本論文主要研究齊次多項式尤拉法應用於切換式模糊控制器設計，以泰勒級數建模產生模糊系統，並使用片段式李亞普諾夫函數(piecewise polynomial Lyapunov function) 作穩定性分析，其高階李亞普夫函數型式為V (x) = min_1<=l<=N(V_l(x))，其中V_l(x)=x^T*P_l(x)*x，N 為分段個數，根據dot(V(x))<0，做穩定性條件分析，在求解的過程中，片段李亞普諾夫函數V_l(x) 對時間微分會產生Pl(x) 之微分項，其化簡過程過於繁瑣複雜，為避免此問題，引用尤拉齊次法，再以平方和方法檢驗穩定度條件，並設計出控制器。

In this thesis, it is mainly to research piecewise fuzzy controller design by homogeneous polynomial Lyapunov Euler method. We use Taylor series to model the fuzzy system, and implement the analysis of stability by piecewise polynomial Lyapunov function,which has the following form:V(x)= min_1<=l<=N(V_l(x))and V_l(x)=x^T*P_l(x)*x. Due to the fact that dot(V_l(x))=dot(x_T)*P_l(x)*x+x^T*dot(P_l(x))*x+x^T*P_l(x)*dot(x), the dot(P_l(x)) will yields complicated form to implement stablility condition. To avoid this problem, we propose a homogeneous polynomial function method and utilize Euler’s theorem for homogeneous function. The next, we solve the stabilization problem under consideration by the method of sum of squares and design its corresponding controller.

[1] 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.
[2] M. Sugeno and G. Kang, “Structure identification of fuzzy model,”Fuzzy Set and Systems, vol. 28, pp. 15–33, 1988.
[3] K. Tanaka and M. Sugeno, “Stability analysis and design of fuzzy control systems,” Fuzzy Set and Systems, vol. 45, pp. 135–156, 1992.
[4] 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.
[5] H.O. Wang and K. Tanaka and M.F. 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.
[6] J.R. Wan and J.C. Lo, “LMI relaxations for nonlinear fuzzy control systems via homogeneous polynomials,” The 2008 IEEE World Congress on Computational Intelligence, FUZZ2008, pp. 134—140, Hong Kong, June 2008.
[7] V.F. Montagner and R.C.L.F Oliveira and P.L.D., “Necessary and sufficient LMI conditions to compute quadratically stabilizing state feedback controller for Takagi-Sugeno systems,” Proc. of the 2007 American Control Conference, pp. 4059–4064, July 2007.
[8] T. M. Guerra and L. Vermeiren, “LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno’s form,” Automatica, vol. 40, pp. 823–829, 2004.
[9] B.C. Ding and H. Sun and P Yang, “Further studies on LMI-based relaxed stabilization conditions for nonlinear systems in Takagisugeno’s form,” Automatica, vol. 43, pp. 503–508, August 2006.
[10] X. Chang and G. Yang, “FA descriptor representation approach to observer-based H1 control synthesis for discrete-time fuzzy systems,” Fuzzy Set and Systems, vol. 185, no. 1, pp.38–51, 2010.
[11] B. Ding, “Homogeneous polynomially nonquadratic stabilization of discrete-time Takagi–Sugeno systems via nonparallel distributed compensation law,” IEEE Transactions on Fuzzy Systems, vol. 18, no. 5, pp. 994–1000, August 2010.
[12] J. Pan, S. Fei, A. Jaadari and T. M. Guerra, “Nonquadratic stabilization of continuous T-S fuzzy models: LMI solution for local approach,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 3, pp.
594–602, 2012.
[13] D. H. Lee, J. B. Park and Y. H. Joo, “Approaches to extended
non-quadratic stability and stabilization conditions for discrete-time
Takagi-Sugeno fuzzy systems,” Automatica, vol. 47,no. 3, pp.534–538, 2011.
