本論文探討如何將混沌系統精確表示成T-S模糊模型，並使用改良型基因演算法用以達成各種常見的控制目標。首先，提出系統化方法，將非線性系統精確地表示成T-S模糊模型。利用這種方法，將非線性系統以數個具線性系統為後件部的模糊子系統表示之。此外，少量模糊的規則就足以在一定區域內完整無誤地表示此非線性系統。本論文中同時說明了，此過程中可將系統之不確定性全由歸屬函數加以刻畫描述。利用已知的系統矩陣用以求解滿足系統穩定性條件的固定回授增益；在此處理過程中，當只有部分狀態是已知時，運用觀測器的設計來估測不可量測的狀態。另一方面，以改良型基因演算法調整模糊控制器的複合控制增益以獲取較快速的暫態響應。
此論文另一重點乃介紹所謂虛擬預期變數合成法來完成許多不同形式的T-S模糊系統控制目標之實現。所達到的控制目標有(一)調節；(二)非線性模型追蹤；(三) 混沌化；(四) 輸出調節；(五) 輸出跟蹤，而目的在於達到零狀態誤差。這裡我們集中討論很多物理系統的模糊集合的歸屬函數是滿足類Lipschitz的特性。實際上的虛擬預期變數合成法的控制增益及估測增益，是透過一組線性矩陣不等式求解所得到的，這線性矩陣不等式的形式與前述穩定問題的線性矩陣不等式形式相同。
論文最後，提出一個全域混合控制方法，除了少數不可控制的點之外可使混沌系統穩定在任意點。這方法包含兩種型式，型式一以低作用控制力為主，當軌跡通過吸引範圍才開始控制；型式二以較少的時間消耗為考量，一開始就使用全域控制力的方式，當軌跡進入吸引範圍，改用區域控制力的方式。論文中以Chua的電路為應用的例子，說明混合型式控制的設計方法。

This dissertation focuses on the issue of representing chaotic systems precisely as Takagi-Sugeno (T-S) fuzzy models and making use of improved genetic algorithm, so that various control objectives can be achieved. First, a method is given to systematically represent a nonlinear system exactly as a T-S fuzzy model. By the method, the nonlinear system is represented by several fuzzy subsystems where the consequent parts are linear dynamical systems. Besides, the derived fuzzy model gives zero modeling errors in the universe of discourse where the fuzzy sets are defined. As the overall inferred output is presented, the method mentioned above is illustrated to put all the system uncertainty into the membership functions. We can make use of the known existing system matrices to obtain the controller gains to satisfy the conditions of system stability. In the process of controller synthesis,
if only partial states are known, an observer is designed to estimate immeasurable states. Moreover, by using improved genetic algorithm to tune composite control gain, a new control scheme is proposed to improve the system performance of the transient response of fuzzy systems.
Another main issue in this dissertation, the new concepts, virtual-desired-variables synthesis and generalized kinematic constraints, are introduced to benefit the various control objectives design. The control objectives achieved, are i) regulation; ii) nonlinear model following; iii) chaotification; iv) output regulation; v) output tracking. Meanwhile, zero tracking error is concluded. Here we focus on a common feature held by many physical systems where their membership functions of fuzzy sets satisfy a Lipschitz-like property. The control gains and observer gains of
virtual-desired-variables synthesis are determined by solving a set of LMIs, the same type of LMIs for stabilization problem.
At the end of the dissertation, a global hybrid control methodology is proposed to stabilize chaotic systems at almost arbitrary points. This method includes two types-hybrid-types 1 and 2. In the hybrid-type 1, the control force is activated when the trajectory passes through the attraction region. The low-effort control force is the main concern. If time consuming is concerned, the hybrid-type 2 is adopted. The global control force is activated at beginning and the local control force is activated when the trajectory passes through the attraction region. Synthesis using the hybrid-type method is applied to the numerical simulations of Chua's circuit.

[1] E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett., vol. 64, no. 11, pp. 1196–1199, 1990.
[2] G. Chen and X. Dong, From chaos to order: methodologies, perspectives and
applications, Singapore: World Scientific, 1998.
[3] K. Y. Lian, P. Liu, T. C. Wu, and W. C. Lin, “Chaotic control using fuzzy
model-based meyhods,” Int. J. Bifurc. Chaos, vol. 12, no. 8, pp. 1827-1841,
2002.
[4] B. M. Ganesh and N. S. Sahjendra, “Modular adaptive control of chaos in
Chua’s circuit,” Int. J. Bifurc. Chaos, vol. 15, no. 9, pp. 2973-2984, 2005.
