本論文主要研究連續模糊控制系統的非二次穩定(non-quadratic
stability) 條件，關於擴展狀態決定於高階的非二次李亞普諾夫函數，其函數形式是V(x) = 1/2(x^TP^(-1)(x)x)，其中條件P^(-1)(x) > 0取決於P(x)x 是一正定的梯度向量(gradient vector)。遺憾的是，此梯度向量P(x)x 是一非凸面體(nonconvex) 的問題。因此可控制的模糊系統之穩定性檢測條件，需要使用尤拉齊次多項式定理，並使用其定理之齊次性質與波雅定理(Pólya theorem) 之代數性質，以平方和方法(sum of squares) 去檢驗非凸面體問題，使得其模擬系統之空間解更寬鬆。最後，模擬其多項式模糊系統，表現出本論文提出之方法是有效的。

Extension of the state dependent Riccati inequalities to non-quadratic Lyapunov function of the form V(x) = 1/2(x^TP^(-1)(x)x), with P^(-1)(x) > 0 requires that P(x)x is a gradient of positive definite function. Unfortunately, the test of P^(-1)(x)x is nonconvex problem. Thus this thesis studies stabilization problems of the polynomial fuzzy systems via homogeneous Lyapunov functions exploiting the Euler’s homogeneity property and algebraic property of Pólya to construct a family of SOS polynomials that solves the nonconvexity problem and releases conservatism as well. Lastly, examples of polynomial fuzzy systems are demonstrated to show the proposed method being effective and effective.

[1] T. Takagi and M. Sugeno. Fuzzy identification of systems and its
applications to modelling and control. IEEE Trans. Syst., Man,
Cybern., 15(1):116–132, January 1985.
[2] M. Sugeno and G.T. Kang. Structure identification of fuzzy model.
Fuzzy Set and Systems, 28:15–33, 1988.
[3] K. Tanaka and M. Sugeno. Stability analysis and design of fuzzy
control systems. Fuzzy Set and Systems, 45:135–156, 1992.
[4] W.M. Haddad and D.S. 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,
4:249–265, 1994.
[5] P.A. Parrilo. Structured Semidefinite Programs and Semialgebraic
Geometry Methods in Robustness and Optimization. PhD thesis,
Caltech, Pasadena, CA., May 2000.
[6] S. Prajna, A. Papachristodoulou, and P. Parrilo. Introducing SOSTOOLS:
a general purpose sum of squares programming solver. In
Proc of IEEE CDC, pages 741–746, Montreal, Ca, July 2002.
[7] S. Prajna, A. Papachristodoulou, and et al. New developments on
sum of squares optimization and SOSTOOLS. In Proc. the 2004
American Control Conference, pages 5606–5611, 2004.
[8] A. Sala and C. Arino. Polynomial fuzzy models for nonlinear control:
A Taylor series approach. IEEE Trans. Fuzzy Systems, 17(6):
1284–1295, December 2009.
[9] H. Ichihara. Observer design for polynomial systems using convex
optimization. In Proc. of the 46th IEEE CDC, pages 5347–5352,
New Orleans, LA, December 2007.
[10] J. Xu, K.Y. Lum, and et al. A SOS-based approach to residual
generators for discrete-time polynomial nonlinear systems. In Proc.
of the 46th IEEE CDC, pages 372–377, New Orleans, LA, December
2007.
[11] J. Xie, L. Xie, and Y. Wang. Synthesis of discrete-time nonlinear
systems: A SOS approach. In Proc. of the 2007 American Control
Conference, pages 4829–4834, New York, NY, July 2007.
[12] 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, pages 4071–4076, New York, NY,
July 2007.
[13] K. Tanaka, H. Yoshida, and et al. 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, pages 160–165, Singapore, October 2007.
[14] 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, pages 932–937, Kyoto, Japan,
2006.
[15] C.W.J. Hol and C.W. Scherer. Sum of squares relaxations for polynomial
semidefinite programming. In Proc.of MTNS, pages 1–10,
2004.
[16] E. Kim and H. Lee. New approaches to relaxed quadratic stability
condition of fuzzy control systems. IEEE Trans. Fuzzy Systems,
8(5):523–534, October 2000.
[17] X.D. Liu and Q.L. Zhang. New approaches to H1 controller designs
based on fuzzy observers for T-S fuzzy systems via LMI. Automatica,
39:1571–1582, June 2003.
[18] C.H. Fang, Y.S. Liu, S.W. Kau, L. Hong, and C.H. Lee. A new
LMI-based approach to relaxed quadratic stabilization of T-S fuzzy
control systems. IEEE Trans. Fuzzy Systems, 14(3):386–397, June
2006.
[19] T. M. Guerra and L. Vermeiren. LMI-based relaxed nonquadratic
stabilization conditions for nonlinear systems in the Takagi-
Sugeno’s form. Automatica, 40:823–829, 2004.
[20] B.C. Ding, H. Sun, and P Yang. Further studies on LMI-based
relaxed stabilization conditions for nonlinear systems in Takagisugeno’s
form. Automatica, 43:503–508, 2006.
[21] 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, 39(5):487–496, 2008.
[22] T. M. Guerra and L. Vermeiren. Conditions for non quadratic stabilization
of discrete fuzzy models. In 2001 IFAC Conference, 2001.
[23] V.F. Montagner, R.C.L.F Oliveira, and P.L.D. 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, pages 4059–4064, July 2007.
[24] V.F. Montagner, R.C.L.F Oliveira, and P.L.D. Peres. Convergent
LMI relaxations for quadratic stabilization and H1 control
of Takagi-sugeno fuzzy systems. IEEE Trans. Fuzzy Systems, (4):
863–873, August 2009.
[25] K. Tanaka, H. Yoshida, H. Ohtake, and H. O. Wang. A sum of
squares approach to modeling and control of nonlinear dynamical
systems with polynomial fuzzy systems. IEEE Trans. Fuzzy Systems,
17(4):911–922, August 2009.
[26] C. Ebenbauer, J. Renz, and F. Allgower. Polynomial Feedback and
Observer Design using Nonquadratic Lyapunov Functions.
[27] G.H. Hardy, J.E. Littlewood, and G. Pólya. Inequalities, second
edition. Cambridge University Press, Cambridge, UK., 1952.
[28] V. Power and B. Reznick. A new bound for Pólya’s Theorem with
applications to polynomials positive on polyhedra. J. Pure Appl.
Algebra, 164:221–229, 2001.
[29] J. de Loera and F. Santos. An effect version of Pólya’s Theorem on
positive definite forms. J. Pure Appl. Algebra, 108:231–240, 1996.
[30] 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, pages 157–165, July
2004.
[31] J.C. Lo and M.Z. Liu. Exact solutions to fuzzy PD-LMIs via SOS.
In The 2010 World Congress on Computational Intelligence; FUZZIEEE
International Conference on Fuzzy Systems, pages 1352–1357,
Barcelona, SP, July 2010.
[32] L.K. Lam and L.D. Seneviratne. Stability analysis for polynomial
fuzzy-model-based control systems under perfect-imperfect premise
matching. IET Control Theory & Appl., 5(15):1689–1697, 2011.
[33] H.K. Lam. Polynomial fuzzy-model-based control systems: stability
analysis via piecewise-linear membership function. IEEE Trans.
Fuzzy Systems, 19(3):588–593, 2011.