本篇論文提出三種非線性系統的控制方法。首先，在第二章先提出一個小波基底函數適應性倒階典型非線性系統的控制器。本文提出小波類神經自適應倒階控制器來控制未知的非線性系統。這個控制器結合小波基底函數與驅動飽和限制。接著，在第三章提出一個小波基底函數適應性倒階非典型非線性系統的控制器。本文中推薦在非典型非線性系統使用小波適應性倒階去控制。此控制方法結合適應性倒階控制器及小波基底函數，並以此方法近似未知的動態系統，小波基底函數擁有良好的近似性能，它較適合線上動態系統藉由調整內部的參數，以均值定理設計參數適應律來避免基底微分，且使用濾波器在控制輸入上減少基底微分計算量。最後，本篇論文提出結合小波適應性倒階與一階濾波器的設計概念來控制非典型非線性系統。而系統的穩定性以李亞普諾夫函數方程式分析說明，再以電腦舉例模擬論證本文所提出方法之控制性能與應用性。

Three control methods for nonlinear systems are proposed in this study. The first controller design is about a wavelet adaptive backstepping controller for affine nonlinear systems. The controller is comprised of a wavelet identifier and actuator saturation. The second controller design is about a wavelet adaptive backstepping controller for nonaffine nonlinear systems. A wavelet adaptive backstepping controller for a nonaffine system is proposed in this paper. The control scheme combines the backstepping technique and adaptive control with wavelet function. The wavelet function has well performance. It is much proper to online compute the system dynamics by tuning its interior parameters. A mean-value estimation method is also proposed to avoid a higher-order derivative problem generated by Taylor linearization expansion.The third controller design is about a wavelet adaptive backstepping controller for nonaffine nonlinear systems with first order filters. Furthermore, the stability of the system with the mean-value theorem is dissected through Lyapunov functions. In the end, simulation results illustrate the application of the proposed scheme.

[1] M. M. Polycarpou, “Stable adaptive neural control scheme for nonlinear systems,” IEEE Trans. Autom. Control, vol. 41, no. 3, pp. 447–451, Mar. 1996.
[2] F. J. Lin, R. J. Wai, and H. P. Chen, “A PM synchronous servo motor drive with an on-line trained fuzzy neural network controller,” IEEE Trans. Energy Conv., vol. 13, no. 4, pp. 319–325, Dec. 1998.
[3] H. Wang and Y. Wang, “Neural-network-based fault-tolerant control of unknown nonlinear systems,” Proc. IEE, Contr. Theory Appl., vol. 146, pp. 389–398, Sep. 1999.
[4] C. M. Lin and C. F. Hsu, “Neural-network-based adaptive control for induction servomotor drive system,” IEEE Trans. Ind. Electron., vol. 49, no. 1, pp. 115–123, Feb. 2002.
[5] D. Wang and J. Huang, “Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form,” IEEE Trans. Neural Netw., vol. 16, no. 1, pp. 195–202, Jan. 2005.
[6] C. K. Lin, “Adaptive tracking controller design for robotic systems using Gaussianwavelet networks,” Proc. IEE, Contr. Theory Appl., vol.149, pp. 316–322, Jul. 2002.
[7] C. D. Sousa, E. M. Hemerly, and R. K. H. Galvao, “Adaptive control for mobile robot usingwavelet networks,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 32, no. 4, pp. 493–504, Aug. 2002.
[8] R. J.Wai, “Development of new training algorithms for neuro-wavelet systems on the robust control of induction servo motor drive,” IEEE Trans. Ind. Electron., vol. 49, no. 6, pp. 1323–1341, Dec. 2002.
[9] F. J. Lin, H. J. Shieh, and P. K. Huang, “Adaptive wavelet neural network control with hysteresis estimation for piezo-positioning mechanism,” IEEETrans. Neural Netw.,vol. 17, no. 2, pp. 432–444, Mar. 2006.
[10] B. Delyon, A. Juditsky, and A. Benveniste, “Accuracy analysis for wavelet approximations,” IEEE Trans. Neural Netw., vol. 6, no. 2, pp. 332–348, Mar. 1995.
[11] Q. Zhang, “Using wavelet network in nonparametric estimation,” IEEE Trans. Neural Netw., vol. 8, no. 2, pp. 227–236, Mar. 1997.
[12] D. W. C. Ho, P. A. Zhang, and J. Xu, “Fuzzy wavelet networks for function learning,” IEEE Trans. Fuzzy Syst., vol. 9, no. 1, pp. 200–211, Feb. 2001.
[13] S. A. Billings and H. L. Wei, “A new class of wavelet networks for nonlinear system identification,” IEEE Trans. Neural Netw., vol. 16, no. 4, pp. 862–874, Jul. 2005.
