一个混沌系统和一个超混沌系统的控制与同步
|
摘要
混沌动力学是一门新兴的学科,混沌本身是不稳定的,对初值非常敏感。混沌吸引子的存在性由两个条件确定:一是有吸引域,保证吸引子的存在,二是在吸引子上存在混沌行为。本文围绕非线性动力系统的混沌系统的控制与同步问题进行了深入的研究与探讨。主要包括以下几个方面的内容:
第一章,介绍了混沌系统的背景和现状。
第二章,阐述了混沌的定义,以及混沌动力学反馈控制与同步的基本概念和基本理论。
第三章,本章讨论最近提出的一个新混沌系统的控制与同步问题。在系统的控制中,设计了在参数已知时的两个线性反馈控制器,得到了将系统控制到任意不稳定平衡点的充分条件,并且对控制前后的情况用图像作了比较,形象地说明了控制的有效性和科学性;在系统的同步中,设计了两个同步问题:1.在参数已知时,设计了两个非线性控制器,得到了实现全局完全同步的充分条件;2.在参数未知时,设计了三个非线性控制器和参数估计的自适应律,得到了实现全局完全同步的充分条件。理论与仿真结果都验证了本章结论的有效性。
第四章,本章讨论最近提出的一个新超混沌系统的控制及同步问题。在参数已知时,设计了一个线性反馈控制器和一个速度反馈控制器,得到将系统控制到不稳定平衡点的充分条件,并且对控制前后的情况用图像作了比较,形象地说明了控制的有效性和科学性;在参数未知的时候,科学地设计了四个非线性控制器和参数估计的自适应律,得到了实现全局完全同步的充分条件。理论与仿真结果都验证了本章结论的有效性和科学性。
第五章,给出了的工作总结和混沌研究及应用的展望。
Chaotic dynamics is a new subject, chaotic is unstable itself, it is sensitive to the initial value. A chaotic attractor is ensured by two things, one is that the system has a trapping region which guarantees the existence of an attractor, the other is that the system displays chaotic behavior on the attractor. The thesis mainly focuses on the control and synchronization of chaos of nonlinear system. The main content is depicted as follows:
In chapter 1, background and study conditions of chaos are introduced.
In chapter 2, the basic theory and the definition of the control and synchronization for chaos dynamics are presented.
In chapter 3, we consider the issues of chaotic control and chaotic synchronization for a new presented chaotic system. In the chaotic control, when the parameter is known, two linear controllers are designed and a sufficient condition is presented to stabilize the chaotic system to any unstable equilibrium points, control the pre-image and control over their images are compared to more graphically illustrate the control of an effective and scientific. Meanwhile, In the chaotic synchronization, two different issues of chaotic synchronization are designed: 1. when the parameter is known, two nonlinear controllers are designed and a sufficient condition is presented to assure globally complete synchronization; 2. when the parameter is unknown, three nonlinear controllers are designed, an adaptive role for estimating the unknown parameter is found and a sufficient condition is presented to assure globally complete synchronization. Theoretical analysis and numerical simulation show that the design is feasible.
In chapter 4, we consider the issues of hyper-chaotic control and hyper-chaotic synchronization for a new presented hyper-chaotic system: In the hyper-chaotic control , when the parameter is known, two controls are designed scientifically, One is linear controller, the other is velocity feedback controller, and sufficient conditions are presented to stabilize the chaotic system to any appointed unstable equilibrium points, control the pre-image and control over their images are compared to more graphically illustrate the control of an effective and scientific. In the hyper-chaotic synchronization, when the parameter is unknown, four nonlinear controllers and an adaptive role for estimating the unknown parameter are designed scientifically, and a sufficient condition is presented to assure globally complete synchronization. Theoretical analysis and numerical simulation show that the design is feasible.
In chapter 5, we give the summary of the work and the prospect of research and application of chaos.
