连续混沌系统的最优控制
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摘要
最优控制理论是从20世纪50年代末60年代初发展起来的现代控制论的一个重要分支,其形成与发展奠定了整个现代控制理论的基础。混沌运动是一种确定性的类随机运动,它广泛存在于客观世界中,混沌系统的优化控制在系统控制中得到广泛的应用。
在本论文中,以混沌基本特征和最优控制基本理论为基础,我们给出了混沌系统最优控制的几种方法。
(1)描述了一种新的混沌系统及其动力学行为;同时根据Pontryagin极小值原理为该混沌系统设计一个线性状态反馈最优控制器,所得到的控制律是最优控制,该控制器能将混沌系统的轨道从任意初始点出发稳定到零点,控制器结构简单且易于实现。基于Lyapunov稳定性理论证明了该系统的稳定性,同时通过对施加闭环控制后的系统状态变化的数值模拟证明了所建议的方法的正确性。
(2)针对Lorenz系统提出了一种最优控制方法,将该混沌系统控制到任意所期望的状态。基于哈密顿-雅可比-贝尔曼方程将构建最优控制器问题归结为解偏微分方程问题,通过巧妙构造Lyapunov函数从而得到最优控制器。仿真结果表明该方法的有效性。
(3)讨论了参数不确定统一混沌系统的最优控制问题。给出了统一混沌系统的平衡点且分析了平衡点的稳定性,根据Pontryagin极小值原理为统一混沌系统设计一个状态反馈最优控制器,理论分析说明控制器能将不稳定的平衡点镇定。通过对参数不确定统一混沌系统的数值模拟表明了方法的有效性。
(4)采用最小时间控制方法为Rossler的同步误差系统设计了一个非线性状态反馈控制器,基于Lyapunov稳定性理论,证明了所设计的控制器能够使受控误差系统全局渐近稳定到同步误差系统的零点。并且使所提出的目标泛函取得极小值。数值仿真表明,所设计的控制器实用有效并且易于实现。
The optimal control theory which developed from the late 1950s to the early 1960s is an important branch of modern control theory; its formation and development have laid the foundation of modern control theory. Chaotic motion is a complex motion, whose equation is certain but the trajectory of the orbits is stochastic. There are lots of chaotic phenomena in real world, the chaotic systems which have a high application value are a kind of the special nonlinear system, and the optimal control of chaotic systems gets widespread application in the control of systems.
In this paper, we present some techniques for the optimal control of chaotic systems based on the basic characteristics of chaotic systems and the basic theory of optimal control.
(1) The dynamic behaviors of a new chaotic system are described. Then a quadratic performance is given and a simple linear state feedback controller is designed based on Pontryagin Minimum Principle and the resulted control law is proved to be optimal control. The system orbit can be controlled to its originally unstable zero equilibrium point by the designed controller. The structure of the controller is simple and the controller is easy to attain. Based on Lyapunov functional method, the stability of system is proved. Simulation results of the transient process of the states of the closed control system are provided to demonstrate the effectiveness of the suggested scheme.
(2)A method of optimal control which takes the orbit of system to any desired point is provided based on Lorenz system. Based on Hamilton-Jacobi-Bellman equation, the problem of constructing the optimal controller comes down to the problem of solving the partial differential equations. The optimal controller is obtained through constructing Lyapunov function. The result of simulation shows the effectiveness of the method.
(3)The paper is devoted to discuss the problem of optimal control of unified chaotic systems with complete unknown parameters. The equilibrium points of unified chaotic systems were given and stability of the equilibrium points was analyzed. Then the state feedback controllers are designed based on Pontryagin Minimum Principle. Theorize analysis show that the equilibrium points of systems, which are essentially unstable, can be stabilized by the optimal controllers. The effectiveness of the method is verified according to numerical simulations of unified chaotic systems with complete unknown parameters.
(4) A nonlinear state feedback controller is designed for controlling the error system of synchronization of Rossler using time minimization control approach. Based on the Lyapunov stability theory, the designed controller is proved enable to globally stabilize asymptotically the controlled system to its zero point and minimize the proposed cost functional. The numerical simulation shows the effectiveness and readiness of the controller.