[5] J. Zhu and Y. P. Tian, “Stabilizing periodic solutions of nonlinear systems and applications in chaos control,” IEEE Trans. Circ. Syst. II, vol. 52, no. 2, pp. 870–874, 2005.
[6] J. H. Park, “Controlling chaotic systems via nonlinear feedback control,”
Chaos, Solitons, Fractals, vol. 23, pp. 1049–1054, 2005.
[7] L. Udawatta, K. Watanabe, K. Kiguchi, and K. Izumi, “ Fuzzy-chaos hybrid
controlling of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 10, pp. 401–
411, 2002.
[8] G. Chen, and X. Dong, “On feedback control of chaotic continuous-time systems,”IEEE Trans. Circ. Syst. I, vol. 40, 591-601, 1993.
[9] ¨ O. Morg¨ul,“A model-based scheme for anticontrol of some chaotic systems,”Int. J. Bifurc. Chaos, vol. 13, no. 11, pp. 3449-3457, 2003.
[10] K. Starkov and G. Chen, “Chaotification of polynomial continuous-time systems and rational normal forms,” Chaos, Solitons, Fractals, vol. 22, pp. 849–856, 2004.
[11] S. K. Sharma, and G. W. Irwin, “Fuzzy coding of genetic algorithms”, IEEE Trans. Evolutionary Computation, vol. 7, no. 4, pp344–355, 2003.
[12] F. Herrera and M. Lozano, “Adaptive genetic operators based on coevolution with fuzzy behaviors”, IEEE Trans. Evolutionary Computation, vol. 5, no. 2, pp149–165, 2001.
[13] S. Harp, T. Samad, and A. Guha, “Designing application specific neural networks using the genetic algorithm,” Neural information processing systems, vol. 2. San Mateo, CA: Morgan Kaufmann, 1990.
[14] J. D. Schaffer, R. A. Caruana, and L. J. Eshelman, “Using genetic search to exploit the emergent behavior of neural networks,” Phys. D, vol. 42, pp. 244–248, 1990.
[15] D. Whitley, S. Dominic, R. Das, and C. W. Anderson, “Genetic reinforcement learning for neurocontrol problems,” Machine Learning, vol. 13, pp. 259–284, 1993.
[16] D. E. Moriarty and R. Miikkulainen, “Efficient reinforcement learning through symbiotic evolution,” Machine Learning, vol. 22, pp. 11–32, 1996.
[17] W. Chang, J. B. Park, and Y. H. Joo, “GA-Based intelligent digital redesign of fuzzy-model-based controllers” IEEE Trans. Neural Networks, vol. 11, no. 1, pp. 35–44, 2003.
[18] C.-T. Lin, and C.-P. Jou, “Controlling chaos by GA-based reinforcement learning neural network” IEEE Trans. Neural Networks, vol. 10, no. 4, pp. 846–859, 1999.
[19] G. P. Jiang, G. Chen and K. S. Tang, “Stabilizing unstable equilibria of chaotic systems from a state observer approach,” IEEE Trans. Circ. Syst. II, vol. 51, no. 6, pp. 281–288, 2004.
[20] H. O. Wang, K. Tanaka, and T. Ikeda, “Fuzzy modeling and control of chaotic systems,” Proc. IEEE Int. Symp. Circ. Syst., vol. 3, pp. 209–212, 1996.
[21] K. Tanaka, T. Ikeda and H. O. Wang, “ A unified approach to controlling
chaos via an LMI-based fuzzy control system design,” IEEE Trans. Circ. Syst.
I: Fundamental Theory and Applications, vol. 45, no. 10, pp. 1021-1040, 1998.
[22] Z. Li, J. B. Park, and Y. H. Joo, “Chaotifying continuous-time TS fuzzy systems via discretization,” IEEE Trans. Circ. Syst. I, vol. 48, no. 10, pp. 1237–1243, 2001.
[23] Z. Li, J. B. Park, Y. H. Joo, Y. H. Choi, and G. Chen , “Anticontrol of chaos for discrete TS fuzzy systems,” IEEE Trans. Circ. Syst. I, vol. 49, no. 2, pp. 249-253, 2002.
[24] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its application to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. 15, no. 1, pp. 116-132, 1985.
[25] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear matrix inequalities in system and control theory. Philadelphia, PA:SIAM, 1994.
[26] Tanaka K. and Sugeno M., “Stability analysis and design of fuzzy control systems,”IEEE Transactions on Automatic Control, vol. 43, pp. 491-500, 1998.
