Study on enhanced Karnik–Mendel algorithms: Initialization explanations and computation improvements

Computing the centroid of an interval type-2 fuzzy set is an important operation in a type-2 fuzzy logic system, and is usually implemented by Karnik–Mendel (KM) iterative algorithms. By connecting KM algorithms and continuous KM algorithms together, this paper gives theoretical explanations on the initialization methods of KM and Enhanced Karnik–Mendel (EKM) algorithms, proposes exact methods for centroid computation of an interval type-2 fuzzy set, and extends the Enhanced Karnik–Mendel (EKM) algorithms to three different forms of weighted EKM (WEKM) algorithms. It shows that EKM algorithms become a special case of the WEKM algorithms when the weights of the latter are constant value. It also shows that, in general, the weighted EKM algorithms have smaller absolute error and faster convergence speed than the EKM algorithms which make them very attractive for real-time applications of fuzzy logic system. Four numerical examples are used to illustrate and analyze the performance of WEKM algorithms.