在本论文中,以混沌基本特征和最优控制基本理论为基础,我们给出了混沌系统最优控制的几种方法。
(1)描述了一种新的混沌系统及其动力学行为;同时根据Pontryagin极小值原理为该混沌系统设计一个线性状态反馈最优控制器,所得到的控制律是最优控制,该控制器能将混沌系统的轨道从任意初始点出发稳定到零点,控制器结构简单且易于实现。基于Lyapunov稳定性理论证明了该系统的稳定性,同时通过对施加闭环控制后的系统状态变化的数值模拟证明了所建议的方法的正确性。
(2)针对Lorenz系统提出了一种最优控制方法,将该混沌系统控制到任意所期望的状态。基于哈密顿-雅可比-贝尔曼方程将构建最优控制器问题归结为解偏微分方程问题,通过巧妙构造Lyapunov函数从而得到最优控制器。仿真结果表明该方法的有效性。
(3)讨论了参数不确定统一混沌系统的最优控制问题。给出了统一混沌系统的平衡点且分析了平衡点的稳定性,根据Pontryagin极小值原理为统一混沌系统设计一个状态反馈最优控制器,理论分析说明控制器能将不稳定的平衡点镇定。通过对参数不确定统一混沌系统的数值模拟表明了方法的有效性。
(4)采用最小时间控制方法为Rossler的同步误差系统设计了一个非线性状态反馈控制器,基于Lyapunov稳定性理论,证明了所设计的控制器能够使受控误差系统全局渐近稳定到同步误差系统的零点。并且使所提出的目标泛函取得极小值。数值仿真表明,所设计的控制器实用有效并且易于实现。
The optimal control theory which developed from the late 1950s to the early 1960s is an important branch of modern control theory; its formation and development have laid the foundation of modern control theory. Chaotic motion is a complex motion, whose equation is certain but the trajectory of the orbits is stochastic. There are lots of chaotic phenomena in real world, the chaotic systems which have a high application value are a kind of the special nonlinear system, and the optimal control of chaotic systems gets widespread application in the control of systems.
In this paper, we present some techniques for the optimal control of chaotic systems based on the basic characteristics of chaotic systems and the basic theory of optimal control.
(1) The dynamic behaviors of a new chaotic system are described. Then a quadratic performance is given and a simple linear state feedback controller is designed based on Pontryagin Minimum Principle and the resulted control law is proved to be optimal control. The system orbit can be controlled to its originally unstable zero equilibrium point by the designed controller. The structure of the controller is simple and the controller is easy to attain. Based on Lyapunov functional method, the stability of system is proved. Simulation results of the transient process of the states of the closed control system are provided to demonstrate the effectiveness of the suggested scheme.
(2)A method of optimal control which takes the orbit of system to any desired point is provided based on Lorenz system. Based on Hamilton-Jacobi-Bellman equation, the problem of constructing the optimal controller comes down to the problem of solving the partial differential equations. The optimal controller is obtained through constructing Lyapunov function. The result of simulation shows the effectiveness of the method.
(3)The paper is devoted to discuss the problem of optimal control of unified chaotic systems with complete unknown parameters. The equilibrium points of unified chaotic systems were given and stability of the equilibrium points was analyzed. Then the state feedback controllers are designed based on Pontryagin Minimum Principle. Theorize analysis show that the equilibrium points of systems, which are essentially unstable, can be stabilized by the optimal controllers. The effectiveness of the method is verified according to numerical simulations of unified chaotic systems with complete unknown parameters.
(4) A nonlinear state feedback controller is designed for controlling the error system of synchronization of Rossler using time minimization control approach. Based on the Lyapunov stability theory, the designed controller is proved enable to globally stabilize asymptotically the controlled system to its zero point and minimize the proposed cost functional. The numerical simulation shows the effectiveness and readiness of the controller.