[27] K. Tanaka, T. Ikeda, and H. O. Wang, “Robust stabilization of a class of
uncertain nonlinear systems via fuzzy control: quadratic stabilizability, H1
control theory, and linear matrix inequalities ,” IEEE Trans. Fuzzy Syst., vol. 4, no. 1, pp. 1–13, 1996.
[28] K. Y. Lian, C. S. Chiu, T. S. Chiang and P. Liu, “LMI-based fuzzy chaotic
synchronization and communications,” IEEE Trans. Fuzzy syst., vol. 9, no. 4,
pp. 539-553, 2001.
[29] K.-Y. Lian, T.-S. Chiang, C.-S. Chiu, and P. Liu, “Synthesis of fuzzy modelbased design to synchronization and secure communication for chaotic systems,”IEEE Trans. Syst. Man, Cybern. Part B., vol. 31, pp. 66-83, 2001.
[30] G. P. Jiang, W. X. Zheng, W. K. S. Tang, and G. Chen, “Integral-observerbased chaos synchronization, ” IEEE Trans. Circ. Syst. II, vol. 53, no. 2, pp. 110-114, 2006.
[31] K. Tanaka, T. Ikeda, and H. O. Wang, “Fuzzy regulators and fuzzy observer:Relaxed stability conditions and LMI-based designs,” IEEE Trans. Fuzzy Syst., vol. 6, pp. 250-265, 1998.
[32] C.-S. Tseng, B.-S. Chen, and H.-J. Uang, “Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model,” proc. Automatic Control
Conf., Taiwan, pp. 581-586.
[33] S. S. Farinwata, D. Filev, and R. Langari, Fuzzy control: synthesis and analysis, New York: Wiley, pp. 246-247, 2000.
[34] H. O. Wang, K. Tanaka and M. Griffin, “An approach to fuzzy control of
nonliner systems: stability and design issues,” IEEE Trans. Fuzzy Syst., vol. 4, no. 1, pp. 14–23, 1996.
[35] F. C. Moon, Chaotic and fractal dynamics :an introduction for applied scientists and engineers, New York: Springer, 1992.
[36] N. Tufillaro, T. Abbott and J. Reilly An experimental approach to nonlinear dynamics and chaos. Redwood City, Calif. : Addison-Wesley, 1992.
[37] J. C. Sprott Chaos and time-series analysis. New York: Oxford, 2003.
[38] A. N. Sharkovsky “Chaos from a time-delayed Chuas circuit,” IEEE Trans.
Circ. Syst. I: Fundamental Theory and Applications, vol. 40, no. 10, pp. 781-
783, 1993.
[39] L. O. Chua, “The genesis of Chua’s circuit,” Archiv fur elektronik und ubertragungstechnik, vol. 46, no. 4, pp. 250-251, 1992.
[40] J. H. Holland, Adaptation in natural and artificial systems. Ann Arbor, MI:Univ. Michigan Press, 1975.
[41] D. T. Pham and D. Karaboga, Intelligent optimization techniques, genetic algorithms, tabu search, simulated annealing and neural networks. New York:
Springer-Verlag, 2000.
[42] Z. Michalewicz, Genetic algorithm+data structures=evolution programs, 3nd
extended ed. New York: Springer-Verlag, 1996.
[43] H. K. Lam, Frank H.Leung, and Peter K. S. Tam, “Design and stability analysis of fuzzy model-based nonlinear controller for nonlinear systems using genetic algorithm,” IEEE Trans. Syst., Man, Cybern. B, vol. 33, pp. 250-257, April. 2003.
[44] F. Janabi-Sharifi, and J. Liu, “Design of a self-adaptive fuzzy tension controller for tandem rolling,” IEEE Trans. Ind. Electronics, Vol. 52, pp.1428–1438, Oct. 2005
[45] C. F. Juang, “Combination of online clustering and Q-value based GA for
reinforcement fuzzy system design,” IEEE Trans. Fuzzy Syst., Vol. 13, pp289–
302, Jun. 2005
[46] C. F. Juang, “A hybrid of genetic algorithm and particle swarm optimization for recurrent network design,” IEEE Trans. Syst., Man, Cybern. B, Cybern., Vol. 34, pp.997–1006, Apr. 2004
[47] W. Y. Wang, C. Y. Cheng, and Y. G. Leu; “An online GA-based outputfeedback direct adaptive fuzzy-neural controller for uncertain nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., Vol. 34, pp334–345, Feb. 2004
[48] A. Alvarez, A. Caiti, and R. Onken “Evolutionary path planning for autonomous underwater vehicles in a variable ocean,” IEEE J. Oceanic Eng, Vol.