第一章,介绍了混沌系统的背景和现状。
第二章,阐述了混沌的定义,以及混沌动力学反馈控制与同步的基本概念和基本理论。
第三章,本章讨论最近提出的一个新混沌系统的控制与同步问题。在系统的控制中,设计了在参数已知时的两个线性反馈控制器,得到了将系统控制到任意不稳定平衡点的充分条件,并且对控制前后的情况用图像作了比较,形象地说明了控制的有效性和科学性;在系统的同步中,设计了两个同步问题:1.在参数已知时,设计了两个非线性控制器,得到了实现全局完全同步的充分条件;2.在参数未知时,设计了三个非线性控制器和参数估计的自适应律,得到了实现全局完全同步的充分条件。理论与仿真结果都验证了本章结论的有效性。
第四章,本章讨论最近提出的一个新超混沌系统的控制及同步问题。在参数已知时,设计了一个线性反馈控制器和一个速度反馈控制器,得到将系统控制到不稳定平衡点的充分条件,并且对控制前后的情况用图像作了比较,形象地说明了控制的有效性和科学性;在参数未知的时候,科学地设计了四个非线性控制器和参数估计的自适应律,得到了实现全局完全同步的充分条件。理论与仿真结果都验证了本章结论的有效性和科学性。
第五章,给出了的工作总结和混沌研究及应用的展望。
Chaotic dynamics is a new subject, chaotic is unstable itself, it is sensitive to the initial value. A chaotic attractor is ensured by two things, one is that the system has a trapping region which guarantees the existence of an attractor, the other is that the system displays chaotic behavior on the attractor. The thesis mainly focuses on the control and synchronization of chaos of nonlinear system. The main content is depicted as follows:
In chapter 1, background and study conditions of chaos are introduced.
In chapter 2, the basic theory and the definition of the control and synchronization for chaos dynamics are presented.
In chapter 3, we consider the issues of chaotic control and chaotic synchronization for a new presented chaotic system. In the chaotic control, when the parameter is known, two linear controllers are designed and a sufficient condition is presented to stabilize the chaotic system to any unstable equilibrium points, control the pre-image and control over their images are compared to more graphically illustrate the control of an effective and scientific. Meanwhile, In the chaotic synchronization, two different issues of chaotic synchronization are designed: 1. when the parameter is known, two nonlinear controllers are designed and a sufficient condition is presented to assure globally complete synchronization; 2. when the parameter is unknown, three nonlinear controllers are designed, an adaptive role for estimating the unknown parameter is found and a sufficient condition is presented to assure globally complete synchronization. Theoretical analysis and numerical simulation show that the design is feasible.
In chapter 4, we consider the issues of hyper-chaotic control and hyper-chaotic synchronization for a new presented hyper-chaotic system: In the hyper-chaotic control , when the parameter is known, two controls are designed scientifically, One is linear controller, the other is velocity feedback controller, and sufficient conditions are presented to stabilize the chaotic system to any appointed unstable equilibrium points, control the pre-image and control over their images are compared to more graphically illustrate the control of an effective and scientific. In the hyper-chaotic synchronization, when the parameter is unknown, four nonlinear controllers and an adaptive role for estimating the unknown parameter are designed scientifically, and a sufficient condition is presented to assure globally complete synchronization. Theoretical analysis and numerical simulation show that the design is feasible.
In chapter 5, we give the summary of the work and the prospect of research and application of chaos.
引文
[1] Lorenz En. Deterministic non-periods flows[J]. Journal of the Atmospheric Science, 1963, 20: 130-141.
[2]吴祥兴,陈忠.混沌学导论[M].上海:上海科学技术文献出版社,1996.
[3]黄润生,黄浩.混沌及其应用(第二版).武汉:武汉大学出版社,2005.
[4]吕金虎,陆军安,陈士华.混沌时间序列分析及其应用[M].武汉:武汉大学出版社,2002.
[5]张建树,管忠,于学文.混沌生物学(第二版)[M].北京:科学出版社,2006.
[6] Hao B L. Chaos[M]. Singapore, World Scientific, 1984.
[7] Hubler A W.Adaptive control of chaotic systems[J].Helv. Phys. Acta,1989,62:343.
[8] Ott, Edward, Grebogi, Celso and Yorke, James A. Controlling chaos[J]. Physical Review Letters,1990, Vol.64, No.11.
[9] L.M. Pecora, T.L. Carroll. Synchronization in chaotic systems[J]. Phys Rev Lett ,1990, 64 :821-824.
[10] En.洛伦兹著,刘式达等译.混沌的本质.北京:气象出版社,1997.