引文
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[2]吴受章.最优控制理论与应用[M].机械工业出版社,2008
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[7] Li C, Chen G. An improved version of the Marotto theorem[J]. Chaos, Soliton and Fractal, 2003,18(1):69-77
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[10] Tsien H S. Engineering Cybernetics[M]. McGraw-Hill Book Company. NewYork,1954
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[18] Elabbasy E M, Agiza H N, Dessoky M M.Adaptive synchronization of Lu system with uncertain parameters[J]. Chaos, Solitons & Fractals , 2004,21:657–667
[19] Park J H.Adaptive synchronization of a unified chaotic system with an uncertain parameter[J]. Int J Nonlinear Sci Numer Simulat, 2005, 6:201-206
[20] Li Z, Chen G, Robust adaptive synchronization of uncertain dynamical networks[J]. Physics Letters A, 2004, 324:166-178
[21] Yang S K, Chen C L, Yau H T. Control of chaos in Lorenz system[J]. Chaos, Solitons & Fractals, 2002, 13:767-780
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[23] Jiang G P, Zheng W X. Chaos control for a class of chaotic systems using PI-type state observer approach[J]. Chaos, Solitons & Fractals, 2004, 21:93-96
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[26] Park J H. Controlling chaotic systems via nonlinear feedback control[J]. Chaos, Solitons & Fractals, 2005(23):1049-1054
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[41] Awad El-Gohary,Ammar Sarhan.Optimal control and synchronization of Lorenz system with complete unknown parameters[J].Chaos,Solitons and Fractals,2006,30(5):1122-1132
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[2]吴受章.最优控制理论与应用[M].机械工业出版社,2008
[3]郝柏林.分岔、混沌、奇怪吸引子、湍流及其他[J].物理学进1983,3(3): 329-416
[4]陈关荣,吕金虎. Lorenz系统族的动力学分析、控制与同步[M].科学出版社,北京,2003
[5]吕金虎,陆君安,陈士华.混沌时间序列分析[M].武汉大学出版社.武汉, 2002
[6] Li T Y,Yorke J A.Period three implies chaos[J].Amer. Math. Monthly,1975,82:985-992
[7] Li C, Chen G. An improved version of the Marotto theorem[J]. Chaos, Soliton and Fractal, 2003,18(1):69-77
[8]陈士华,陆君安.混沌动力学初步.[M]武汉水利电力大学出版社.武汉,1993
[9] Chen G, Dong X. From chaos to order: Methodologies, Perspectives and Application[M]. Singapore: World Scientific, 1998
[10] Tsien H S. Engineering Cybernetics[M]. McGraw-Hill Book Company. NewYork,1954
[11]王光瑞,于熙龄,陈式刚.混沌的控制、同步与利用[M].国防科技大学出版社.北京,2001
[12]邹恩,李详飞,陈建国.混沌控制及其优化应用[M].国防科技大学出版社.长沙,2002
[13]胡岗,萧井华,郑志刚.混沌控制[M].上海科技教育出版社,上海,2000
[14]陈明杰.连续混沌系统及其应用研究[D].哈尔滨工程大学博士论文,2005
[15] Frison T W.Controlling chaos with a neural network[J] , Proc. of Int Conference on Neural Networks ,Baltimore,MO,June,75-80
[16] Yassen M T.Adaptive synchronization of Rosssler and Lu systems with fully uncertain parameters[J].Chaos, Solitons & Fractals, 2005, 23:1245–1251
[17] Han X, Lu J A, Wu X.Adaptive feedback synchronization of Lu systems[J]. Chaos, Solitons & Fractals , 2004,22:221–227
[18] Elabbasy E M, Agiza H N, Dessoky M M.Adaptive synchronization of Lu system with uncertain parameters[J]. Chaos, Solitons & Fractals , 2004,21:657–667
[19] Park J H.Adaptive synchronization of a unified chaotic system with an uncertain parameter[J]. Int J Nonlinear Sci Numer Simulat, 2005, 6:201-206
[20] Li Z, Chen G, Robust adaptive synchronization of uncertain dynamical networks[J]. Physics Letters A, 2004, 324:166-178
