本篇論文主要研究連續時間強健 (Robust) 控制系統及連續時間 Takagi-Sugeno(T-S)模糊控制系統的非二次(non-quadratic)穩定寬鬆條件; 我們利用波雅定理(Polya Theorem)的代數性質加上寬鬆矩陣變數(slack matrix variables)來建立一組寬鬆的線性矩陣不等式(LMI)，因為非二次(non-quadratic)穩定的分析加上寬鬆矩陣變數 (slack matrix variables) 的使用，使得此組線性矩陣不等式(LMI)的求解保守性更進一步的降低，亦即當使用波雅定理(Polya Theorem)時，齊次多項式的階數不用太高，就可以找到解，這是本論文最大的優點；最後會提出幾個例子來證明我們理論的優越性。
關鍵字：強健(Robust)控制系統 Takagi-Sugeno(T-S)模糊控制系統、 非二次 (non-quadratic)穩定、 波雅定理 (Polya Theorem)、寬鬆矩陣變數(slack matrix variables)、 線性矩陣不等式 (LMI)

In this thesis, we investigate non-quadratic ralaxation for continuous-time robust control systems
and continuous-time fuzzy control systems, which are characterized by parameter-dependent LMIs (PD-LMIs), exploiting
the algebraic property of Polya Theorem to construct a family of finite-dimensional LMI relaxations with righ-hand-side
slack matrices that release conservatism. Lastly, numerical
experiments to illustrate the advantage of relaxations, being less conservative and effective, are provided.
it keyword: Robust control systems; Takagi-Sugeno fuzzy control systems; Non-quadratic relaxations;
Parameter-dependent LMIs (PD-LMIs); Polya Theorem; Slack matrices; Linear matrix inequality (LMI).

[1] T. Taniguchi, K. Tanaka, H. Ohatake, and H. Wang, “Model construction, rule reduction and robust compensation for generalized form of Takagi-Sugeno fuzzy systems,” IEEE Trans. Fuzzy Systems, vol. 9, no. 4, pp. 525–538, Aug. 2001.
[2] H. 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,” in Proc. of 18th Int’l Conf. of the North American Fuzzy Information Processing Society, 2000.
[3] K. Tanaka, T. Taniguchi, and H. Wang, “Generalized Takagi-Sugeno fuzzy systems: rule reduction and robust control,” in Proc. of 7th IEEE Conf. on Fuzzy Systems, 2000.
[4] H. Wang, K. Tanaka, and M. Griffin, “An approach to fuzzy control of nonlinear systems:
stability and design issues,” IEEE Trans. Fuzzy Systems, vol. 4, no. 1, pp. 14–23, Feb.
1996.
[5] K. Tanaka and H. Wang, Fuzzy Control Systems Design: A Linear Matrix Inequality
Approach. New York, NY: John Wiley & Sons, Inc., 2001.
[6] 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.
[7] J. Lo and M. Lin, “Observer-based robust H∞ control for fuzzy systems using two-step procedure,” IEEE Trans. Fuzzy Systems, vol. 12, no. 3, pp. 350–359, Jun. 2004.
[8] ——, “Robust H∞ nonlinear control via fuzzy static output feedback,” IEEE Trans. Cir- cuits and Syst. I: Fundamental Theory and Applications, vol. 50, no. 11, pp. 1494–1502, Nov. 2003.
[9] M. de Oliveira, J. Bernussou, and J. Geromel, “A new discrete-time robust stability con- dition,” Syst. & Contr. Lett., vol. 37, pp. 261–265, 1999.
66
[10] D. Peaucelle, D. Arzelier, O. Bachelier, and J. Bernussou, “A new robust D-stability condition for real convex polytopic uncertainty,” Syst. & Contr. Lett., vol. 40, pp. 21–23,
2000.
[11] J. Geromel and M. de Oliveira, “H2 and H∞ robust filtering for convex bounded uncertain system,” IEEE Trans. Automatic Control, vol. 46, no. 1, pp. 100–107, Jan. 2001.
