A Triangulation Approach to Asymptotically Exact Conditions for Fuzzy Summations

Many Takagi-Sugeno (T-S) fuzzy control-synthesis problems in the literature are expressed as the problem of finding decision variables in a double convex sum (fuzzy summation) of positive definite matrices. Matricespsila coefficients in the summation take values in the standard simplex. This paper presents a triangulation approach to the problem of generating simplicial partitions of the standard simplex in order to set up a family of sufficient conditions and, in parallel, another family of necessary ones for fuzzy summations. The conditions proposed in this paper are asymptotically exact as the size of the involved simplices decreases; its conservativeness vanishes for a sufficiently fine partition (sufficiently dense mesh of vertex points). The set of conditions is in the form of linear matrix inequalities (LMIs), for which efficient software is available.