[11] O.E. Rossler. Equation for continuous chaos[J]. Physics Letters A, 1976 , 57:397-398.
[12] T. Matsumoto. A chaotic attractor from Chua's circuit[J]. Circuits and Systems, IEEE Transactions on , 1984, 31:1055-1058.
[13] G. Chen, T. Ueta. Yet another chaotic attractor[J]. International Journal of Bifurcation and Chaos ,1999, 9: 1465-1466.
[14] J. Liu, G. Chen. A new chaotic attractor coined[J]. International Journal of Bifurcation and Chaos, 2002, 12:659–661.
[15] J. Liu. Bridge the gap between the Lorenz system and the Chen system[J]. International Journal of Bifurcation and Chaos, 2002,12: 2917–2926.
[16] J. Starrett. Control of chaos by occasional bang-bang[J].Phys Rev E Stat Nonlin Soft Matter Phys, 2003, 67: 036203.
[17] P.M.Alsing, A Gavrielides V.Kovanis. Using Neural Networkes For Controlling Chaos[M].Chaos Nonlinear Dynamics Methods and Commercialization, 1994,126-138.
[18] G. Chen, X. Dong. On feedback control of chaotic continuous-time systems[J]. Circuits and Systems I: Fundamental Theory and Applications, 1993, 591-601.
[19] D.Q. Wei, X.S. Luo, J.Q. Fang, B.H. Wang. Controlling chaos in permanent magnet synchronous motor based on the differential geometry method[J]. Acta Physica Sinica , 2006,55: 54-59.
[20] S. Boccaletti, F.T. Arecchi. Adaptive-control of chaos[J]. Europhysics Letters, 1995,11:127-132.
[21]叶慧,姚洪兴.一类新混沌系统的自适应控制与异结构同步在保密通信中的应用[J].江苏科技大学学报(自然科学版), 2008,5:90-94.
[22] M.T. Yassen. Controlling chaos and synchronization for new chaotic system using linear feedback control[J]. Chaos Solitons & Fractals ,2005,26 :913-920.
[23] X.X. Liao, G.R. Chen. Chaos synchronization of general Lur'e systems via time-delay feedback control[J]. International Journal of Bifurcation and Chaos ,2003,13:207-213.
[24] R. Kilic, Experimental study on impulsive synchronization between two modified Chua's circuits[J]. Nonlinear Analysis-Real World Applications, 2006,7: 1298-1303.
[25]舒永录,孔昭毅,张青.一个新混沌系统的滑模变结构控制[J].重庆工商大学学报(自然科学版),2010, 1 :1-4.
[26] M. Haeri, M.S. Tavazoei, M.R. Naseh, Synchronization of uncertain chaotic systems using active sliding mode control[J]. Chaos Solitons & Fractals , 2007,33: 1230-1239.
[27] W. Xianyong, G. Zhi-Hong, W. Zhengping, Adaptive synchronization between two different hyperchaotic systems[J]. Nonlinear Analysis|Nonlinear Analysis 2008,13:46-51.
[28]张青,舒永录,张万欣.一个新超混沌系统的自适应控制与同步[J].安徽师范大学大学学报(自然科学版), 2010,3 :235-242.
[29] Shu Yong-lu, Wang Meng. Boundedness of the solusions of the T-system and its control[J]. Journal of East China Normal University(Natural science),2009,1:43-47.
[30] Shu Yong-lu, Zhang Yong. Gloabally attractive set and feedback synchronization of the BLDCM system[J]. Journal of East China Normal University(Natural science), 2010,1:62-78.
[31] Zhang Qing, Shu Yong-lu, Zhang Wan-xin. Controlling and synchronization of a new Lorenz-like chaotic system[J]. Journal of East China Normal University(Natural science), 2010,1:53-61.
[32]杨高翔,舒永录.利用周期函数对弦振动方程混沌化[J].西南大学学报(自然科学版), 2010,1:29-32.
[33]高普云.非线形动力学[M].长沙:国防科技大学出版社,2005.
[34]关新平,范正平,陈彩莲.混沌控制及其在保密通信中的应用[M].第一版,北京:国防工业出版社,2002.
[35]廖晓昕.稳定性的数学理论及应用[M].武汉:华中师范大学出版社,2001.
