Using information on membership function shapes in asymptotically exact triangulation approaches

Many Takagi-Sugeno (TS) synthesis and analysis conditions can be expressed as positive/negative definiteness conditions of fuzzy summations. In this regard, several sufficient conditions have been proposed in the literature that are asymptotically exact (i.e. necessary and sufficient as their number tends to infinity). However, since these conditions do not take the membership function shapes into account (aside from the fact that they add up to one and are greater than zero), the exactness of these conditions is only true when the membership functions can assume any possible combination of values inside the standard simplex. By making use of the membership functions shapes, in this paper we propose a triangulation (or simplicial partition) based approach that generates asymptotically exact conditions for fuzzy summations with a possibly smaller growing rate of added simplices per iteration than that observed in recently published algorithms. The set of conditions are Linear Matrix Inequalities (LMIs), for which efficient numerical routines are available.