[12] M. de Oliveira, J. Geromel, and J. Bernussou, “Extended H2 and H∞ norm characteriza- tions and controller parameterizations for discrete-time systems,” Int. J. Contr., vol. 75, no. 9, pp. 666–679, 2002.
[13] R. Oliveira and P. Peres, “LMI conditions for robust stability analysis based on polynomi- ally parameter-dependent Lyapunov functions,” Syst. & Contr. Lett., vol. 55, pp. 52–61,
2006.
[14] ——, “Parameter-dependent LMIs in robust analysis: characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations,” IEEE Trans. Automatic Control, vol. 52, no. 7, pp. 1334–1340, Jul. 2007.
[15] ——, “LMI conditions for the existence of polynomially parameter-dependent Lyapunov functions assuring robust stability,” in Proc. of 44th IEEE Conf. on Deci and Contr, Seville, Spain, Dec. 2005, pp. 1660–1665.
[16] V. Montagner, R. Oliveira, P. Peres, and P.-A. Bliman, “Linear matrix inequality charac- terization for H∞ and H2 guaranteed cost gain-scheduling quadratic stabilization of linear time-varying polytopic systems,” IET Control Theory & Appl., vol. 1, no. 6, pp. 1726–1735,
2007.
[17] 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.
[18] ——, “Convergent LMI relaxations for quadratic stabilization and H∞ control of Takagi- sugeno fuzzy systems,” IEEE Trans. Fuzzy Systems, no. 4, pp. 863–873, Aug. 2009.
[19] 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.
[20] C. Fang, Y. Liu, S. Kau, L. Hong, and C. Lee, “A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems,” IEEE Trans. Fuzzy Systems, vol. 14, no. 3, pp. 386–397, Jun. 2006.
67
[21] X. Liu and Q. Zhang, “New approaches to H∞ controller designs based on fuzzy observers for T-S fuzzy systems via LMI,” Automatica, vol. 39, pp. 1571–1582, 2003.
[22] B. Ding, H. Sun, and P. Yang, “Further studies on LMI-based relaxed stabilization con- ditions for nonlinear systems in Takagi-sugeno’s form,” Automatica, vol. 43, pp. 503–508,
2006.
[23] B. Ding and B. Huang, “Reformulation of LMI-based stabilization conditions for non-linear systems in Takagi-Sugeno’s form,” Int’l J. of Systems Science, vol. 39, no. 5, pp. 487–496,
2008.
[24] T. Guerra and L. Vermeiren, “Conditions for non quadratic stabilization of discrete fuzzy models,” in 2001 IFAC Conference, 2001.
[25] ——, “LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno’s form,” Automatica, vol. 40, pp. 823–829, 2004.
[26] J. Wan and J. Lo, “LMI relaxations for nonlinear fuzzy control systems via homoge- neous polynomials,” in The 2008 IEEE World Congress on Computational Intelligence, FUZZ2008, Hong Kong, CN, Jun. 2008, pp. 134–140.
[27] A. Sala and C. Arin˜o, “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.
[28] H. Zhang and X. Xie, “Relaxed Stablity Conditions for Continuous-Time T-S Fuzzy- Control Systems Via Augmented Multi-Indexed Matrix Approach,” pp. 478–492, June
2011.
[29] J. Lo and C. Tsai, “LMI relaxations for non-quadratic discrete stabilization via Polya theorem,” in Proc. of the 48th IEEE Conference on Decision and Control, Shanghai,CH, Dec. 2009, pp. 7430–7435.
[30] V. Montagner, R. Oliveira, T. Calliero, R. Borges, P. Peres, and C. Prieur, “Robust absolute stability and nonlinear state feedback stabilization based on polynomial Lur’e functions,” Nonlinear Analysis, vol. 70, pp. 1803–1812, 2009.
[31] I. Abdelmalek, N. Gol’ea, and M. Hadjili, “A new fuzzy Lyapunov approach to non- quadratic stabilization of Takagi-Sugeno fuzzy models,” Int. J. Appl. Math. Comput. Sci, vol. 17, no. 1, pp. 39–51, Feb. 2007.