[36]杨晓松,李清都.混沌系统与混沌电路[M].北京:科学出版社,2007.
[37] Yau Ht, Chen Cl. Chaos control of Lorenz system using Adaptive controller with input saturation[M]. Chaos, Solitons &Fractals, 2006.
[38] Zeng Y.sing SN. Adaptive control of chaos in Lorenz system[J]. Dynamo Control1997,7:143-154.
[39]师礼,孔金生.反馈控制系统导论[M].北京:科学出版社,2005.
[40] M.R.Molaei Generalized synchronization of nuclear spin generator system[M]. Chaos, Solitons&Fractals, 2006.
[41]王奔,庄圣贤.非线性控制系统[M].北京:电子工业出版社,2005.
[42] Chen W C. Nonlinear dynamics and chaos in a fractional-order financial system[M]. Chaos, Solitons&Fractals, 2006.
[43] Wu XQ, Lu J. Adaptive feedback synchronization of Liu system[M]. Chaos, Solitons& Fractals, 2004.
[44]李言俊,张科.自适应控制理论及应用[M].西安:西北工业大学出版社,2005.
[45] Lu Jin-hua, Chen Guan-rong, Zhang Suo-chun. A Unified Chaotic System and Its Research[J]. Journal of Graduate School of the Chinese Academy of Sciences,2003,20(1):123-127.
[46] Huberman A, et al. Dynamics of adaptive systems[J].IEEE Trans,CAS,1990,37:547-550.
[47] John K,et al. Synchronization of unstabale orbits using adaptive control[J]. Phys Rev E,1994,49(6):4843-4848.
[48] Chen G, Ueta T. Yet another chaotic attractor [J]. International Journal of Bifurcation and Chaos,1999, 9: 1465–1466.
[49] Tigan Gh. Analysis of a dynamical system derived from the Lorenz system[J]. Sci Bull Politehnica University of Timisoara, Tomul50(64), Fascicola 2005,1:61–72.
[50]王光义,丘水生,陈辉等.一个新的混沌系统及其电路设计与实现[J].电路与系统学报, 2008,13: 58-60.
[51] O.E. Rossler, EQUATION FOR CONTINUOUS CHAOS[M]. Physics Letters A 57 (1976) 397-398.
[52]武相军,王兴元.基于非线性控制的超混沌Chen系统的混沌同步[J].物理学报,2006,55(12):6261-6266.
[53] Chen Zengqiang, Yang Yong, Qi Guoyuan, et al. A novel hyperchaos system only with one equilibrium[J].Physics Letters A, 2007,360 (6): 696-701.
[54] Min Xiao,Jinde Cao Synchronization of a chaotic electronic circuit system with cubic term via adaptive feedback control [J].Commun Nonlinear Sci Numer Simulat ,2009,14: 3379–3388.
[55] G. Chen, and X. Dong, On feedback control of chaotic continuous-time systems[M]. 1993, 591-601.
[56]张帅,刘文清,张玉钧等.数字滤波方法在TDLAS气体检测中的应用[J].光学学报,2010,5:1362-1367.
[57] D.Q. Wei, X.S. Luo, J.Q. Fang, and B.H. Wang, Controlling chaos in permanent magnetsynchronous motor based on the differential geometry method[J]. Acta Physica Sinica , 2006,55:54-59.
[58]倪雨,许建平.准滑模控制开关DC-DC变换器分析[J].中国电机工程学报,2008,28(21):1-9.
[59] Chung S Cy,Lin C L. A transformed Lure problem for sliding mode control and chattering reduction[J]. International Journal of Control, 1999,3:563-568.
[60]高为炳.变结构控制的理论及设计方法[M].北京:科学出版社,1996.
[2]吴祥兴,陈忠.混沌学导论[M].上海:上海科学技术文献出版社,1996.
[3]黄润生,黄浩.混沌及其应用(第二版).武汉:武汉大学出版社,2005.
[4]吕金虎,陆军安,陈士华.混沌时间序列分析及其应用[M].武汉:武汉大学出版社,2002.
[5]张建树,管忠,于学文.混沌生物学(第二版)[M].北京:科学出版社,2006.
[6] Hao B L. Chaos[M]. Singapore, World Scientific, 1984.
