在本篇論文中針對Takagi-Sugeno (T-S) 模糊模型[1]的系統架構提出了控制器以及估測器的設計方式，並對系統中考慮收斂速度影響時作探討與設計。在控制器部分，我們利用平行分配補償（Parallel Distributed Compensation; PDC）[2]的設計觀念對以T-S 模糊模型描述的非線性受控體作控制器設計，在每一個子系統屬於可控標準式時，我們引用李亞普諾夫(Lyapunov)漸近穩定理論並以李亞普諾夫等式取代李亞普諾夫不等式後，我們可直接獲得李亞普諾夫方程式的解，透過此解便可進一步求到最終的控制器對受控體做控制並達到系統穩定的要求。對於估測器的部分，針對T-S 模糊模型描述的非線性受控體，當每一個子系統的狀態矩陣用可估測標準式描述時，我們同樣利用李亞普諾夫漸近穩定理論分析估測誤差與系統輸出的收斂情形，推導並求解李亞普諾夫方程式後，我們能夠直接設計此系統的狀態估測器。除此之外，本論文尚且針對T-S模糊系統在考慮系統輸出收斂速度的要求下，進行控制器的設計與解決，並使系統輸出的響應收斂情形有大幅度的改善。

In the past, LMI（Linear Matrix Inequalities） technique is used to solve the fuzzy controller, observer and to analyze the stability of Lyapunov inequalities of the T-S (Takagi-Sugeno) fuzzy systems. However, there are still many difficulties in systematically designing the T-S fuzzy controller and the observer. In this thesis, we provide a fuzzy controller and observer design method for the nonlinear plant whose structure is represented by T-S fuzzy model. The model-based fuzzy controller and observer are designed by the concept of the so-called “PDC (Parallel Distributed Compensation)” [2]. Applying the Lyapunov asymptotic stability theorem instead of Lyapunov inequality, we can solve the Lyapunov equation via the algorithm provided in this work. The final stable controller and observer of the nonlinear plant can be obtained directly. This method is mainly based on the fact that each subsystem of T-S fuzzy model can be represented by the controllable and the observable canonical form. Besides, the decay rate of system state can also be improved.

Table of content
Page
Abstract Ⅰ
Table of contentⅡ
List of figuresⅤ
Chapter 1: Introduction 1
1-1. Research background 1
1-2. Motivation 1
1-3. Organization 2
Chapter 2: An algorithm of designing controllers for SISO T-S fuzzy
systems 3
2-1. Introduction 3
2-2. T-S fuzzy model and the stability conditions 5
2-3. Fuzzy controller design 6
2-4. T-S fuzzy controller design algorithm 9
2-5. Simulation 11
2-6. Discussion13
Chapter 3: Fuzzy controller design with decay rate for SISO T-S fuzzy
systems 18
3-1. Introduction18
3-2. T-S fuzzy model and the PDC concept18
3-2-1. T-S fuzzy model 18
3-2-2. Parallel distributed compensation (PDC)19
3-3. Decay rate and the stability conditions20
3-4. T-S fuzzy controller design with decay rate22
3-5. Fuzzy controller design algorithm with decay rate23
3-6. Simulation 26
3-7. Discussion 27
Chapter 4: Output feedback gain design with decay rate for SISO T-S fuzzy systems32
4-1. Introduction 32
4-2. T-S fuzzy model and the PDC concept32
4-3. Stability conditions and the output feedback gain design with decay
rate 34
4-3-1. Stability conditions of system34
4-3-2. Output feedback gain design36
4-4. T-S fuzzy output feedback gain design algorithm37
4-5. Simulation40
4-6. Discussion42
Chapter 5: Observer design for SISO T-S fuzzy system46
5-1. Introduction 46
5-2. T-S fuzzy model and the T-S fuzzy observer47
5-3. The stability conditions of the fuzzy system with the observer
feedback 49
5-4. Fuzzy controller and the observer design algorithm 53
5-5. Simulation 59
5-6. Discussion 61
Chapter 6: Conclusions 65
6-1. The fuzzy controller design 65
6-2. The analysis of fuzzy observer feedback control 65
6-3. Future direction66
References 67

[1]T. Takagi and M. Sugeno, “Fuzzy Identification of System and Its Applications to Modeling and Control”, IEEE Transactions on Systems,Man, and Cybernetics, Vol. SMC-15, No. 1, pp. 116-132, 1985.[2]H. O. Wang, K. Tanaka and M. F. Griffin, “An Approach to Fuzzy Control of Nonlinear Systems: Stability and Design Issues”, IEEE Transactions on Fuzzy Systems, Vol. 4, No. 1, pp. 14-23, 1996.[3]C. S. Chen and W. L. Chen, “Analysis and Design of a Stable Fuzzy Control System”, Fuzzy Sets and Systems, Vol. 96, No. 1, pp. 21-35, 1998. [4]J. Park, J. Kim and D. Park, “LMI-Based Design of Stabilizing Fuzzy Controller for Nonlinear Systems Described by Takagi-Sugeno Fuzzy Model”, Fuzzy Sets and Systems, Vol. 122, No. 1, pp. 73-82, 2001.