Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya's theorem

Stability and performance requirements in fuzzy control of Takagi–Sugeno systems are usually stated as fuzzy summations, i.e., sums of terms, related to Lyapunov functions, which are weighted by membership functions or products of them. This paper presents an application to fuzzy control of Polya's theorems on positive forms on the standard simplex. The result provides a set of progressively less conservative sufficient conditions for proving positivity of fuzzy summations; such conditions are less and less conservative as a complexity parameter, n, increases. Particular cases of such conditions are those in [C.-H. Fang, Y.-S. Liu, S.-W. Kau, L. Hong, C.-H. Lee, A new LMI-based approach to relaxed quadratic stabilization of T–S fuzzy control systems, IEEE Trans. Fuzzy Systems 14 (2006) 286–397; X. Liu, Q. Zhang, New approaches to H∞ controller designs based on fuzzy observers for T–S fuzzy systems via LMI, Automatica 39 (9) (2003) 1571–1582], with n=2 and 3, respectively. The proposed conditions are asymptotically exact, i.e., necessary and sufficient when n tends to infinity or, equivalently, when a tolerance parameter tends to zero.