[7] Hubler A W.Adaptive control of chaotic systems[J].Helv. Phys. Acta,1989,62:343.
[8] Ott, Edward, Grebogi, Celso and Yorke, James A. Controlling chaos[J]. Physical Review Letters,1990, Vol.64, No.11.
[9] L.M. Pecora, T.L. Carroll. Synchronization in chaotic systems[J]. Phys Rev Lett ,1990, 64 :821-824.
[10] En.洛伦兹著,刘式达等译.混沌的本质.北京:气象出版社,1997.
[11] O.E. Rossler. Equation for continuous chaos[J]. Physics Letters A, 1976 , 57:397-398.
[12] T. Matsumoto. A chaotic attractor from Chua's circuit[J]. Circuits and Systems, IEEE Transactions on , 1984, 31:1055-1058.
[13] G. Chen, T. Ueta. Yet another chaotic attractor[J]. International Journal of Bifurcation and Chaos ,1999, 9: 1465-1466.
[14] J. Liu, G. Chen. A new chaotic attractor coined[J]. International Journal of Bifurcation and Chaos, 2002, 12:659–661.
[15] J. Liu. Bridge the gap between the Lorenz system and the Chen system[J]. International Journal of Bifurcation and Chaos, 2002,12: 2917–2926.
[16] J. Starrett. Control of chaos by occasional bang-bang[J].Phys Rev E Stat Nonlin Soft Matter Phys, 2003, 67: 036203.
[17] P.M.Alsing, A Gavrielides V.Kovanis. Using Neural Networkes For Controlling Chaos[M].Chaos Nonlinear Dynamics Methods and Commercialization, 1994,126-138.
[18] G. Chen, X. Dong. On feedback control of chaotic continuous-time systems[J]. Circuits and Systems I: Fundamental Theory and Applications, 1993, 591-601.
[19] D.Q. Wei, X.S. Luo, J.Q. Fang, B.H. Wang. Controlling chaos in permanent magnet synchronous motor based on the differential geometry method[J]. Acta Physica Sinica , 2006,55: 54-59.
[20] S. Boccaletti, F.T. Arecchi. Adaptive-control of chaos[J]. Europhysics Letters, 1995,11:127-132.
[21]叶慧,姚洪兴.一类新混沌系统的自适应控制与异结构同步在保密通信中的应用[J].江苏科技大学学报(自然科学版), 2008,5:90-94.
[22] M.T. Yassen. Controlling chaos and synchronization for new chaotic system using linear feedback control[J]. Chaos Solitons & Fractals ,2005,26 :913-920.
[23] X.X. Liao, G.R. Chen. Chaos synchronization of general Lur'e systems via time-delay feedback control[J]. International Journal of Bifurcation and Chaos ,2003,13:207-213.
[24] R. Kilic, Experimental study on impulsive synchronization between two modified Chua's circuits[J]. Nonlinear Analysis-Real World Applications, 2006,7: 1298-1303.
[25]舒永录,孔昭毅,张青.一个新混沌系统的滑模变结构控制[J].重庆工商大学学报(自然科学版),2010, 1 :1-4.
[26] M. Haeri, M.S. Tavazoei, M.R. Naseh, Synchronization of uncertain chaotic systems using active sliding mode control[J]. Chaos Solitons & Fractals , 2007,33: 1230-1239.
[27] W. Xianyong, G. Zhi-Hong, W. Zhengping, Adaptive synchronization between two different hyperchaotic systems[J]. Nonlinear Analysis|Nonlinear Analysis 2008,13:46-51.
[28]张青,舒永录,张万欣.一个新超混沌系统的自适应控制与同步[J].安徽师范大学大学学报(自然科学版), 2010,3 :235-242.
[29] Shu Yong-lu, Wang Meng. Boundedness of the solusions of the T-system and its control[J]. Journal of East China Normal University(Natural science),2009,1:43-47.
[30] Shu Yong-lu, Zhang Yong. Gloabally attractive set and feedback synchronization of the BLDCM system[J]. Journal of East China Normal University(Natural science), 2010,1:62-78.
[31] Zhang Qing, Shu Yong-lu, Zhang Wan-xin. Controlling and synchronization of a new Lorenz-like chaotic system[J]. Journal of East China Normal University(Natural science), 2010,1:53-61.
