臺灣博碩士論文加值系統

本研究的主旨是探討如何將類神經網路及模糊控制方法，以及高頻抖動訊號應用到混沌系統。在本研究中，我們首先建立混沌系統的類神經網路模型，並將其轉換成類似模糊形式的線性為分函式。但是不可避免地，模型與真實系統間存在著近似誤差的問題。此時強健模糊控制設計來克服類神經網路模型與混沌系統中誤差問題。然而如果我們所設計出的模糊控制器不能使系統穩定，則加入高頻抖動訊號來協助穩定系統。當高頻抖動訊號的頻率愈高，系統的輸出就會與其相對應之模型－類神經網路鬆弛系統的輸出愈接近，所以我們可以利用類神經網路鬆弛系統的穩定度，來探討加入高頻抖動訊號後混沌系統的穩定度來探討加入高頻抖動訊號後混沌系統的穩定度。最後，藉著找出適當的高頻抖動訊號波形、振幅及頻率，來有效控制混沌運動到一個期望的週期性軌跡或是一個穩定的狀態。

This thesis presents an effective approach for controlling chaos. First, a neural-network (NN) model is employed to approximate the chaotic system. Then, a linear differential inclusion (LDI) state-space representation is established for the dynamics of NN model. Based on the LDI state-space representation, a robustness design of fuzzy control is proposed to overcome the effect of modeling error between chaotic systems and NN models. If the designed fuzzy controller cannot asymptotically stabilize the chaotic system, a high frequency signal, commonly called dithers, into the chaotic systems. According to the relaxed method, an appropriate dither is introduced to suppress chaotic motion. If the frequency of dither is high enough, the outputs described by the dithered chaotic system and the outputs of its corresponding mathematical model — the relaxed system would be made as close as desired. Thus, the closed-loop dithered chaotic system’s behavior can be rigorously predicted by establishing that of the closed-loop relaxed system. Finally, a numerical example with simulations is given to illustrate the concepts discussed throughout this thesis

摘要………………………………………………………………………i
Abstract………………………………………………………………ii
誌謝……………………………………………………………………………iv
Contents…………………………………………………………………………v
Contents of Figures…………… ………………………………………………vi
Chapter 1 Introduction………………………………………………………1
Chapter 2 Introduction to chaos………………………………………………4
Chapter 3 Neural-Network Based Approach and Fuzzy Control…………8
3.1. Neural-Network Model of Chaotic System………………………………8
3.2. Fuzzy Controller………………………………………………………11
Chapter 4 Robustness Control Design and Stability Analysis……………13
4.1. Modeling Error…………………………………………………………13
4.2. Stability Analysis………………………………………………………15
Chapter 5 Dithered System and Stability Analysis…………………………18
5.1. Dithered Chaotic System and Relaxed Model…………………………18
5.2. Closed-loop Relaxed System and Modeling Error……………………20
5.3. Stability Analysis………………………………………………………21
Chapter 6 Simulation Results………………………………………………25
Chapter 7 Brief Discussion and Conclusion…………………………………35
7.1 Discussion………………………………………………………………35
7.2 Conclusion……………………………………………………………….36
Bibliography…………………………………………………………………37

[1] T. Kapitaniak, L. J. Kocarev, and L. O. Chua, “Controlling chaos without feedback and control signals,” Int. J. Bifurcation Chaos, vol 3, pp. 459-468, 1993.
[2] C. C. Hwang, H. Y. Chow, and Y. K. Wang, “A new feedback control of a modified Chua’s circuit system,” Physica D, vol 92, pp. 95-100, 1996.
[3] G. Chen and X. Yu, “On time-delayed feedback control of chaotic systems,” IEEE Trains. Circuit and systems I: Fundamental Theory and Applications, vol 46, pp. 767-772, 1999.
[4] B. F. Feeny and F. C. Moon, “Quenching stick-slip chaos with dither,” Journal of Sound and Vibration, vol. 237, pp. 173-180, 2000.
