Construction of non-convex fuzzy sets and its application

Abstract Although non-convex fuzzy set (FS) has the high potential of great performance in data modeling and controlling, it is seldom used and discussed because the lack of linguistic explanation and normative construction way. To address this problem, we propose a method named “parametric qualitative fuzzy set (PQ FS) plus choice strategy” for the construction and linguistic explanation of non-convex FS, in which PQ FS is a collection of convex FSs with special structure, and choice strategy is an approach to choose convex FSs from PQ FS. Based on this method, a non-convex FS is obtained as the trajectory of a collection of convex FS by choosing specific convex FS under specific situation. Thus, the linguistic explanation of non-convex FS is obtained: using non-convex FSs to represent linguistic variables does not violate the routine of using convex FSs, because it shows that the linguistic variable is just represented by different convex FS at different situation. Theorems are shown to demonstrate that the “PQ FS plus choice strategy” can effectively construct a non-convex FS. Furthermore, “Why a fuzzy logic system (FLS) adopting non-convex FSs may have a higher approximation capability” is discussed by introducing a parametric qualitative FLS (PQ FLS) that is compared with a typical Mamdani FLS as a function approximator. This indicates that non-convex FSs can approximate more extrema in a given universe with smaller partition numbers or fewer rules than convex FSs. Finally, the experimental results verify that a PQ FLS designed with the proposed non-convex FS construction method can outperform traditional convex fuzzy logic controllers (FLCs). Meanwhile, using parallel computing in the model training phase of PQ FLSs can reduce the calculation time compared to single-thread mode.