Monotone Centroid Flow Algorithm for Type Reduction of General Type-2 Fuzzy Sets

Recently, type-2 fuzzy logic systems (T2 FLSs) have received increased research attention due to their potential to model and cope with the dynamic uncertainties ubiquitous in many engineering applications. However, because of the complex nature and the computational intensity of the inference process, only the constrained version of T2 FLSs, i.e., the interval T2 FLSs, was typically used. Fortunately, the very recently introduced concepts of alpha-planes and zSlices allow for efficient representation, as well as a computationally fast inference process, with general T2 (GT2) FLSs. This paper addresses the type-reduction phase in GT2 FLSs, using GT2 fuzzy sets (FSs) represented in the alpha-plane framework. The monotone property of centroids of a set of alpha-planes is derived and leveraged toward developing a simple to implement but fast algorithm for type reduction of GT2 FSs-i.e., the monotone centroid flow (MCF) algorithm. When compared with the centroid flow (CF) algorithm, which was previously developed by Zhai and Mendel, the MCF algorithm features the following advantages. 1) The MCF algorithm computes numerically identical centroid as the Karnik-Mendel (KM) iterative algorithms, unlike the approximated centroid which is obtained with the CF algorithm; 2) the MCF algorithm is faster than the CF algorithm, as well as the independent application of the KM algorithms; 3) the MCF algorithm is easy to implement, unlike the CF algorithm, which requires computation of the derivatives of the centroid; and 4) the MCF algorithm completely eliminates the need to apply the KM iterative procedure to any alpha-planes of the GT2 FS. The performance of the algorithm is presented on benchmark problems and compared with other type-reduction techniques that are available in the literature.