[5] M. Aldawod, B. Samali, F. Naghdy, and K. C.S. Kwok, “Active control of along wind response of tall building using a fuzzy controller,” Engineering Structures, vol 23, pp. 1512-1522, 2001.
[6] Y. C. Hsiao and P. C. Tung, "Controlling Chaos for Nonautonomous Systems by Decting Unstable Periodic Orbits," Chaos, Solitons and Fractals, Vol. 13, pp. 1043-1051, 2002.
[7] J. S. Lin, J. J. Yan, T. L. Liao, “Robust control of chaos in Lorenz systems subject to mismatch uncertainties”, Chaos, Solitons and Fractals, vol 27, pp. 501–510, 2006.
[8] H. O. Wang and E. H. Abed, “Bifurcation control of a chaotic system,” Automatica, vol 31, pp. 1213-1226, 1995.
[9] P. Colet and Y. Braiman, “Control of chaos in multimode solid state lasers by the use of small periodic perturbations,” Phys. Rev. E, vol 53, pp. 200-206, 1996.
[10] E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett,. vol 64, pp. 1196-1199, 1990.
[11] A. R. Tose, “Nonlinear feedback for controlling the Lorenz equation,” Phys. Rev. E, vol 50, pp. 2339-2342, 1994.
[12] D. Kuo, “Chaos and its computing paradigm,” IEEE Potentials, vol. 24, pp. 13-15, 2005.
[13] H. Nijmeijer, and H. Berghuis, “On Lyapunov of control of the Duffing equation,” IEEE Trans. Circuits system, vol. 42, pp. 473-477, 1995.
[14] Y. Zeng, and S. N. Singh, “Adaptive control of chaos in Lorenz system,” Dynamic control, vol. 7, pp. 143-154, 1997.
[15] K. Tanaka, “An approach to stability criteria of neural-network control systems,” IEEE Trans. Neural Networks, vol. 7, pp. 629-643, 1996.
[16] S. Limanond and J. Si, “Neural-network-based control design: an LMI approach,” IEEE Trans. Neural Networks, vol. 9, pp. 1422-1429, 1998.
[17] A. Tascillo and N. Bourbakis, “Neural and Fuzzy Robotic Hand Control,” IEEE Trans. Systems, Man, and Cybernetics- Part B: Cybernetics, vol. 29, pp. 636-642, 1999.
[18] S. Limanond, J. Si, and Y. L. Tseng, “Production data based optimal etch time control design for a reactive ion etching process,” IEEE Trans. Semiconductor Manufacturing, vol. 12, pp.139-147, 1999.
[19] D. Zhang and S. K. Pal, “Parallel system design for time-delay neural networks,” IEEE Trans. Systems, Men, Cybernetics-C, vol. 30, pp. 265-275, 2000.
[20] X. B. Liang and J. Si, “Global exponential stability of neural networks with globally Lipschitz continuous activations and its application to linear variational inequality problem,” IEEE Trans. Neural Networks, vol. 12, pp.349-359, 2001.
[21] S. F. Su and F. Y. P. Yang, “On the Dynamical Modeling with Neural Fuzzy Systems,” IEEE Trans. Neural Networks, vol.13, no.6, pp. 1548-1553, 2002.
[22] R. Enns and J. Si, “Helicopter trimming and tracking control using direct neural dynamic programming,” IEEE Trans. Neural Networks, vol. 14, pp.929-939, 2003.
[23] C. Alippi, C. de Russis, and V. Piuri, “A neural-network based control solution to air-fuel ratio control for automotive fuel-injection systems,” IEEE Trans. Systems, Men, Cybernetics-C, vol. 33, pp. 259-268, 2003.
[24] 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, pp. 69-80, 2006.
[25] F. J. Lin, H. J. Shieh, P. H. Shieh, and P. H. Shen, “An Adaptive recurrent-neural-network motion controller for X-Y table in CNC machine,” IEEE Trans. Systems, Men, Cybernetics-B, vol. 36, no. 2, pp. 286-299, 2006.