29, pp.418–429, Apr. 2004
[49] L.-H. Chen, C.-H. Chiang “New approach to intelligent control systems with self-exploring process,” IEEE Trans. Syst., Man, Cybern. B, Vol. 33, pp.56–66, Feb. 2003
[50] D. P. Muni, N. R. Pal, and J. Das, “Genetic programming for simultaneous
feature selection and classifier design,” IEEE Trans. Syst., Man, Cybern. B,
Vol. 36, pp.106–117, Feb. 2006
[51] M. Haseyama, and D. Matsuura, “A filter coefficient quantization method with genetic algorithm including simulated annealing,” IEEE Letters, Signal Process. Vol. 13, pp189–192, Apr. 2006
[52] K. Tanaka and C. S. Wang, Fuzzy control systems analysis and design: A linear matrix inequality approach, New York: Wiley, 2001.
[53] Frank H. F. Leung, H. K. Lam, S. H. Ling, and Peter K. S. Tam, “Tuning of
the structure and parameters of a neural network using an improved genetic
algorithm,” IEEE Trans. Neural Network, vol. 14, pp. 79-88, Jan. 2003.
[54] L. Davis, Handbook of genetic algorithms. New York: Van Nostrand Reinhold,
1991.
[55] X. Wang and M. Elbuluk, “Neural network control of induction machines using genetic algorithm training,” in Conf. Record 31st IAS Annual Meeting, vol. 3, 1996, pp. 1733-1740.
[56] M. Srinivas and L. M. Patnaik, “Genetic algorithms: A survey,” IEEE Computer, vol. 27, pp. 17-27, June 1994.
[57] J. T. Tsai,J. H. Chou,and T. K. Liu,“Tuning the structure and parameters
of a neural network by using hybrid taguchi-genetic algorithm,” IEEE Trans.
Neural Networks, Vol 17, Issue 1, pp.69-80, Jan. 2006.
[58] M. Sugeno and G. T. Kang, “Fuzzy modeling and control of multilayer incinerator,” Fuzzy Sets Syst., no. 18, pp. 329-346, 1986.
[59] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Sets Syst., vol. 45, pp. 135–156, 1992.
[60] H. Ying, “General SISO Takagi-Sugeno fuzzy systems with linear rule consequent are universal approximators,” IEEE Trans. Fuzzy Syst., vol. 6, no. 4, pp. 582-587, 1998.
[61] Q. Gan and C. J. Harris, “Fuzzy local linearization and local basis function expansion in nonlinear system modeling,” IEEE Trans. Syst., Man, Cybern. B, vol. 29, pp. 559-565, Aug. 1999.
[62] L. X.Wang and J. M. Mendel, “Fuzzy basis functions, universal approximation, and orthogonal least-squares learning,” IEEE Trans. Neural Networks, vol.3, Issue 5, pp.807-814 Sept. 1992.
[63] A. Jadbabaie, Robust, non-fragile controller synthesis using model-based fuzzy systems: A linear matrix inequality approach, Master Thesis, Univ. New Mexico, pp. 28-41, 1997.
[64] L. M. Pecora, and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett., vol. 64, no. 8, pp. 821-824, 1990.
[65] K. Y. Lian, P. Liu, T. S. Chiang, and C. S. Chiu “Adaptive synchronization design for chaotic systems via a scalar driving signal,” IEEE Trans. Circ. Syst. I, vol. 49, pp. 17-27, 2002.
[66] C.-C. Wu, “Graph coloring via synchronization of coupled oscillators,” IBM Research Report, RC 20965, 1997.
[67] O. M. Aamo and M. Krstic, Flow control by feedback:stabilization and mixing, New York: Springer, 2003.
[68] C.-S. Tseng, B.-S. Chen, and H.-J. Uang, “Fuzzy tracking control design for nonlinear dynamic systems via T-S fuzzy model,” IEEE Trans. Fuzzy Syst.,
vol. 9, pp.381-392, 2001.
[69] G. Feng, and T. Zhang, “Output regulation of discrete-time piecewise-linear systems with application to controlling chaos,” IEEE Trans. Circ. Syst. II, vol. 53, no.4, pp. 249-253, 2006.
[70] B. S. Chen, C. S. Tseng, and H. J. Uang, “Mixed H2/H1 fuzzy output feedback control design for nonlinear dynamic systems: an LMI approach,” IEEE Trans. Fuzzy Syst., vol. 8, pp. 249-265, 2000.
[71] C. Chen, Linear system themy and design, New York: Oxford University Press; 1999.
[72] F. Cuesta, F. Gordillo, J. Aracil, and A. Ollero, ”Stability analysis of nonlinear multivariable Takagi-Sugeno fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 508–519, Oct. 1999.
