本論文中利用適應性步階迴歸追蹤控制技術成功地使得兩個具有匹配與非匹配擾動的不同混沌系統得以達到混沌同步。在所提出的控制法則中引入適應性調控機制，因此可以輕易調適擾動的上界，而其資訊不需事先知道。所提出引入適應性調控機制的強健性步階迴歸追蹤控制器確實可將slave混沌系統的軌跡追蹤到master混沌系統的軌跡。最後，兩個數值實例及電腦模擬可驗證理論分析的正確性。

In this thesis the synchronization of two different chaotic systems with matched and mismatched perturbations are developed by utilizing adaptive backstepping control technique. The adaptive mechanisms embeded in the proposed control scheme is used to adapt the unknown upper bounds of the perturbations. The resultant robust backstepping tracking controller with adaptive mechanisms can indeed drive the trajectories of the slave system to track those of the master system. Two numerical examples and simulations are given to illustrate the correctness of theoretical analyses.

Contents

Abstract i
List of Figures iv
List of Notations vi
Chapter 1 Introduction 1
1.1 Motivation……………………………………………1
1.2 Brief Sketch of the Contents………………………3
Chapter 2 Robust Control Design for Perturbed CS
Systems 5
2.1 Systems Description and General Assumptions…….7
2.2 Stability Analysis and Design Tracking Controller
for the Mismatched Perturbed Cases………………..9
2.3 Stability Analysis and Design Tracking Controller
for the Matched Perturbed Cases……………………27
Chapter 3 Numerical Examples and Simulations 35
3.1 Chaotic synchronization with mismatched
perturbations……………………………………….35
3.2 The Matched Perturbations Cases Example………..37
Chapter 4 Conclusions and Future Works 53
Bibliography 53
Appendix A 59
Appendix B 64
Appendix C 65