[14] R. J. Wai and H. H. Chang, “Backstepping wavelet neural network control for indirect field-oriented induction motor drive,” IEEE Trans. Neural Netw., vol. 15, no. 2, pp. 367–382, Mar. 2004.
[15] M. A. S. K. Khan and M.A. Rahman, “A novel neuro-wavelet-based self-tuned wavelet controller for IPM motor drives,” IEEE Transactions on Industray Applications, vol. 46, no. 3, pp. 1194–1203, May 2010.
[16] C. K. Lin, “Nonsingular terminal sliding mode control of robot manipulators using fuzzy wavelet networks,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 6, pp. 849-859, April 2006.
[17] C. Kwan and F. L. Lewis, “Robust backstepping control of nonlinear systems using neural networks,” IEEE Transactions Syst., Man, Cybern. A, vol. 30, pp. 753-765, 2000.
[18] C. M. Kwan and F. L. Lewis, “Robust backstepping control of induction motors using neural networks,” IEEE Transactions Neural Networks, vol. 11, pp.1178-1187, 2000.
[19] J. Y. Choi and J. A. Farrell, “Adaptive observer backstepping control using neural networks,” IEEE Transactions Neural Networks, vol.12, pp.1103-1112, 2001.
[20] T. Zhang, S. S. Ge, and C. C. Hang, “Adaptive neural network control for strict-feedback nonlinear systems using backstepping design,” Automatica, vol. 36, pp. 1835–1846, 2000.
[21] O.Kuljaca, N.Swamy, F.L.Lewis, and C.M.Kwan, “Design and implementation of industrial neural network controller using backstepping,” IEEE Trans. Ind. Electron., vol. 50, no. 1, pp. 193–201, Feb. 2003.
[22] C. M. Lin and C. F. Hsu, “Recurrent-neural-network-based adaptive backstepping control for induction servomotor,” IEEE Trans. Ind. Electron., vol. 52, no. 6, pp. 1677–1684, Dec. 2005.
[23] Y.G. Leu, “Backstepping nonlinear control using nonlinear parametric fuzzy systems,” International Journal of Fuzzy Systems, vol. 11, no. 4, pp.225-231, 2009.
[24] M. Hojati and S. Gazor, “Hybrid adaptive fuzzy identification and control of nonlinear systems,” IEEE Transactions on Fuzzy Systems, vol. 10, no. 2, April 2002.
[25] J. Y. Choit and J. Farrell, “Nonlinear adaptive control using networks of piecewise linear approximators,” Proceedings of 38th Conference on Decision & Control Phoenix, Arizona USA. December 1999.
[26] L. X. Wang, “Stable adaptive fuzzy controllers with application to inverted pendulum tracking,” IEEE Transactions on Fuzzy Systems, man, and Cybernetics-Part B: Cybernetics, vol.26, no. 5,October 1996
[27] J. E. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991.
[28] A. Isidori, Nonlinear Control System. New York: Springer-Verlag, 1989
[29] M. Krstic, I. Kanellakopoulos, and P.V.Kokotovic, Nonlinear and Adaptive Control Design. New York: Wiley, 1995.
[30] I Kellakopoulos, P. V. Kokotovic, and A. S.Morse, “Systematic design of adaptive controller for feedback linearizable system,” IEEE Transactions Automat. Contr., vol. 36, pp. 1241–1253, 1991.
[31] R. Larson, B.H. Edwards, and D.E. Heyd, Calculus 7th, Houghton Mifflin ompany, 2002.
[32] C.F. Hsu, C.M. Lin, and T.T. Lee, “Wavelet adaptive backstepping control for a class of nonlinear systems,” IEEE Transactions on Neural Networks, vol. 17, no. 5, September 2006.
[33] C. Y. Chen, J.-Y. Lin, Z. H. Lee, Y. G. Leu, and W. Y. Wang, “Adaptive backstepping control of nonlinear systems using B-spline neural networks,” National Conference on Fuzzy Theory and Its Applications, pp. 436-440, 2008
[34] M. Sharma, A. Kulkarni, and S. Puntambekar, “Wavelet based adaptive tracking control for uncertain nonlinear systems with input constraints,” International Conference on Advances in Recent Technologies in Communication and Computing, pp. 694-698, 2009.
[35] M. Sharma, A. Kulkarni, and A. Verma , “Wavelet adaptive output tracking control for a class of delayed uncertain MIMO nonlinear systems subjected to actuator saturation,” International Conference on Advances in Computing, Control, and Telecommunication Technologies, pp. 705-710, 2009.