第一部分論述非線性系統轉換成廣義Takagi-Sugeno模糊系統後, 將其結合廣義動態輸出回授控制器, 並在上述架構下與圓定理整合, 求得雙線性矩陣不等式, 再簡化至動態輸出回授控制器, 求得在此系統架構下之二次穩定條件。
第二部份說明二次穩定LMI 檢測條件的演化, 從傳統LMI 檢測法到寬鬆矩陣變數和波雅定理的出現, 而重點在如何演進至平方和二次穩定檢測法。
第三部份討論圓定理與耗散性控制理論兩者的關聯性。

In this thesis, three topics are addressed. First, we investigate a general control problem via the Circle criterion borrowed from system theory; We show a fuzzy version of Circle criterion and then provide a synthesis result based on the Circle criterion, establishing closed-loop stabilizability for dynamic output feedback controllers and state-feedback controllers. Second, we show how to solve LMI representation numerically by SOSTOOLS. Third, we solve a stabilization problem for systems with sector-bounded nonlinearities at their input. Then, based on dissipative control and Circle criterion, we get the same stabilization conditions for a state feedback stabilizing controller. Finally, we integrate Circle theorem and dissipative control theorem into one unified structure.

[1] 林玉婷, “強健模糊動態輸出回饋控制- Circle 與Popov 定理,” 國立中央大學, 碩士論文, 民國92 年.
[2] 萬嘉仁, “寬鬆耗散性模糊控制- 波雅定理,” 國立中央大學, 碩士論文, 民國97 年.
[3] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modelling and control,” IEEE Trans. Syst., Man, Cybern., vol. 15, no. 1, pp. 116–132, Jan. 1985.
[4] M. Sugeno and G. Kang, “Structure identification of fuzzy model,” Fuzzy Set and Systems, vol. 28, pp. 15–33, 1988.
[5] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,”Fuzzy Set and Systems, vol. 45, pp. 135–156, 1992.
[6] W. Haddad and D. Bernstein, “Explicit construction of quadratic Lyapunov functions for the small gain, positive, circle and Popov theorems and their application to robust stability. Part II: discrete-time theory,” Int’l J. of Robust and Nonlinear Control, vol. 4, pp. 249–265, 1994.
[7] K. Tanaka, T. Ikeda, and H. Wang, “Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs,” IEEE Trans. Fuzzy Systems, vol. 6, no. 2, pp. 250–265, May 1998.
[8] E. Kim and H. Lee, “New approaches to relaxed quadratic stability condition of fuzzy control systems,” IEEE Trans. Fuzzy Systems, vol. 8, no. 5, pp. 523–534, Oct. 2000.
[9] X. Liu and Q. Zhang, “New approaches to H1 controller designs based on fuzzy observers for T-S fuzzy systems via LMI," Automatica, vol. 39, pp. 1571–1582, 2003.
[10] C.-H. Fang, Y.-S. Liu, S.-W. Kau, L. Hong, and C.-S. Lee, “A new LMI-based approach to relaxed quadratic stabilzation of T-s fuzzy control systems,” IEEE Trans. Fuzzy Systems, vol. 14, no. 3, pp. 386–397, 2006.
[11] M. Teixeira, E. Assuncao, and R. Avellar, “On relaxed LMI-based design for fuzzy regulators and fuzzy observers,” IEEE Trans. Fuzzy Systems, vol. 11, no. 5, pp.613–623, 2003.
[12] G. Hardy, J. Littlewood, and G. P´olya, Inequalities, second edition. Cambridge, UK.: Cambridge University Press, 1952.
[13] R. Oliveira and P. Peres, “Stability of polytopes of matrices via affine parameterdependent Lyapunov functions: Asymptotically exact LMI conditions,” Linear Algebra and its Applications, vol. 405, pp. 209–228, 2005.
[14] V. Montagner, R. Oliveira, P. Peres, and P.-A. Bliman, “Linear matrix inequality characterisation for H1 and H2 guaranteed cost gain-scheduling quadratic stabiilisation of linear time-varying polytopic systems,” IET Control Theory Appl., vol. 1, no. 6, pp. 1726–1735, 2007.
[15] R. Oliveira and P. Peres, “Parameter-dependent LMIs in robust analysis: characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxatiions,”IEEE Trans. Automatic Control, vol. 52, no. 7, pp. 1334–1340, Jul. 2007.
[16] V. Montagner, R. Oliveira, and P. Peres, “Necessary and sufficient LMI conditions to compute quadratically stabilizing state feedback controller for Takagi-sugeno systems,”in Proc. of the 2007 American Control Conference, Jul. 2007, pp. 4059–4064.
[17] J. Lo and J. Wan, “Studies on LMI relaxations for fuzzy control systems via homogeneous polynomials,” IET Control Theory Appl., 2010, doi:10.1049/iet-cta.2009.0192.
[18] J. Wan and J. Lo, “LMI relaxations for nonlinear fuzzy control systems via homogeneous polynomials,” in The 2008 IEEE World Congress on Computational Intelligence, FUZZ2008, Hong Kong, CN, Jun. 2008, pp. 134–140.