[32]杨高翔,舒永录.利用周期函数对弦振动方程混沌化[J].西南大学学报(自然科学版), 2010,1:29-32.
[33]高普云.非线形动力学[M].长沙:国防科技大学出版社,2005.
[34]关新平,范正平,陈彩莲.混沌控制及其在保密通信中的应用[M].第一版,北京:国防工业出版社,2002.
[35]廖晓昕.稳定性的数学理论及应用[M].武汉:华中师范大学出版社,2001.
[36]杨晓松,李清都.混沌系统与混沌电路[M].北京:科学出版社,2007.
[37] Yau Ht, Chen Cl. Chaos control of Lorenz system using Adaptive controller with input saturation[M]. Chaos, Solitons &Fractals, 2006.
[38] Zeng Y.sing SN. Adaptive control of chaos in Lorenz system[J]. Dynamo Control1997,7:143-154.
[39]师礼,孔金生.反馈控制系统导论[M].北京:科学出版社,2005.
[40] M.R.Molaei Generalized synchronization of nuclear spin generator system[M]. Chaos, Solitons&Fractals, 2006.
[41]王奔,庄圣贤.非线性控制系统[M].北京:电子工业出版社,2005.
[42] Chen W C. Nonlinear dynamics and chaos in a fractional-order financial system[M]. Chaos, Solitons&Fractals, 2006.
[43] Wu XQ, Lu J. Adaptive feedback synchronization of Liu system[M]. Chaos, Solitons& Fractals, 2004.
[44]李言俊,张科.自适应控制理论及应用[M].西安:西北工业大学出版社,2005.
[45] Lu Jin-hua, Chen Guan-rong, Zhang Suo-chun. A Unified Chaotic System and Its Research[J]. Journal of Graduate School of the Chinese Academy of Sciences,2003,20(1):123-127.
[46] Huberman A, et al. Dynamics of adaptive systems[J].IEEE Trans,CAS,1990,37:547-550.
[47] John K,et al. Synchronization of unstabale orbits using adaptive control[J]. Phys Rev E,1994,49(6):4843-4848.
[48] Chen G, Ueta T. Yet another chaotic attractor [J]. International Journal of Bifurcation and Chaos,1999, 9: 1465–1466.
[49] Tigan Gh. Analysis of a dynamical system derived from the Lorenz system[J]. Sci Bull Politehnica University of Timisoara, Tomul50(64), Fascicola 2005,1:61–72.
[50]王光义,丘水生,陈辉等.一个新的混沌系统及其电路设计与实现[J].电路与系统学报, 2008,13: 58-60.
[51] O.E. Rossler, EQUATION FOR CONTINUOUS CHAOS[M]. Physics Letters A 57 (1976) 397-398.
[52]武相军,王兴元.基于非线性控制的超混沌Chen系统的混沌同步[J].物理学报,2006,55(12):6261-6266.
[53] Chen Zengqiang, Yang Yong, Qi Guoyuan, et al. A novel hyperchaos system only with one equilibrium[J].Physics Letters A, 2007,360 (6): 696-701.
[54] Min Xiao,Jinde Cao Synchronization of a chaotic electronic circuit system with cubic term via adaptive feedback control [J].Commun Nonlinear Sci Numer Simulat ,2009,14: 3379–3388.
[55] G. Chen, and X. Dong, On feedback control of chaotic continuous-time systems[M]. 1993, 591-601.
[56]张帅,刘文清,张玉钧等.数字滤波方法在TDLAS气体检测中的应用[J].光学学报,2010,5:1362-1367.
[57] D.Q. Wei, X.S. Luo, J.Q. Fang, and B.H. Wang, Controlling chaos in permanent magnetsynchronous motor based on the differential geometry method[J]. Acta Physica Sinica , 2006,55:54-59.
[58]倪雨,许建平.准滑模控制开关DC-DC变换器分析[J].中国电机工程学报,2008,28(21):1-9.
[59] Chung S Cy,Lin C L. A transformed Lure problem for sliding mode control and chattering reduction[J]. International Journal of Control, 1999,3:563-568.
[60]高为炳.变结构控制的理论及设计方法[M].北京:科学出版社,1996.