[5]J. Yoneyama, M. Nishikawa, H. Katayama and A. Ichikawa, “ Control for T-S Fuzzy Descriptor System”, Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, Vol. 3, pp. 28-33, 1999.[6]K. Tanaka, T. Ikeda and H. O. Wang, “An Approach to Stabilization of Uncertain Fuzzy Systems”, Proceedings of the IEEE International Conference on Fuzzy Systems, Vol. 1, pp. 72-77, 1996.[7]K. Tanaka, T. Taniguchi and H. O. Wang, “Model-Based Fuzzy Control of TORA System: Fuzzy Regulator and Fuzzy Observer Design via LMIs that Represent Decay Rate, Disturbance Rejection, Robustness, Optimality”, Proceedings of the IEEE International Conference on Fuzzy Systems, Vol. 1, pp. 313-318, 1998.[8]K. Tanaka, T. Taniguchi and H. O. Wang, “Generalized T-S Fuzzy System: Rule Reduction and Robust Control”, Proceedings of the IEEE International Conference on Fuzzy Systems, Vol. 2, pp. 688-693, 2000.[9]T. Taniguchi, K. Tanaka, K. Yamafuji and H. O. Wong, “Nonlinear Model Following Control via T-S Fuzzy Model”, Proceedings of the American Control Conference, Vol. 3, pp. 1837-1841, 1999.[10]T. Taniguchi, “Modeling and Model Reduction Using Generalized Form of T-S Fuzzy System”, Proceedings of the American Control Conference, Vol. 4, pp. 2854-2858, 2000.[11]T. Taniguchi, K. Tanaka and H. O. Wang, “Fuzzy Descriptor Systems and Nonlinear Model Following Control”, IEEE Transactions on Fuzzy Systems, Vol. 8, No. 4, pp. 442-452, 2000.[12]W. J. Chang and C. C. Sun, “Fuzzy Controller Design for Nonlinear TORA Systems”, International Journal of Fuzzy Systems, Vol. 2, No. 1, pp. 60-66, 2000.[13]W. J. Chang, C. C. Sun and C. C. Fuh, “Applications of Continuous TS-Type Fuzzy Control with Common Controllability Gramian Assignment”, Proceedings of the International Conference on Automation Technology, Vol. 1, pp. 569- 574, Taipei, Taiwan, May 9-11, 2000.[14]W. J. Chang, “Model-Based Fuzzy Controller Design with Common Observability Gramian Assignment”, ASME, J. Dynamic Systems, Measurement and Control, Vol. 123, No. 1, pp. 113-116, 2001.[15]J. Joh, R. Langari, E. T. Jeung and W. J. Chung, “A New Design Method for Continuous Takagi-Sugeno Fuzzy Controller with Pole Placements: An LMI Approach”, Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, Vol. 3, pp. 2969-2974, 1997.[16]J. Joh, Y. H. Chen and R. Langari, ”On the Stability Issues of Linear Takagi-Sugeno Fuzzy Models”, IEEE Transactions on Fuzzy System, Vol. 6, No. 3, pp. 402-410, 1998.[17]J. M. Zhang, R. H. Li and P. A. Zhang, “Stability Analysis and Systematic Design of Fuzzy Control System”, Fuzzy Sets and Systems, Vol. 120, No. 1, pp. 65-72, 2001.[18]K. Tanaka and M. Sugeno, “Stability Analysis and Design of Fuzzy Control Systems”, Fuzzy Sets and Systems, Vol. 45, No. 2, pp. 135-156, 1992.[19]K. Tanaka, T. Ikeda and H. O. Wang “Design of Fuzzy Control Systems Based on Relaxed LMI Stability Conditions”, Proceedings of the Decision and Control, Vol. 1, pp. 598 -603, 1997.[20]K. Tanaka, T. Ikeda and H. O. Wang, “Fuzzy Control System Design via LMIs”, Proceedings of the American Control Conference, Vol. 5, pp. 2873- 2877, 1997.[21]M. A. L. Thathachar and P. Viswanath, “On the Stability of Fuzzy Systems”, IEEE Transactions on Fuzzy Systems, Vol. 5, No. 1, pp. 145-151, 1997.[22]J. Li, D. Niemann, H. O. Wong and K. Tanaka, “Parallel Distributed Compensation for T-S Fuzzy Models Multiobjective Controller Design”, Proceedings of the American Control Conference, Vol. 3, pp. 1832-1836, 1999.[23]K. Tanaka, T. Ikeda and H. O. Wang “An LMI Approach to Fuzzy Controller Designs Based on Relaxed Stability Conditions”, Proceedings of the IEEE International Conference on Fuzzy Systems, Vol. 1, pp. 171 -176, 1997.[24]K. Tanaka, T. Ikeda and H. O. Wang, “A Unified Approach to Controlling Chaos via an LMI-Based Fuzzy Control System Design”, IEEE Transactions on Circuits and Systems, Vol. 45, No. 10, pp. 1021-1040, 1998.[25]W. Chang, Y. H. Joo, J. B. Park and G. Chen, “Robust Fuzzy-Model-Based Controller for Uncertain Systems”, Proceedings of the IEEE International Conference on Fuzzy Systems, Vol. 1, pp. 486-491, 1999.