[1] L. M. Pecora, and T. L. Carroll, ``Synchronization in chaotic systems", Phys. Rev. Lett. Vol.64, No.8, pp.821-824, 1990.
[2] H. T. Yau, ``Design of adaptive sliding mode controller for chaos synchronization with uncertainties", Chaos, Solitons & Fractals, Vol.22, Issue 2, pp.341-347, 2004.
[3] J. J. Yan, Y. S. Yang, T. Y. Chiang, and C. Y. Chen, ``Robust synchronization of unified chaotic systems via sliding mode control", Chaos, Solitons & Fractals, Vol.34, Issue3, pp.947-954,2007.
[4] Z. Li, and S. Shi, ``Robust adaptive synchronization of Rossler and Chen chaotic systems via slide technique", Physics Letters A, Vol.311, Issue 4-5, pp.389-395, 2003.
[5] M. Haeri, M. S. Tavazoei, and M. R. Naseh, ``Synchronization of uncertain chaotic systems using active sliding mode control", Chaos, Solitons & Fractals, Vol.33, Issue 4, pp.1230-1239, 2007.
[6] G. H. Li, ``Generalized projective synchronization of two chaotic systems by using active control", Chaos, Solitons & Fractals, Vol.30, Issue 1, pp.77-82, 2006.
[7] H. N. Agiza, and M. T. Yassen, ``Synchronization of Rossler and Chen chaotic dynamical systems using active control", Physics Letters A, Vol.278, Issue 4, pp.191-197, 2001.
[8] Y. L. Liu, J. Zhu, and D. W. Ding, ``Adaptive synchronization of Chen chaotic system with uncertain parameters", Journal of China Universities of Posts and Telecommunications, Vol.14, No.2, pp.103-105, 2007.
[9] Y. Wang, Z. H. Guan, and T. S. Zhang, ``Adaptive synchronization for Chen chaotic system with fully unknown parameters", Chaos, Solitons & Fractals, Vol.19, Issue 4, pp.899-903, 2004.
[10] F. M. Moukam Kakmeni, S. Bowong, and C. Tchawoua, ``Nonlinear adaptive synchronization of a class of chaotic systems", Physics Letters A, Vol.355, Issue 1, pp.47-54, 2006.
[11] T. Botmart, and P. Niamsup, ``Adaptive control and synchronization of the perturbed Chua's system", Mathematics and Computers in Simulation, Vol.75, Issues 1-2, pp.37-55, 2007.
[12] J. Zhou, and M. J. Er, ``Adaptive output control of a class of uncertain chaotic systems", Systems & Control Letters, Vol.56, Issue 6, pp.452-460, 2007.
[13] B. Wang, and G. Wen, ``On the synchronization of a class of chaotic systems based on backstepping method", Physics Letters A, Vol.370, Issue 1, pp.35-39, 2007.
[14] J. H. Park, ``Synchronization of Genesio chaotic system via backstepping approach", Chaos, Solitons & Fractals, Vol.27, Issue 5, pp.1369-1375, 2006.
[15] Y. Yu, and S. Zhang, ``Adaptive backstepping synchronization of uncertain chaotic system", Chaos, Solitons & Fractals, Vol.21, Issue 3, pp.643-649, 2004.
[16] C. Wang, and S. S. Ge, ``Adaptive synchronization of uncertain chaotic systems via backstepping design", Chaos, Solitons & Fractals, Vol.12, Issue 7, pp.1199-1206, 2001.
[17] S. Bowong, ``Adaptive synchronization of chaotic systems with unknown bounded uncertainties via backstepping approach", Nonlinear Dynamics, Vol.49, No.1-2, pp.59-70, 2007.
%[18]bs
[18] C. Wang, and S. S. Ge, ``Synchronization of uncertain chaotic systems via adaptive backstepping", Proceedings of the IEEE Conference on Decision and Control, Vol.5, pp.4827-4832, 2000.
[19] M. Krstic, I. Kaneltakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, Inc., New York, Ny, 1995.
[20] S. Bowong, ``Adaptive synchronization between two different chaotic dynamical systems", Communications in Nonlinear Science and Numerical Simulation, Vol.12, Issue 6, pp.976-985, 2007.
[21] J. H. Park, ``Chaos synchronization between two different chaotic dynamical systems", Chaos, Solitons & Fractals, Vol.27, Issue 2, pp.549-554, 2006.
[22] X. Wu, Z. H. Guan, and T. Li, ``Chaos Synchronization Between Unified Chaotic System and Genesio System", Lecture Notes in Computer Science, Vol.4492, pp.8-15, 2007.
[23] M. C. Ho, and Y. C. Hung, ``Synchronization of two different systems by using generalized active control", Physics Letters A, Vol.301, Issue 5-6, pp.424-428, 2002.
[24] H. K. Khalil, Nonlinear Systems, Third ediction, New Jersey: Prentice-Hall, Inc., 2000.
[25] J. Y. Hung, W. Gao, and J. C. Hung, ``Variable structure control: a survey", IEEE Trans. Industrial Electronic, Vol.40, No.1, pp.2-22, 1993.
[26] S. Sastry, Nonlinear system: analysis, stability, and control, New York: Springer, 1999.
[27] M. Rafikov, and J. Manoel Balthazar, ``On control and synchronization in chaotic and hyperchaotic systems via linear feedback control", Communications in Nonlinear Science and Numerical Simulation, Vol.13, Issue 7, pp.1246~-1255, 2008.
[28] J. H. Kim, C. W. Park, E. Kim, and M. Park, ``Fuzzy adaptive synchronization of uncertain chaotic systems", Physics Letters A, Vol.334, Issue 4, pp.295-305, 2005.
[29] M. Bucolo, S. Fazzino, M. La Rosa, and L. Fortuna, ``Small-world networks of fuzzy chaotic oscillators", Chaos, Solitons & Fractals, Vol.17, Issue 2-$3, pp.557-565, 2003.
[30] W. He, and J. Cao, ``Adaptive synchronization of a class of chaotic neural networks with known or unknown parameters", Physics Letters A, Vol.372, Issue 4, pp.408-416, 2008.
[31] Z. He, K. Li, L. Yang, and Y. Shi, ``A Robust Digital Secure Communication Scheme Based on Sporadic Coupling Chaos Synchronization", IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, Vol.47, No.3, pp.397-403, 2000.
[32] K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos : an introduction to dynamical systems, New York : Springer, 1996.
[33] D. W. Moore, and E. A. Spiegel, ``A Thermally Excited Non-Linear Oscillator", Astrophs. J., Vol.143, No.3, pp.871-887, 1966.
[34] J. C. Sprott, ``Simple chaotic systems and circuits", American Journal of Physics, Vol.68, Issue. 8, pp.758-763, Aug. 2000.
[35] G. H. Hardy, J. E. Littlewood, and G. Polya, A Theorem of W. H. Young. Section 8.3, in: Inequalities, second ed., Cambridge Univ. Press, Cambridge, UK, pp.198-200, 1988.
[35] Q. Chen, H. Chen, and Y. Wang, ``Robust Adaptive Trajectory Tracking Independent of Models for Robotic Manipulators", Journal of Robotic Systems, Vol.18, Issue. 9, pp.545-551, 2001.
[36] G. Tao, Adaptive control design and analysis, New Jersey, John Wiley & Sons, 2003.