為了探討基於最佳化的強健控制概念應用於學習控制系統的關鍵和其他的可能性，我們針對一種含不確定性的非線性控制系統來對這兩種控制理論進行整合性研究。首先，為了以更普遍的觀念來理解這基於最佳化的強健控制理論，我們利用較早期的Dissipative控制理論來設計一個適用於指定系統的強健控制器，這強健控制器擁有所謂的H∞追蹤效果和隱含的L2-gain的能量收斂等特性。其中最重要的是系統輸入和輸出能量的關係能用可控的能量衰減概念來表示，這強健的概念可被利用來壓制學習控制系統中許多不受歡迎的現象，而應用於學習控制系統的強健設計概念成型於此。接著，這得到的強健控制器被使用L2-gain的能量收斂特性重新解釋，而得到一個L2-gain狀態回饋控制器並且證明一個積分控制項的加入是可以改善系統穩定性。我們將這控制器應用於直接適應模糊控制系統來確認這強健控制器對這類學習控制系統的影響，由模擬結果可以發現到數個重要的學習特性是可被滿足的。另外，對於自然存在於L2-gain狀態回饋控制器的初始控制量振盪問題，我們利用了線上遺傳演化系統取代一般常見的自適應設計方式來解決這個問題。這設計概念有可能是以演化為核心的自適應控制系統的設計參考。最後，對於一個新式的自適應模糊滑動控制系統，我們提出了一個可以有效加速系統學習速度的自適應法則，這新的自適應法則讓系統的穩定性和學習收斂性都被明顯地增強。

In this dissertation, we report our study on the relationship between the optimization based robust control mechanisms and the learning based control systems. In this study, a class of uncertain nonlinear control systems is considered and key points of integrating these two control systems are proposed and discussed. In the literature, it can be observed that the optimization based robust control mechanisms have been widely used in the learning control system and have nicely fits with the properties of the learning concept. Our first study is to understand the optimization based robust control notions generally. Dissipative control theory is studied and a dissipative controller is proposed. The obtained dissipative controller owns the H∞ tracking ability and has the energy convergence property of L2-gain. Therefore, the relationship between the input and output energies can be represented by a controllable attenuate parameter. The selection of the attenuate parameter becomes a means of coping with some undesirable phenomena of the learning control systems. It is also the original idea in robust learning control systems. Next, the dissipative controller is re-derived in terms of the normal L2-gain property. In this study, an L2-gain state feedback controller with an additional integral control term is designed and applied to the direct adaptive fuzzy control system. Besides, based on the property of L2-gain, the high initial gain problem of the L2-gain state feedback controller is resolved using a genetic adaptive scheme. With the use of the proposed adaptive scheme, not only the high initial gain problem is resolved effectively but also the initial tracking performance is not sacrificed. Finally, a novel adaptive law is proposed for the adaptive fuzzy control system to speed-up the learning process. From our simulations, it is evident that the learning speed of the adaptive fuzzy control system is significantly improved.

[1] G. Zames, “On the Input/Output Stability of Nonlinear Time-Varying Feedback Systems, Parts I and II,” IEEE Trans. Automatic control, vol. AC-11, pp. 228-238, and 465-477, 1966.
[2] J. C. Willems, “Least squares stationary optimal control and the algebraic Riccati equation,” IEEE Trans. Automatic control, vol. AC-16, no.6, pp. 621-634, Dec. 1971.
[3] J. C. Willems, “Dissipative dynamical systems - Part I: General theory, Part II: Linear systems with quadratic supply rates,” Arch. Rational Mechanics and Analysis, vol.45, pp. 321-351 and pp. 352-393, 1972.
[4] D. Hill, P. Moylan, “The stability of nonlinear dissipative systems,” IEEE Trans. Automatic control, vol. 21, pp. 708-700, Oct. 1976.
[5] V. Utkin, “Variable structure systems with sliding modes,” IEEE Trans on Automat. Control, vol. 22, no. 2, pp. 212-222, April 1977.
[6] D. Hill and P. Moylan, “Dissipative dynamical systems: basic input-output and state properties,” J. Franklin Inst., vol.309, pp.327-357, 1980.
[7] J. J. E. Slotine and W. Li, Applied Nonlinear Control, Englewood Cliffs, NJ: Prentice Hall, 1991.
[8] A. J. van der Schaft, “ L2-gain analysis of nonlinear systems and nonlinear state feedback control,” IEEE Trans. Automatic control, vol. 37, no.6, pp. 770-784, June 1992.