[26] H. O. Wang, K. Tanaka, and M. F. Griffin, “An approach to fuzzy control of nonlinear systems: stability and design issues,” IEEE Trans. Fuzzy Systems, vol. 4, pp. 14-23, 1996.
[27] X. J. Ma, Z. O. Sun, and Y. Y. He, “Analysis and design of fuzzy controller and Fuzzy observer,” IEEE Trans. Fuzzy Systems, vol. 6, pp. 41-51, 1998.
[28] Y. Y. Cao and P. M. Frank, “Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach,” IEEE Trans. Fuzzy Systems, vol. 8, pp. 200-211, 2000.
[29] C. S. Tseng, B. S. Chen, and H. J. Uang, “On maximum stability margin design on nonlinear uncertain systems: fuzzy control approach,” Asian Journal of Control, vol. 3, pp. 190-203, 2001.
[30] S. H. Liu and C. T. Lin, “A model-based fuzzy logic controller with Kalman filtering for tracking mean arterial pressure,” IEEE Trans. Systems, Men, Cybernetics-A, vol. 31, pp. 676-686, 2001.
[31] S. J. Wund and C. T. Lin, “Discrete-time optimal fuzzy controller design: global concept approach,” IEEE Trans. Fuzzy Systems, vol. 10, no. 1, pp. 21-38, 2002.
[32] J. J. Wang, C. T. Lin, S.H. Liu, and Z. C. Wen, “Model-based synthetic fuzzy logic controller for indirect blood pressure measurement,” IEEE Trans. Systems, Men, Cybernetics-B, vol. 32, pp. 306-315, 2002.
[33] R. J. Wai, “Hybrid fuzzy neural-network control for nonlinear motor-toggle servomechanism,” IEEE Trans. Control Systems Technology, vol. 10, no. 4, pp. 519-532, 2002.
[34] C. M. Lin and C. F. Hsu, “Self-learning fuzzy sliding-mode control for antilock braking systems, “IEEE Trans. Control Systems Technology, vol. 11, pp. 273-278, 2003.
[35] T. H. S. Li and S. J. Chang, “Autonomous fuzzy parking control of a car-like mobile robot,” IEEE Trans. Systems, Men, Cybernetics-A, vol. 33, no. 4, pp. 451-465, 2003.
[36] Y. L. Sun and M. J. Er, “Hybrid fuzzy control of robotics systems,” IEEE Trans. Fuzzy Systems, vol. 12, pp. 755-765, 2004.
[37] K. Y. Lian, J. J. Liou, and C. Y. Huang, “LMI-Based Integral Fuzzy Control of DC-DC Converters,” IEEE Trans. Fuzzy Systems, vol. 14, pp. 71-80, 2006.
[38] K. Kiriakidis, “Fuzzy model-based control of complex plants,”IEEE Trans. Fuzzy Systems, vol. 6, pp. 517-529, 1998.
[39]B. S. Chen, C. S. Tseng, and H. J. Uang, “Robustness design of nonlinear dynamic systems via fuzzy linear control,”IEEE Trans. Fuzzy Systems, vol. 7, pp. 571-585, 1999.
[40] B. S. Chen, C. S. Tseng, and H. J. Uang, “Mixed fuzzy output feedback control design for nonlinear dynamic systems: an LMI approach,” IEEE Trans. Fuzzy Systems, vol. 8, pp. 249-265, 2000.
[41] Y. Y. Cao and P. M. Frank, “Robust disturbance attenuation for a class of uncertain discrete-time fuzzy systems,” IEEE Trans. Fuzzy Systems, vol. 8, pp. 406-415, 2000.
[42] Y. Y. Cao and Z. Lin, “Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation,” IEEE Trans. Fuzzy Systems, vol. 11, no. 1, pp. 57-67, 2003.
