A way to improve results for the stabilization of continuous-time fuzzy descriptor models

Non-quadratic local stabilization for continuous-time Takagi–Sugeno models

“…Unfortunately, obtaining non-conservative LMIs from (11) for global stabilization is no longer possible since the…”

“…Nonetheless, the quadratic approach presents serious limitations because its solutions are inherently pessimistic, i.e., there are stable or stabilizable models which do not have a quadratic solution (see [10] and references therein). Conservativeness comes from different sources: the type of TS model [11,12], the way the membership functions are dropped-off to obtain LMI expressions [13][14][15], the integration of membership-function information [16,17], or the choice of Lyapunov function [18,19]. This work is concerned with a relaxation in the latter sense which demands a change of perspective from global to local conditions.…”

Robust fuzzy Lyapunov stabilization for uncertain and disturbed Takagi–Sugeno descriptors

“…Indeed, R 1 and R 2 being free slack matrices, (27) is a particular case of (25) where R 1 = −X 1 rk and R 2 = −X 1 jl . Note also that the quadratic cases [34,35,37] are included in Theorem 1 by considering X 1 jk = X 1 common matrix for all i, j and R 1 = R 2 = −X 1 .…”

“…Robust quadratic stability conditions for uncertain T-S descriptor systems have been proposed in terms of BMI [33] or LMI [34][35][36]. Some relaxed quadratic conditions introducing fuzzy inferred slack variables have been proposed in [37]. And more recently, with the intention to reduce the conservatism of LMI based descriptor stability analysis, NQLF based analysis have been firstly proposed in [38].…”

Stability analysis of polynomial fuzzy models via polynomial fuzzy Lyapunov functions

“…Several directions have been explored to relax the inherent conservativeness of the quadratic approach, for instance: using a more general class of Lyapunov functions like the piecewise [Johansson99,Feng04] or the fuzzy ones [Tanaka03,Guerra04], handling in a less conservative way the membership-function information [Sala07,Sala08,Bernal09], or employing a class of models broader than the TS ones [Guerra07,Tanaka07a,Tanaka07b]. Among the latter direction, polynomial fuzzy (PF) models have established a new paradigm that overcomes many of the aforementioned problems of conservativeness since they are convex combinations of polynomial models instead of convex combinations of linear ones [Tanaka09a,Tanaka09b].…”