[73] M. C. M. Teixeira and S. H. Zak, “Stabilizing controller design for uncertain nonlinear systems using fuzzy models,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 133–142, Apr. 1999.
[74] O. Begovich, E. N. Sanchez, and M. Maldonado, “T-S scheme for trajectory
tracking of an underactuated robot,” in Proc. IEEE Fuzzy Conf., San Antonio,
TX, May 2000, pp. 798–803
[75] K. Tanaka, T. Hori, and H. O. Wang, “A multiple Lyapunov function approach to stabilization of fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 11, no. 4, pp. 582-589, 2003.
[76] H. J. Lee, H. Kim, Y. H. Joo, W. Chang, and J. B. Park, “A new intelligent digital redesign for T-S fuzzy systems: global approach,” IEEE Trans. Fuzzy Syst., vol. 12, no. 2, pp. 274-284, 2004
[77] C. H. Wang, T. C. Lin, T. T. Lee, and H. L. Liu, “Adaptive hybrid intelligent control for uncertain nonlinear dynamical systems,” IEEE Trans. Syst., Man, Cybern., B, vol. 32, pp. 583–597, 2002.
[78] T. Taniguchi, K. Tanaka, and H. O. Wang, “Fuzzy descriptor systems and
nonlinear model following control ,” IEEE Trans. Fuzzy Syst., vol. 8, pp.442–
452, 2000.
[79] S. J. Wu and C. T. Lin, “Global optimal fuzzy tracker design based on local concept approach,” IEEE Trans. Fuzzy Syst., vol. 10, pp.128–143, 2002.
[80] F. Zheng, Q. G. Wang, and T. H. Lee, “Output tracking control of MIMO fuzzy nonlinear systems using variable structure control approach ,” IEEE Trans. Fuzzy Syst., vol. 10, pp.686–697, 2002.
[81] S. J. Russell and P. Norvig, Artificial intelligence: a modern approach. Upper Saddle River, NJ: Prentice-Hall, 2002.
[82] P. J. Antsaklis and K. M. Passino, An Introduction to intelligent and autonomous control. Boston: Kluwer Academic, 1993.
[83] L. Yu and Y. Q. Zhang; “ Evolutionary fuzzy neural networks for hybrid financial prediction,” IEEE Trans. Syst., Man, Cybern., C, vol. 35, no. 2, pp.244–249, 2005.
[84] S. Mitra, S. K. Pal, and P. Mitra, “Data mining in soft computing framework:a survey,” IEEE Trans. Neural Networks, vol. 13, no. 1, pp. 3–14, 2002.
[85] A. Blanco, M. Delgado, and M. C. Pegalajar, “A real-coded genetic algorithm for training recurrent neural networks,” IEEE Trans. Neural Networks , vol. 14, pp.93–105, 2001.
[86] J. Zhang, “Modeling and optimal control of batch processes using recurrent neuro-fuzzy networks,” IEEE Trans. Fuzzy Syst., vol. 13, no. 4, pp.417–427, 2005.
[87] C. M. Bishop, Neural networks for pattern recognition. Oxford: Clarendon
Press, 1995.
[88] E. Cant´u-Paz and C. Kamath, “An empirical comparison of combinations of
evolutionary algorithms and neural networks for classification problems,” IEEE
Trans. Syst., Man, Cybern., B, vol. 35, no. 5, pp. 915–927, 2005.
[89] C. H. Lee and C. C. Teng, “Identification and control of dynamic systems using recurrent fuzzy neural networks,” IEEE Trans. Fuzzy Syst., vol. 8, no. 4, pp. 349–366, 2000.
[90] C. Y. Lee and J. J. Lee, “ Adaptive control for uncertain nonlinear systems based on multiple neural networks,” IEEE Trans. Syst., Man, Cybern., B, vol. 34, no. 1, pp. 325–333, 2004.
[91] D. E. Rumelhart and J. L. Mclelland, Eds., Parallel and Distributed Processing (PDP): Explorations in the Microstructure of the Cognitron, Cambridge, MA:MIT Press, 1986.
[92] X. Yu, M. O. Efe, and O. Kaynak, “A general backpropagation algorithm for
feedforward neural networks learning,” IEEE Trans. Neural Networks, vol. 13,
no. 1, pp. 251–254, 2002.
[93] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
[94] E. H. Mamdani, “Application of fuzzy algorithms for control of simple dynamic plant,” Proc. IEE, pp. 1585–1588, 1974.
[95] K.-Y. Lian and J. J. Liou, “Output tracking control for fuzzy systems via output feedback design,” IEEE Trans. Fuzzy Syst., vol. 14, no. 5, pp.628-639, 2006.