[26]W. J. Chang, C. C. Sun and C. C. Fuh, “Discrete Output Fuzzy Controller Design for Achieving Common Controllability Gramian”, Asian Journal of Control, Vol. 2, No. 4, pp. 284-289, 2000.[27]K. Tanaka and M. SANO, “On the Concepts of Regulator and Observer of Fuzzy Control Systems”, Proceedings of the IEEE International Conference on Fuzzy Systems, Vol. 2, pp. 767 -772, Taipei, Taiwan, May 9-11, 1994.[28]K. Tanaka and H. O. Wang, “Fuzzy Regulators and Fuzzy Observers: A Linear Matrix Inequality Approach”, Proceedings of the Decision and Control, Vol. 2, pp. 1315-1320, 1997.[29]H. O. Wang and K. Tanaka, “An LMI-Based Stable Fuzzy Control of Nonlinear Systems and Its Application to Control of Chaos”, Proceedings of the IEEE International Conference on Fuzzy Systems, Vol. 2, pp. 1433-1438, 1996.[30]K. Tanaka, T. Ikeda and H. O. Wang, “Fuzzy Regulators and Fuzzy Observers: Relaxed Stability Conditions and LMI-Based Design”, IEEE Transactions on Fuzzy Systems, Vol. 6, No. 2, pp. 250-265, 1998.[31]K. Tanaka, T. Ikeda and H. O. Wang, “Robust Stabilization of a Class of Uncertain Nonlinear System via Fuzzy Control: Quadratic Stabilizability, Control Theory, and Linear Matrix Inequalities”, IEEE Transactions on Fuzzy Systems, Vol. 4, No. 1, pp. 1-13, 1996.[32]S. Boyd et al., Linear Matrix Inequalities in Systems and Control Theory, Philadelphia, PA: SIAM, 1994.[33]W. J. Chang and C. C. Sun, “Fuzzy Control with Common Observability Gramian Assignment for Continuous Takagi-Sugeno Models”, Proceedings of the American Control Conference, Vol. 2, pp. 1366-1370, 1999.[34]W. J. Chang, “Common Observability Gramian Assignment Using Discrete Fuzzy Control”, Proceedings of the IEEE International Conference on Fuzzy Systems, Vol. 1, pp. 84-89, 1999.[35]W. J. Chang, C. C. Sun and C. C. Fuh, “Continuous Output Feedback Fuzzy Controller Design with a Specified Common Controllability Gramian”, International Journal of Fuzzy Systems, Vol. 3, No. 1, pp. 356-363, 2001.[36]J. Zhao, V. Wertz and R. Gorez, “Fuzzy Gain Scheduling Controllers Based on Fuzzy Models”, Proceedings of the IEEE International Conference on Fuzzy Systems, Vol. 3, pp. 1670-1676, 1996.[37]P. Spångéus, “A Negative Result on Piecewise Quadratic Lyapunov Functions for Decay Rate Analysis”, Proceedings of the Decision and Control, Vol. 3, pp. 2312-2313, 1999.[38]P. A. Cook, Nonlinear Dynamical Systems, Prentice-Hall, New York, 1994.[39]C. S. Xiao, Z. M. Feng and X. M. Shan, “On the Solution of the Continuous- Time Lyapunov Matrix Equation in Two Canonical Forms”, Proceedings of the IEE Control Theory and Applications, Vol. 139, No. 3, pp. 286-290, 1992.[40]H. O. Wang, K. Tanaka and M. Griffin, “Parallel Distributed Compensation of Nonlinear Systems by Takagi-Sugeno’s Fuzzy Model,” Proceedings of the IEEE International Conference on Fuzzy Systems, Vol. 2, pp.531-538, 1995.[41]H. O. Wang, J. Li, D. Niemann and K. Tanaka “T-S Fuzzy Model with Linear Rule Consequence and PDC Controller: A Universal Framework for Nonlinear Control Systems”, Proceedings of the Decision and Control, Vol. 1, pp. 549 - 554, 2000.[42]X. Yu, Z. Man and B. Wu, “Design of Fuzzy Sliding-Mode Control Systems”, Fuzzy Sets and Systems, Vol. 95, No. 3, pp. 295-306, 1998.[43]W. J. Rugh, Linear System Theory. Prentice-Hall, New Jersey, 1996.[44]K. N. Hong, J. D. Park and T. Y. Kuc, “Pole Assignment and Tracking of Uncertain Linear Systems with State/Output Feedback: A Model Reference Adaptive Control Approach”, Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, Vol. 2, pp. 1120-1125, 1996.[45]W. J. Chang, “Dynamic Multi-Constrained Output Feedback Controller Design for Linear Discrete Systems”, Journal of the Chinese Institute of Engineers, Vol. 21, No. 5, pp. 563-573, 1998.[46]W. J. Chang, C. C. Fuh and C. C. Sun, “A New Method of Designing Discrete Output Feedback Fuzzy Controllers”, Proceedings of 2000 Automatic Control Conference, R. O. C., p. 816, 2000.