chung. Robust reli-able H∞ control for uncertain nonlinear systems via LMI approach [J]. Applied Mathematics and Computation,2008,198:453-462.
[30]J.H. Park, O. Kwon. Novel stability criterion of time delay systems with nonlinear uncertainties [J]. Applied Mathematics Letters,2005,18:683-688.
[31]X.H. Jiao, T.L. Shen, Y.Z. Sun. Further result on robust stabilization for uncertain nonlinear time-delay systems [J]. Acta Automatica Sinica, 2007,33:164-169.
[32]R. Postoyan, T.A. Ali, L. Burlion. On the Lyapunov-based adaptive control redesign for a class of nonlinear sampled-data systems [J]. Auto-matica,2008,44:2099-2107.
[33]Angus E. Taylor, David C. Lay. Introduction to functional analysis (sec-ond edition) [M]. New York:Chichester Brisbane Toronto,1980.
[34]L.Y. Wang, W. Zhan. Robust disturbance attenuation with stability for linear systems with norm-bounded nonlinear uncertainties [J]. IEEE Transactions on Automatic Control,1996, AC-41:886-888.
[35]闵颖颖,刘允刚.Barbalat引理及其在系统稳定性分析中的应用[J].山东大学学报(工学版),2007,37：51-55.
[36]Z. Li, G.R. Chen, S.J. Shi. Robust adaptive tracking control for a class of uncertain chaotic systems [J]. Physics Letters A,2003,310:40-43.
[37]X.Z. Jin, G.H. Yang. Adaptive tracking and disturbance rejection for a class of distributed systems [J]. Acta Automatica Sinica,2009,35:1114-1120.
[38]L. Magni, G.D. Nicolao. Output feedback and tracking of nonlinear sys-tems with model predictive control [J]. Automatica,2001,37:1601-1607.
[39]X.D. Ye, Z.T. Ding. Robust tracking control of uncertain nonlinear sys-tems with unknown control directions [J]. Systems & Control Letters, 2001,42:1-10.
[40]Y.G. Niu, C.W. Yang. An adaptive neural robust controller for trajectory tracking of robot manipulators [J]. Control Theory and Applications, 2000,17:924-928.
[41]H.S. Wu. Adaptive robust tracking and model following of uncertain dynamical systems with multiple time delays [J]. IEEE Transactions on Automatic Control,2004,49:611-666.
[42]H.S. Wu. Adaptive robust control of uncertain nonlinear systems with nonlinear delayed state perturbations [J]. Automatica,2009,45:1979-1984.
[43]H. K. Khali著.朱义胜,董辉,李作洲等译.非线性系统(第三版)[M].北京：电子工业出版社,2002.
[44]Fang Mei. Delay-dependent robust H∞ control for uncertain singular systems with a state delay [J]. Acta Automatica Sinica,2009,35:65-70.
[2]梅生伟,申铁龙,刘康志.现代鲁棒控制理论与应用(第2版)[M].北京：清华大学出版社,2008.
[3]薛安克.鲁棒最优控制理论与应用[M].北京：科学出版社,2008.
[4]E. Fridman, U. Shaked. An improved stabilization method for linear time-delay systems [J]. IEEE Transactions on Automatic Control,2002, 47:1931-1937.
[5]J.H. Kim. Delay and its time-derivative dependent robust stability of time-delayed linear systems with uncertainty [J]. IEEE Transactions on Automatic Control,2001,46:789-792.
[6]郑大钟.线性系统理论(第2版)[M].北京：清华大学出版社,2007.
[7]Boyd S, Ghaoui E, Feron E, et al. Linear matrix inequalities in systems and control theory [M]. Philadel phia:SIAM,1994.
[8]L. Xie. Output feedback H∞ control of systems with parameter uncer-tainty [J]. International Journal of Control,2000,63:741-750.
[9]E. Fridman. Delay-dependent stability and H∞ control:constant and time-varying delays [J]. International Journal of Control,2003,76:48-60.
[10]C. Peng, Y.C. Tian. Delay-dependent robust H∞ control for uncertain systems with time-varying delay [J]. Information Sciences,2009,179: 3187-3197.
[11]SHIEH CHENG-SHION. Robust control of uncertain systems with time-varying state and control input delays [J]. Journal of Mathematical Con-trol and Information,2002,19:353-361.
[12]R.M. Palhares, C.D. Campos, P.Y. Ekel. Delay-dependent robust H∞ control of uncertain linear systems with time-varying delays [J]. Com-puters and Mathematics with Applications,2005,50:13-32.
[13]D. Yue, S.C. Won. An improvement on delay and its time-derivative de-pendent robust stability of time-delayed linear systems with uncertainty [J]. IEEE Transactions on Automatic Control,2002,47:407-408.
[14]V.N. Phat. New stabilization criteria for linear time-varying systems with state delay and norm-bounded uncertainties [J]. IEEE Transactions on Automatic Control,2002,47:2095-2098.
[15]X.F. Jiang, Q.L. Han. New stability criteria for linear systems with in-terval time-varying delay [J]. Automatica,2008,44:2680-2685.
