主要分為三大部分。第一部分討論李雅普諾夫穩定度判斷，以及考慮波雅理論後，系統是否較容易求解(寬鬆性)；第二部分考慮性能指標 H_∞，也就是系統受雜訊干擾時的穩定度檢測條件，最後則以平方和寬鬆方法去達到完整解，使系統穩定度滿足充分條件，並以模擬結果加以驗證。
前兩部分所探討的為一般所熟悉的現有成果，採循序漸近的方式加以回顧。在較早的模糊控制文獻中，大部分研究都只著重於找出滿足二次穩定的共同李雅普諾夫函數，2000年左右由於寬鬆變數矩陣的概念出現，加速了求解過程；2005年波雅理論的發展已趨成熟，當隨著波雅冪次增加到足夠大時，可使模糊系統穩定度滿足充分條件，對寬鬆性有很大幫助；在2008年時，萬嘉仁學長的研究中[1]，將寬鬆變數矩陣概念及波雅理論加以結合，模擬結果顯示所需的波雅冪次小於波雅理論所建議的值，並且展現了更大的解空間，但隨著波雅冪次的增加，寬鬆變數量會呈指數遞增，造成電腦運算上的負擔，因此，提出了平方和寬鬆法以解決變數上的問題，並探討其寬鬆性。另一方面，也提出了以芬斯勒定理為架構的穩定度條件加以比較，第二部分主要探討考慮H∞性能在一般及芬斯勒架構下的穩定度檢測條件。
第三部分，我們將一個考慮閉迴路的模糊系統轉換成偕正矩陣，並且以平方和分解去近似偕正矩陣的錐體，使變數相依的線性矩陣不等式達到完整解(非保守解)。經由系統化的挑選單項式，我們即可以偕正矩陣去建立一個偕正寬鬆家族，使系統達漸近穩定且收斂。最後的數值分析結果顯示，平方和寬鬆法與線性矩陣不等式寬鬆法皆有相同的寬鬆結果，但是值得注意的是，偕正寬鬆方法並不會因寬鬆階數的提高，而增加額外的寬鬆變數，故偕正寬鬆法有其優點，值得推廣。

In this thesis, we convert a double fuzzy summation into cone of copositive matrices, proposing an SOS decomposition to approximate the cone of copositive matrices, leading to exact relaxation for fuzzy PD-LMIs. Based on suitable monomials which can be generated systematically, we show that one can work with copositive matrices to construct asymptotically exact copositive relaxation families with certificates of convergence. Numerical experiments show that both LMI and SOS relaxations reach the exact solutions, but the copositive relaxation demands NO slack variables when the level of relaxation increases.

[1] 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.
[2] P. Parrilo, Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. Caltech, Pasadena, CA.: PhD thesis, 2000.
[3] S. Prajna, A. Papachristodoulou, and P. Parrilo, “Introducing SOSTOOLS: a general purpose sum of squares programming solver,” in Proc of IEEE CDC, Montreal, Ca, Jul. 2002, pp. 741–746.
[4] S. Prajna, A. Papachristodoulou, and et al, “New developments on sum of squares optimization and SOSTOOLS,” in Proc. the 2004 American Control Conference, 2004, pp. 5606–5611.
[5] 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.
[6] K. Tanaka, H. Yoshida, “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.
[7] H. Ichihara and E. Nobuyama, “A computational approach to state feedback synthesis for nonlinear systems based on matrix sum of squares relaxations,” in Proc. 17th Int’l
Symposium on Mathematical Theory of Network and Systems, Kyoto, Japan, 2006, pp. 932–937.
[8] H. Ichihara, “Observer design for polynomial systems using convex optimization,” in Proc. of the 46th IEEE CDC, New Orleans, LA, Dec. 2007, pp. 5347–5352.
[9] 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.
[10] 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.
[11] 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.
[12] C. Hol and C. Scherer, “Sum of squares relaxations for polynomial semidefinite programming,” in Proc.of MTNS, 2004, pp. 1–10.
[13] C. Scherer and C. Hol, “Asymptotically exact relaxations for robust LMI problems based on matrix-valued sum-of-squares,” in Proc. 16th Int’l Symposium on Mathematical Theory of Network and Systems, 2004, pp. 1–11.
[14] 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.
[15] V. Montagner, R. Oliveira, and P. Peres, “Convergent LMI relazations for quadratic stabilization and H1 controll of Takagisugeno fuzzy systems,” IEEE Trans. Fuzzy Systems, pp. 863–873, Aug. 2009.
[16] A. Sala and C. Ari˜no, “Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya’s theorem,” Fuzzy Set and Systems, vol. 158, pp. 2671–2686, 2007.
[17] A. Sala and C. Ari˜no, “Design of multiple- parameterization PDC controllers via relaxed condition for
multi- dimensional fuzzy summations,” in Proc. of Fuzz-IEEE’07., London, UK, 2007, doi:10.1109/FUZZY.2007.4295495.
[18] A. Kruszewski, A. Sala, T. M. Guerra, and C. Ari˜no, “A triangulation approach to asymptotically exact conditions for fuzzy summations,” IEEE Trans. Fuzzy Systems, vol. 17, no. 5, pp. 985–994, Oct. 2009.
[19] J. Lo and J. Wan, “Studies on LMI relaxations for nonlinear fuzzy control systems via homogeneous polynomials,” IET Control and Applicaitons, Apr. 2010, in press.
[20] 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.
[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] I. M. Bomze and E. D. Klerk, “Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming,” in Journal of Global Optimization, vol. 24, no. 2, Netherlands, Oct. 2002, pp. 163–185.
[23] M. C. de Oliveira and R. E. Skelton, “Stability Tests for Constrained Linear Systems,” in Perspectives in Robust Control, vol. 268, California, 2001, pp. 241–257.
[24] M. de Oliveira, J. Bernussou, and J. Geromel, “A new discrete-time robust stability condition,” Syst. & Contr. Lett., vol. 37, pp. 261–265, 1999.
[25] 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.
[26] 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.
[27] G. Hardy, J. Littlewood, and G. P′olya, Inequalities, second edition. Cambridge, UK.: Cambridge University Press, 1952.
[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] C. Scherer, “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] K. Tanaka and H. Wang, Fuzzy Control Systems Design: A Linear Matrix Inequality Approach. New York, NY: John Wiley & Sons, Inc., 2001.
[33] S. Boyd, L. E. Ghaoui, and et al, Linear Matrix Inequalities in System and Control Theory. Philadelphia: PA:SIAM, 1994.
[34] 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.
[35] A. Papachristodoulou and S. Prajna, “On the construction of Lyapunov functions using the sum of squares decomposition,” in Proc of IEEE CDC, Montreal, Ca, Jul. 2002, pp. 3482–3487.
[36] J. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimization Methods and Software, vol. 11-12, pp. 625–653, 1999.
[37] B. Reznick, “Some concrete aspects of Hilbert’s 17th problem,” in Contemporary Mathematics, vol. 253, American Mathematical Society, 2000, pp. 251–272.
[38] 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.
[39] A. Kruszewski and T. M. Guerra, “New approaches for the stabilization of discrete Takagi-Sugeno fuzzy models,” in Proc. 44th IEEE Conf. Decision Contr. and Eur. control Conf. ECC, Seville, Spain, Dec. 2005, pp. 3255–3260.
[40] H. Lee, J. Park, and G. Chen, “Robust fuzzy control of nonlinear systems with parametric uncertainties,” IEEE Trans. Fuzzy Systems, vol. 9, no. 2, pp. 369–379, Apr. 2001.