[19] K. Tanaka, H. Yoshida, and et al, “A sum of squares approach to stability analysis of polynomial fuzzy systems,” in Proc. of the 2007 American Control Conference, New York, NY, Jul. 2007, pp. 4071–4076.
[20] ——, “Stabilization of polynomial Fuzzy systems via a sum of squares approach,”in Proc. of the 22nd Int’l Symposium on Intelligent Control Part of IEEE Multi-conference on Systems and Control, Singapore, Oct. 2007, pp. 160–165.
[21] S. Prajna, A. Papachristodoulou, and F. Wu, “Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based Approach,” in Proc. 5th Asian Control Conference, 2004, pp. 157–165.
[22] W. Haddad and D. Bernstein, “Explicit construction of quadratic Lyapunov functions for the small gain, positive, circle and Popov theorems and their application to robust stability. Part I: continuous-time theory,” Int’l J. of Robust and Nonlinear Control, vol. 3, pp. 313–339, 1993.
[23] S. Yuliar and M. James, “Stabilization of linear systems with sector bounded nonlinearities at the input and output,” in Proc. of the 36th Conf. on Deci. & Contr., vol. 3, Kobe, JP, Dec. 1996, pp. 4759–4764.
[24] S. Xie, L. Xie, and C. de Souza, “Robust dissipative control for linear systems with dissipative uncertainty,” Int. J. Contr., vol. 70, no. 2, pp. 169–191, 1998.
[25] H. K. Khalil, Nonlinear Systems. New York, NY.: Macmillian Publishing Co., 1992.
[26] J. Lo and E. Chen, “State feedback control via circle criterion for uncertain fuzzy systems,” in Proc. 2004 Conf. Auto. Contr., vol. 1, Taipei, TW, Mar. 2004.
[27] J. Lo and Y. Lin, “Robust control for uncertain fuzzy systems via circle criterion,”in Proc. Joint Conf on AI, Fuzzy and Grey, vol. 1, Taipei, TW, Nov. 2003.
[28] V. Power and B. Reznick, “A new bound for P´olya’s Theorem with applications to polynominals positive on polyhedra,” J. Pure Appl. Algebra, vol. 164, pp. 221–229, 2001.
[29] J. de Loera and F. Santos, “An effective version of P´olya’s theorem on positive definite forms,” Journal of Pure and Applied Algebra, vol. 108, pp. 231–240, 1996.
[30] C. Scherer, “Higher-order relaxations for robust LMI problems with verifications for exactness,” in Proc. of 42th IEEE Conf. on Deci and Contr, Dec. 2003, pp. 4652–4657.
[31] ——, “Relaxations for robust linear matrix inequality problems with verification for exactness,” SIAM J. Matrix Anal.Appl., vol. 27, no. 2, pp. 365–395, 2005.
[32] E. de Klerk and D. V. Pasechnik, “Approximation of the Stability Number of a Graph via Copositive Programming,” SIAM Journal of Optimization, vol. 12, pp. 875–892, 2002.
[33] P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. Caltech, Pasadena, CA.: PhD thesis, 2000.
[34] J. Lo and M. Liu, “Exact Solutions to Fuzzy PD-LMIs via SOS,” in The 2010 World Congress on Computational Intelligence; FUZZ-IEEE International Conference on Fuzzy Systems, Barcelona, SP, Jul. 2010, pp. 1352–1357.
[35] J. Li, H. Wang, D. Niemann, and K. Tanaka, “Dynamic parallel distributed compensation for Takagi-Sugeno fuzzy systems: An LMI approach,” Information Sciences, vol. 123, pp. 201–221, 2000.
[36] Y. Cao and P. Frank, “Robust H1 disturbance attenuation for a class of uncertain discrete-time fuzzy systems,” IEEE Trans. Fuzzy Systems, vol. 8, no. 4, pp. 406–415, Aug. 2000.
[37] K. Tanaka and M. Sano, “A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer,” IEEE Trans. Fuzzy Systems, vol. 2, no. 2, pp. 119–133, May 1994.
[38] G. Feng and J. Ma, “Quadratic stabilization of uncertain discrete-time fuzzy dynamic system,” IEEE Trans. Circuits and Syst. I: Fundamental Theory and Applications, vol. 48, no. 11, pp. 1137–1344, Nov. 2001.
[39] K. Tanaka and H. Wang, Fuzzy Control Systems Design: A Linear Matrix Inequality Approach. New York, NY: John Wiley & Sons, Inc., 2001.
[40] J. Lo and J. Wan, “Dissipative control to fuzzy systems with nonlinnearity at the input,”in The 2007 CACS International Automatic Control Conference, Taichung,Tw, Nov. 2007, pp. 329–334.
[41] 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, Feb. 2003.
[42] B. Ding, H. Sun, and P. Yang, “Further studies on LMI-based relaxed stabilization conditions for nonlinear systems in Takagi-sugeno’s form,” Automatica, vol. 43, pp.503–508, 2006.