[21] Yang S K, Chen C L, Yau H T. Control of chaos in Lorenz system[J]. Chaos, Solitons & Fractals, 2002, 13:767-780
[22] Yau H T, Yan J J. Design of sliding mode controller for Lorenz chaotic system with nonlinear input[J]. Chaos, Solitons & Fractals, 2004, 19:891-898
[23] Jiang G P, Zheng W X. Chaos control for a class of chaotic systems using PI-type state observer approach[J]. Chaos, Solitons & Fractals, 2004, 21:93-96
[24] Tsai H H, Fuh C C, Chang C N. A robust controller for chaotic systems under external excitation[J]. Chaos, Solitons & Fractals, 2002, 14:627-632
[25] Wu X, Lu J. Adaptive control of uncertain Lu system[J]. Chaos, Solitons & Fractals, 2004, 22:375-381
[26] Park J H. Controlling chaotic systems via nonlinear feedback control[J]. Chaos, Solitons & Fractals, 2005(23):1049-1054
[27] Yu X. Controlling chaos using input-output linearization approach[J]. Int J Bifurc Chaos,1997,7(7):1664-1695
[28] Wan C J, Sbernstein D. Nonlinear feedback control with global stabilization[J]. Dyn Contr, 1995, 5:321-346
[29] K Pyragas. Continous control of chaos by self-controlling feedback[J]. Phys Rev A, 1992, 170(6):421-428
[30] Sun J. Delay-dependent stability criteria for time-delay chaotic systems via time-delay feedback control[J]. Chaos, Solitons & Fractals, 2004, 21:143–150
[31] T.Kapitaniak Controlling Chaos: Theoretical and Practical Methods in Non-linear Dynamics[M]. Academic, New York, 1996
[32] Yu X. Variable structure control approach for controlling chaos[J]. Chaos, Solitons & Fractals, 1997,8(9):1577-1586
[33] Ott E,Grebogi C,Yorke J A.Controlling chaos[J].Phys. Rev. Lett,1990,64:1196-1199
[34] Pcora L M ,Carroll T L.Synchronization in chaotic systems[J].Phys. Rev. Lett,1990,64(8):821-824
[35] Carroll T L,Pecora L M.Synchronizing chaotic circuits[J].IEEE Trans Circ Syst I,1991,38:453-456
[36]单梁,李军,王执铨.基于统一混沌系统的广义同步[J].信息与控制,2004,33(6):689-693
[37] Yassen M T.The optimal control of Chen chaotic dynamical system[J].Applied Mathematics and Computation,2002,131:171-180
[38] Awad El-Gohary,F.Bukhari.Optimal control of Lorenz system during different time intervals[J].Applied Mathematics and Computation,2003,144:337-351
[39] Marat Rafikov,Jose Manoel Balthazar.On an optimal control design for Rossler system[J].Physics Letters A,2004,333(3-4):241-245
[40] Awad El-Gohary.Optimal synchronization of Rossler system with complete uncertain parameters[J].Chaos,Solitons and Fractals,2006,27(2):345-355
[41] Awad El-Gohary,Ammar Sarhan.Optimal control and synchronization of Lorenz system with complete unknown parameters[J].Chaos,Solitons and Fractals,2006,30(5):1122-1132
[42]高洁,陆君安.不确定参数下的四维超混沌吕系统的最优同步[J].动力学与控制学报,2006,4(4):320-325
[43] Awad El-Gohary.Chaos and optimal control of cancer self-remission and tumor system steady states[J].Chaos,Solitons and Fractals,2008,37:1305-1316
[44] Freeman R A,Kokotovic P V.Inverse optimality in robust stabilization [J].SIAM Journal on Control and Optimization,1996,34:1365-1391
[45] Krstic M,Li Z H.Inverse optimal design of input-to-state stabilizing nonlinear controllers [J].IEEE Trans Automatic Control,1998,43(3):336-350
[46] Sanchez E N,Perez J P,Martinez M,Chen G.Chaos stabilization:an inverse optimal control approach [J].Latin Amer App Res:Int J,2002,32:111-114
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