[14] M. Johansson and A. Rantzer and K.-E. Arzen, “Piecewise quadratic stability of fuzzy systems,” IEEE Transactions on Fuzzy Systems, vol. 7, no. 6, pp. 713–722, December 1999.
[15] G. Feng, “Controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions,” IEEE Trans. Circuits and Syst. I: Fundamental Theory and Applications, vol. 11, no. 5, pp. 605–612, August 2003.
[16] G. Feng, C. Chen, D. Sun and Y. Zhu, “H1 controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions and bilinear matrix inequalities,” IEEE Trans. Circuits and Syst. I: Fundamental Theory and Applications, vol. 13, no. 1, pp. 94–103, 2005.
[17] K. Tanaka and H. Yoshida and H. Ohtake and H. O. Wang, “A sum of squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 4, pp. 911–922, August 2009.
[18] Xie, Lin; Shishkin, Serge; Fu, Minyue. “Piecewise Lyapunov functions for robust stability of linear time-varying systems,” Systems Control Letters, pp. 165-171, 1997.
[19] Chen, Ying-Jen, et al. “Relaxed stabilization criterion for T–S fuzzy systems by minimum-type piecewise-Lyapunov-function-based switching fuzzy controller,” IEEE Transactions on Fuzzy Systems,
pp. 1166-1173, 2012.
[20] Chen, Ying-Jen, et al. “Stability analysis and region-of-attraction estimation using piecewise polynomial Lyapunov functions: polynomial fuzzy model approach,” IEEE Transactions on Fuzzy Systems, pp. 1314-1322, 2015.
[21] Lam, H. K. “Stabilization of nonlinear systems using sampled-data output-feedback fuzzy controller based on polynomial-fuzzy-modelbased control approach,” IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), pp. 258-267, 2012.
[22] S. Prajna, A. Papachristodoulou , “New developments on sum of squares optimization and SOSTOOLS,” in Proc. the 2004 American Control Conference, pp. 5606–5611, 2004.
[23] H. Ichihara, “Observer design for polynomial systems using convex optimization,” in Proc. of the 46th IEEE CDC, New Orleans, LA, pp. 5347–5352, Dec. 2007.
[24] J. Xu and K.Y. Lum and et al, “A SOS-based approach to residual generators for discrete-time polynomial nonlinear systems,” Proc. of the 46th IEEE CDC, New Orleans, pp. 372–344, December 2007.
[25] J. Xie and L. Xie and Y. Wang, “Synthesis of discrete-time nonlinear systems: A SOS approach,” Proc. of the 2007 American Control Conference, New York, pp. 4829–4834, July 2007.
[26] Sala, Antonio, and C. Ario. ”Polynomial fuzzy models for nonlinear control: a Taylor series approach,” IEEE Transactions on Fuzzy Systems, pp. 1284-1295, 2009.
[27] K. Tanaka and H. Yoshida and et al, “Stabilization of polynomial fuzzy systems via a sum of squares approach,” Proc. of the 22nd Int’l Symposium on Intelligent Control Part of IEEE Multi-conference
on Systems and Control, pp. 160–165, Singapore, October 2007.
[28] H. Ichihara and E. Nobuyama, “A computational approach to state feedback synthesis for nonlinear systems based on matrix sum of squares relaxations,” Proc. 17th Int’l Symposium on Mathematical Theory of Network and Systems, pp. 932–937, Kyoto, Japan, 2006.
[29] C.W.J. Hol and C.W. Scherer, “Sum of squares relaxations for polynomial semidefinite programming,” Proc.of MTNS, pp. 1–10, 2004.
[30] C. Ebenbauer and J. Renz and F. Allgower, “Polynomial feedback and observer design using nonquadratic lyapunov functions,” 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, pp.7587–7592, 2005.
[31] 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, pp. 157–165, 2004.
[32] P.A. Parrilo, “Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization,” Caltech, Pasadena, CA, May 2000.
[33] 林承緯, “Observer and controller design of fuzzy systems via homogeneous euler’s method,” June 2015.