[9] B. Kosko, “Fuzzy systems as universal approximates,” IEEE Int’l Conf. on Fuzzy Systems, pp. 1153-1162, 1992.
[10] L. X. Wang, “Fuzzy systems are universal approximators,” IEEE Int’l Conf. On Fuzzy Systems, pp. 1163-1170, 1992.
[11] L. X. Wang, “Stable adaptive fuzzy control of nonlinear systems,” IEEE Trans. Fuzzy Systems, vol. 1, pp. 146-155, May 1993.
[12] C. Wang, S. S. Venkatesh, and J. S. Judd, “Optimal stopping and efficient machine complexity in learning,” NIPS6, pp. 303-310, 1994.
[13] D. Par, A. Kandel, and G. Langholz, “Genetic-based new fuzzy reasoning models with application to fuzz control,” IEEE Trans on Systems, Man, and Cybernetics, vol. 25, no. 1, pp. 39-47, Jan. 1994.
[14] K. J. Astrom and B. Wittenmark, Adaptive Control 2/e, New York: Addison-Wesley, 1994.
[15] L. X. Wang, “A supervisory controller for fuzzy control systems that guarantees stability,”IEEE Trans. on Automatic Control, vol.39, no.9, pp.1845-1847, 1994.
[16] Y. S. Lu, J. S. Chen, “A self-organizing fuzzy sliding-mode controller design for a class of nonlinear servo systems,”IEEE Trans. Industrial Electronics, vol. 41, no. 5, Oct. 1994.
[17] A. Tesi, F. Viloresi, and R. Genesio, “On the stability domain estimation via a quadratic Lyapunov function: Convexity and optimality properties for polynomial systems,” IEEE Trans. Automatic control, vol. 41, no.11, pp. 770-784, Nov. 1996.
[18] B. S. Chen, C. H. Lee, and Y. C. Chang, “H∞ Tracking Design of Uncertain Nonlinear SISO Systems: Adaptive Fuzzy Approach,” IEEE Trans. Fuzzy Systems, vol. 4, no. 1, Feb. 1996.
[19] L. X. Wang, “Stable adaptive fuzzy controllers with application to inverted pendulum tracking,” IEEE Trans. Systems, Man, and Cybernetics, Part B, vol. 26, no. 5, pp. 677-691, Oct. 1996.
[20] M. T. Hagan, H. B. Demuth, M. H. Beale, Neural Network Design, Thomson, 1996.
[21] P. Kachroo and M. Tomizuka, “Chattering reduction and error convergence in the sliding-mode control of a class of nonlinear systems,” IEEE Trans. Automatic Control, vol. 41, no. 7, pp. 1063-1068, July 1996.
[22] H. L. Trentelman and J. C. Willems, “Every storage function is a state function,” Systems & Control Letters, vol. 32, pp. 249-259, 1997.
[23] L. X. Wang, A Course in Fuzzy Systems and Control, Prentice-Hall, 1997.
[24] S. Xie, L. Xie, and S. S. Ge, “Dissipative control for linear systems with time-varying uncertainty,” American Control Conference, New Mexico, vol. 4. pp. 2531-2535, 1997.
[25] S. Xie and L. Xie, “Robust dissipative control for linear systems with dissipative uncertainty and nonlinear perturbation,” Systems & Control Letters, vol. 29, pp. 255-268, 1997.
[26] L. K. Wang, F. H. F. Lenug, and P. K. S. Tam, “A chattering elimination algorithm for sliding mode control of uncertain non-linear systems,” Mechatronics, vol.8, pp. 765-775, 1998.
[27] J. C. Principe, N. R. Euliano, and W. C. Lefebvre, Neural and Adaptive Systems-Fundamentals Through Simulations, John Wiley and Sons, 1999.
[28] K. Fischle and D. Schröder, “An Improved Stable Adaptive Fuzzy Control Method,” IEEE Trans. Fuzzy Systems, vol. 7, no. 1, Feb. 1999.
[29] P. Kachroo, “Existence of solutions to a class of nonlinear convergent chattering-free sliding mode control systems,” IEEE Trans. on Automatic Control, vol. 44, no. 8, pp.1620-1624, Aug. 1999.
[30] V. Utkin, J. Guldner, and J. X. Shi, Sliding Mode Control in Electromechanical Systems, London, U.K., Taylor and Francis, 1999.
[31] W. Su, L. Xie, and C. E. de Souza, “Global robust disturbance attenuation and almost disturbance decoupling for uncertain cascaded nonlinear systems,” Automatica, vol. 35, pp. 697-707, 1999.