[43] C. S. Tseng, “Model Reference Output Feedback Fuzzy Tracking Control Design for Nonlinear Discrete-Time Systems with Time-Delay,” IEEE Trans. Fuzzy Systems, vol. 14, pp. 58-70, 2006.
[44] A. M. Steinberg and I. Kadushin, “Stabilization of nonlinear systems with dither control,” J. Math. Analysis and Application, vol. 43, pp. 273-284, 1973.
[45] G. Zames and N. A. Shneydor, “Dither in Nonlinear Syatems,” IEEE Trans. Automatic Control, vol. 21, no. 5, pp. 660-667, 1976.
[46] G. Zames and N. A. Shneydor, “Structural Stabilization and Quenching by Dither in Non-linear Systems,” IEEE Trans. Automatic Control, vol. 22, no. 3, pp. 352-361, 1977.
[47] S. Mossaheb, “Application of a Method of Averaging to the Study of Dither in Non-linear Systems,” International Journal of Control, vol. 38, no. 3, pp. 557-576, 1983.
[48] C. A. Desoer and S. M. Shahruz, “Stability of the dithered nonlinear system with backlash or hysteresis,” International Journal of Control, 43, 1045-1060, 1986.
[49] F. H. Hsiao and J.D. Hwang, “Stabilization of Nonlinear Singularly Perturbed Multiple Time-delay Systems by Dither”, ASME Trans. Journal of Dynamic Systems, Measurement, and Control, Vol. 118, No. 1, pp. 176-181, 1996.
[50] F. H. Hsiao and J. D Hwang, “Dither in linear systems with memoryless nonlinearity and optimal control,” IEEE Trans. Circuits and Systems-I, vol. 44, no. 5, 1997.
[51] B. F. Feeny and F. C. Moon, “Quenching stick-slip chaos with dither,” Journal of Sound and Vibration, vol. 237, pp. 173-180, 2000.
[52] A. A. Pervozvanski and C. Canudas-de-wit, “Asymptotic analysis of the dither effect in systems with friction,” Automatica, vol. 38, pp. 105-113, 2002.
[53] L. Iannelli, K. H. Johansson, U. T. Jonsson, and F. Vasca “Dither for smoothing relay feedback systems,” IEEE Trans. Circuits and Systems-I, vol. 50, pp. 1025-1035, 2003.
[54] L. Iannelli, K. H. Johansson, U. T. Jonsson, and F. Vasca “Averaging of nonsmooth systems using dither,” Automatica, vol. 42, pp. 669-676, 2006.
[55] K. K. Shyu and Y. Y. Lee, “Compensation and control of dither-smoothed nonlinearities,” JSME International Journal.Series C, Mechanical Systems, machine elements and manufacturing, vol. 49, pp. 512-519, 2006.
[56] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, “Linear matrix inequalities in system and control theory,” Philadelphia, PA: SIAM, vol. 15, 1994.
[57] M. Akar and Ü. Özgüner, “Decentralized parallel distributed compensator design for Takagi-Sugeno fuzzy systems,” IEEE. Conf. on Decision and Control, vol. 5, pp. 4834-4839, 1999.
[58] W. J. Wang and C. F. Cheng, “Stabilising controller and observer synthesis for uncertain large-scale systems by the Riccati equation approach,” IEE Proceeding-D, vol. 139, pp. 72-78, 1992.
[59] H. O. Wang, and K Tanaka, “An LMI-based stable fuzzy control of nonlinear systems and its application to control of chaos,” IEEE Int. Conf. on Fuzzy Systems, pp. 1433-1438, 1996.
[60] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox User’s Guide. The MathWorks, Inc., 1995.
[61] L. Weiss and E. F. Infante, “Finite time stability under perturbing forces and on product spaces,” IEEE Trans. Automatic Control, vol. 12, pp. 54-59, 1967.
[62] K. Tanaka, I. Takayuki, and H. O. Wang, “A unified approach controlling chaos via an LMI-based fuzzy control system design,” IEEE Trans. Circuits and Systems-I, vol. 45, pp. 1021-1040, 1998.