[16]J. Richard. Time-delay systems:an overview of some recent advances and open problems [J]. Automatica,2003,39:1667-1694.
[17]Dong Yue. Robust stabilization of uncertain systems with unknow input delay [J]. Automatica,2004,40:331-336.
[18]X.M. Zhang, M. Wu, J.H. She, Y. He. Delay-dependent stabilization of linear systems with time-varying state and input delays [J]. Automatica, 2005,41:1405-1412.
[19]X.M. Zhang, M. Wu, Q.L. Han, J.H. She. A new integral inequality approach to delay-dependent robust H∞ control [J]. Asian Journal of Control,2006,8:153-160.
[20]W.H. Chen, Z.H. Guan, X.M. Lu. Delay-dependent output feedback guaranteed cost control for uncertain time-delay systems [J]. Automat-ica,2004,40:1263-1268.
[21]N. Xie, G.Y. Tang. Delay-dependent nonfragile guaranteed cost control for nonlinear time-delay systems [J]. Nonlinear Analysis,2006,64:2084-2097.
[22]H.Y. Zhao, Q.W. Chen, S.Y. Xu. H∞ guaranteed cost control for uncer-tain Markovian jump systems with mode-dependent distributed delays and input delays [J]. Journal of the Franklin Institute,2009,346:945-957.
[23]J.C. Wang. Robust H∞ output feedback controller design for linear time-varying uncertain systems with delayed state [J]. Control Theory and Applications,1999,16:334-344.
[24]Le V. Hien, Vu N. Phat. Exponential stability and stabilization of a class of uncertain linear time-delay systems [J]. Journal of the Franklin Institute,2009,346:611-625.
[25]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.
[26]X.F. Zhang, Z.L. Cheng. Output feedback stabilization of nonlinear sys-tems with delays in the input [J]. Applied Mathematics and Computa-tion,2005,167:1026-1040.
[27]O.M. Kwon, J.H. Park, S.M. Lee. On robust stability criterion for dy-namic systems with time-varying delays and nonlinear perturbations [J]. Applied Mathematics and Computation,2008,203:937-942.
[28]C.C. Hua, X.P. Guan, P. Shi. Robust stabilization of a class of nonlinear time-delay systems [J]. Applied Mathematics and Computation,2004, 155:737-752.
[29]C.H. Lien, K.W. Yu, Y.F. Lin, Y.J. Chung, L.Y. Chung. Robust reli-able H∞ control for uncertain nonlinear systems via LMI approach [J]. Applied Mathematics and Computation,2008,198:453-462.
[30]J.H. Park, O. Kwon. Novel stability criterion of time delay systems with nonlinear uncertainties [J]. Applied Mathematics Letters,2005,18:683-688.
[31]X.H. Jiao, T.L. Shen, Y.Z. Sun. Further result on robust stabilization for uncertain nonlinear time-delay systems [J]. Acta Automatica Sinica, 2007,33:164-169.
[32]R. Postoyan, T.A. Ali, L. Burlion. On the Lyapunov-based adaptive control redesign for a class of nonlinear sampled-data systems [J]. Auto-matica,2008,44:2099-2107.
[33]Angus E. Taylor, David C. Lay. Introduction to functional analysis (sec-ond edition) [M]. New York:Chichester Brisbane Toronto,1980.
[34]L.Y. Wang, W. Zhan. Robust disturbance attenuation with stability for linear systems with norm-bounded nonlinear uncertainties [J]. IEEE Transactions on Automatic Control,1996, AC-41:886-888.
[35]闵颖颖,刘允刚.Barbalat引理及其在系统稳定性分析中的应用[J].山东大学学报(工学版),2007,37：51-55.
[36]Z. Li, G.R. Chen, S.J. Shi. Robust adaptive tracking control for a class of uncertain chaotic systems [J]. Physics Letters A,2003,310:40-43.
[37]X.Z. Jin, G.H. Yang. Adaptive tracking and disturbance rejection for a class of distributed systems [J]. Acta Automatica Sinica,2009,35:1114-1120.
[38]L. Magni, G.D. Nicolao. Output feedback and tracking of nonlinear sys-tems with model predictive control [J]. Automatica,2001,37:1601-1607.
[39]X.D. Ye, Z.T. Ding. Robust tracking control of uncertain nonlinear sys-tems with unknown control directions [J]. Systems & Control Letters, 2001,42:1-10.
[40]Y.G. Niu, C.W. Yang. An adaptive neural robust controller for trajectory tracking of robot manipulators [J]. Control Theory and Applications, 2000,17:924-928.
[41]H.S. Wu. Adaptive robust tracking and model following of uncertain dynamical systems with multiple time delays [J]. IEEE Transactions on Automatic Control,2004,49:611-666.
[42]H.S. Wu. Adaptive robust control of uncertain nonlinear systems with nonlinear delayed state perturbations [J]. Automatica,2009,45:1979-1984.
[43]H. K. Khali著.朱义胜,董辉,李作洲等译.非线性系统(第三版)[M].北京：电子工业出版社,2002.
[44]Fang Mei. Delay-dependent robust H∞ control for uncertain singular systems with a state delay [J]. Acta Automatica Sinica,2009,35:65-70.