[32] A. J. van der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control 2/e, Springer, 2000.
[33] S. Tong, T. Wang, and J. T. Tang, “Fuzzy adaptive output tracking control of nonlinear systems,” Fuzzy Sets and systems, vol. 111, pp. 169-182, 2000.
[34] W. Y. Wang, M. L. Chan, T. T. Lee, and C. H. Liu, “Adaptive Fuzzy Control for Strict-Feedback Canonical Nonlinear Systems with H∞ Tracking Performance,”IEEE Trans. on Systems, Man, and Cybernetics-Part B: Cybernetics, vol. 30, no. 6, Dec. 2000.
[35] X. Bao, Z. Lin, and E. D. Sontag, “Finite gain stabilization of discrete-time linear systems subject to actuator saturation,” Automatica, vol. 36, pp. 269-277, 2000.
[36] A. C. Huang and Y. S. Kuo, “Sliding control of non-linear systems containing time-varying uncertainties with unknown bounds,” Int. J. Control, vol. 74, no. 3, pp.252-264, 2001.
[37] L. K. Wong, F. H. F. Leung, P. K. S. Tam, “A fuzzy sliding controller for nonlinear systems,” IEEE Trans on Industrial Electronics, vol. 48, no. 1, pp. 32-37, Feb. 2001.
[38] Y. C. Chang, “Adaptive Fuzzy-Based Tracking Control for Nonlinear SISO Systems via VSS and H∞ Approaches,” IEEE Trans. Fuzzy Systems, vol. 9, no. 2, April 2001.
[39] Z. Li, H. Shao, and J. Wang, “Robust dissipative control for a class of uncertain time-delay systems,” Proceedings of the American Control Conference, Arlington, pp.4004-4005, 2001.
[40] E. Kim, “A Fuzzy Disturbance Observer and Its Application to Control,” IEEE Trans. Fuzzy Systems, vol. 10, no. 1, Feb. 2002.
[41] G. D. Buckner, “Intelligent bounds on modeling uncertainty: applications to sliding mode control,” IEEE Trans. on Systems, Man, and Cybernetics - Part C: Applications and Reviews, vol. 32, no. 2, May 2002.
[42] I. G. Polushin and H. J. Marquez, “On the existence of a continuous storage function for dissipative systems,” Systems & Control Letters, vol. 46, pp. 85-90, 2002.
[43] I. Fantoni and R. Lozano, Non-linear Control for Underactuated Mechanical Systems, Springer, 2002.
[44] S. Tong and H. X. Li, “Direct adaptive fuzzy output tracking control of nonlinear systems,” Fuzzy Sets and Systems, vol. 128, pp. 107-115, 2002.
[45] W. D. Chang, R. C. Hwang, and J. G. Hsieh, “Application of an auto-tuning neuron to sliding mode control,” IEEE Trans. on Systems, Man, and Cybernetics - Part C: Applications and Reviews, vol. 32, no. 4, pp. 517-522, Nov. 2002.
[46] W. Perruquetti and J. P. Barbot, Sliding Mode Control in Engineering, Marcel Dekker, 2002.
[47] W. Y. Wang, M. L. Chan, C. C. Hsu, and T. T. Lee, “H∞ Tracking-Based Sliding Mode Control for Uncertain Nonlinear Systems via an Adaptive Fuzzy-Neural Approach,” IEEE Trans. on Systems, Man, and Cybernetics—Part B: Cybernetics, vol. 32, no. 4, Aug. 2002.
[48] D. Rerkpreedapong, A. Hasanović, and A. Feliachi, “Robust load frequency control using genetic algorithms and linear matrix inequalities,” IEEE Trans. on Power Systems, vol. 18, no. 2, pp. 855-861, May 2003.
[49] H. J. Marquez, Nonlinear Control Systems: Analysis and Design, John Wiley & Sons, 2003.
[50] R. J. Wai, “Supervisory genetic evolution control for indirect field-oriented induction motor drive,” IEE Proc.-Electr Power Appl., vol. 150, no. 2, pp. 215-226, March 2003.
[51] C. M. Lin and C. F. Hsu, “Adaptive Fuzzy Sliding-Mode Control for Induction Servomotor Systems,” IEEE Trans. on Energy Conversion, vol. 19, no. 2, June 2004.
[52] J. H. Ryu, D. S. Kwon, and B. Hannaford, “Stability guaranteed control: Time domain passivity approach,” IEEE Trans. Control System Technology, vol. 12, no. 6, Nov. 2004.
[53]R. J. Wai, C. M. Lin, and C. F. Hsu, “Adaptive fuzzy sliding-mode control for electrical servo drive,” Fuzzy Sets and Systems, vol. 143, pp. 295-310, 2004.
[54] R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms 2/e, John Wiley &Sons, Inc., 2004.
[55] W. Y. Wang, C. Y. Cheng, and Y. G. Leu, “An online GA-based output-feedback direct adaptive fuzzy-neural controller for uncertain nonlinear controller for uncertain nonlinear systems,” IEEE Trans. on System, Man, and Cybernetics-Part B: Cybernetics, vol. 34, no. 1, pp. 334-345, Feb. 2004.
[56] Y. T. Kim and Z. Zenn Bien, “Robust adaptive fuzzy control in the presence of external disturbance and approximation error,” Fuzzy Sets and Systems, vol. 148, pp.377-393, 2004.
[57] C. C. Sun, H. Y. Chung, and W. J. Chang, “H2/H∞ robust static output feedback control design via mixed genetic algorithm and linear matrix inequalities,” Journal of Dynamic Systems, Measurement, and Control, vol. 127, pp. 715-722, Dec. 2005.
[58] C. F. Hsu, C. M. Lin and T. Y. Chen, “Neural-network-identification-based adaptive control of wing rock motions,” IEE proc. Control Theory and Applications, vol. 152, no. 1, pp. 65-71, Jan. 2005.
[59] Y. Yang, “Direct robust adaptive fuzzy control (DRAFC) for uncertain nonlinear systems using small gain theorem,” Fuzzy Sets and Systems, vol. 151, pp. 79-97, 2005.
[60] C. F. Juang and C. F. Lu, “Load-frequency control by hybrid evolutionary fuzzy PI controller,” IEE Proc.-Gener. Transm. Distrib., vol. 153, no. 2, pp. 196-204, March 2006.
[61] H. Zhang, D. Liu, Fuzzy Modeling and Fuzzy Control, Birkhäuser Boston, 2006.
[62] J. K. Liu, “Global sliding mode control with adaptive fuzzy chattering free method for nonlinear system,” IMACS Multiconference on Computational Engineering in Systems Applications, vol. 1, pp. 541-546, Oct. 2006.
[63] J. D. Chen, “LMI based to robust genetic control design for a class of uncertain neutral systems with state and input delays,” Journal of Dynamic Systems, Measurement, and Control, vol. 128, pp. 675-680, Dec. 2006.
[64] J. A. Farrell, M. M. Polycarpou, Adaptive Approximation Based Control – Unifying Neural, Fuzzy and Traditional Adaptive Approximation Approaches, John Wiley & Sons, 2006.
[65] K. J. Kim, J. B. Park, and Y. H. Choi, “Chattering free sliding mode control,” SICE-ICASE, Inter’l Joint Conference, pp. 732-735, Oct, 2006.
[66] L. Qin and G. R. Duan, “Robust dissipative control for uncertain descriptor linear systems with time delay,” Proceedings of the 6th World Congress on Intelligent Control and Automation, China, pp. 2327-2333, 2006.
[67] P. C. Chen and A. C. Huang, “Adaptive sliding control of active suspension systems with uncertain hydraulic actuator dynamics,” Vehicle System Dynamics, vol. 44, no. 5, pp. 357-368, May 2006.
[68] R. J. Wai and K. H. Su, “Supervisory control for linear Piezoelectric Ceramic motor driver using genetic algorithm,” IEEE Trans. on Industrial Electronics, vol. 53, no. 2, pp. 657-673, April 2006.
[69] S. R. Hebertt and S. O. Ramn, Control Design Techniques in Power Electronics Devices, London, U.K. Springer, 2006.
[70] S. F. Su, J. C. Chang, and S. S. Chen, “The study on direct adaptive fuzzy controllers,” International Journal of Fuzzy Systems, vol. 8, no.3, pp.150-159, 2006.
[71] S. C. Tong, “Indirect adaptive fuzzy backstepping control for nonlinear systems,” The Fifth IEEE Int’l Conf. on Machine Learning and Cybernetic, pp. 468-473, 2006.
[72] V. Giordano, D. Naso, and B. Turchiano, “Combining Genetic Algorithms and Lyapunov-Based Adaptation for Online Design of Fuzzy Controllers,” IEEE Trans. System, Man, and Cybernetics-Part B: Cybernetics, vol. 36, no. 5, Oct. 2006.
[73] Y. J. Li, Y. M. Fu, and G. R. Duan, “Robust dissipative control for T-S fuzzy systems with time-varying delays,” IEEE International Symposium on Industrial Electronics (ISIE), Canada, pp. 97-101, 2006.
[74] X. S. Cai and Z. Z. Han, “Robust stabilisation and passivity of nonlinear systems with structural uncertainty,” IEE Proc.-Control Theory Appl., vol. 153, no. 6, Nov. 2006.
[75] B. Brogliato, R. Lozano, B. Maschle, and O. Egeland, Dissipative Systems Analysis and Control – Theory and Applications 2/e, Springer, 2007.
[76] C. Y. Liang1 and S. W. Tan, “A new approach to chattering reduction in the sliding mode controls,” Second International Conference on Innovative Computing, Information and Control, pp. 334-337, Sept. 2007.
[77] C. F. Hsu, G. M. Chen, T. T. Lee, “Robust intelligent tracking control with PID-type learning algorithm,” Neurocomputing, vol.71, pp. 234-243, 2007.
[78] C. H. Lee, B. R. Chang, F. K. Chang, and S. K. Chang, “Adaptive backstepping control for a class of nonlinear uncertain system using fuzzy neural networks,” The Sixth IEEE Int’l Conf. on Machine Learning and Cybernetic, pp. 431-436, 2007.
[79] D. Xinzhuang, “Robust strictly dissipative control for singular systems with time-delay and parameter uncertainties,” Proceedings of the 26th Chinese Control Conference, China, pp. 578 - 582, 2007.
[80] E. Iglesias, Y. Garca, M. Sanjuan, O. Camacho, and C. Smith, “Fuzzy surface-based sliding mode control,” ISA Transactions, vol.46, pp.73-83, 2007.
[81] F. Geng and X. Zhu, “A Novel Adaptive Fuzzy Control of the Inverted Pendulum System,” IEEE Int’l Conf. On Control and Automation, China, pp. 284-288, 2007.
[82] F. Lin, Robust Control Design – An Optimal Control Approach, John Wiley & Sons, 2007.
[83] H. P. Geering, Optima Control with Engineering Applications, Springer, 2007.
[84] J. C. Willems and K. Takaba, “Dissipative and stability of interconnections,” Int. J. Robust and Nonlinear Control, vol.17, pp. 563-586, 2007.
[85] J. C. Willems, “Dissipative dynamical systems,” European Journal of Control, vol. 13, pp.134-151, 2007.
[86] K. Kamali, L. J. Jiang, J. Yen, and K. W. Wang, “Using Q-learning and genetic algorithms to improve the efficiency of weight adjustments for optimal control and design problems,” Journal of Computing and Information Science in Engineering, vol. 7, pp. 302-308, Dec. 2007.
[87] P. A. Phan and T. Gale, “Two-Mode Adaptive Fuzzy Control With Approximation Error Estimator,” IEEE Trans. Fuzzy Systems, vol. 15, no. 5, Oct. 2007.
[88] H. Du and N. Zhang, “H∞ control for buildings with time delay in control via linear matrix inequalities and genetic algorithms,”Engineering Structures, vol. 30, pp. 81-92, 2008.
[89] P. A. Phan and T. J. Gale, “Direct adaptive fuzzy control with a self-structuring algorithm,” Fuzzy Sets and Systems, vol. 159, pp. 871-899, 2008.
[90] R. Shahnazi, H. M. Shanechi, and N. Pariz, “Position Control of Induction and DC Servomotors: A Novel Adaptive Fuzzy PI Sliding Mode Control,” IEEE Trans. Energy Conversion, vol. 23, no. 1, March 2008.
[91] R. Shahnazi and Mohammand-R. Akbarzadeh-T, “PI Adaptive Fuzzy Control With Large and Fast Disturbance Rejection for a Class of Uncertain Nonlinear Systems,” IEEE Trans. Fuzzy Systems, vol. 16, no. 1, Feb. 2008.
[92] R. M. Jan, C. S. Tseng, and R. J. Liu, “Robust PID control design for permanent magnet synchronous motor: A genetic approach,” Electric Power Systems Research, vol. 78, pp. 1161-1168, 2008.
[93] Y. J. Huang, T. C. Kuo, and S. H. Chang, “Adaptive sliding-mode control for nonlinear systems with uncertain parameters,” IEEE Trans. on Systems, Man, and Cybernetics - Part B: Cybernetics, vol. 38, no. 2, pp. 534-539, April 2008.
[94] Y. C. Hsueh and S. F. Su, “Dissipative control for a class of uncertain nonlinear control systems,” submitted for publication in Asia Journal of Control, 2008.