Abstract
General type-2 fuzzy logic systems (GT2 FLSs) have become a hot topic in current academic field. Computing the centroids of general type-2 fuzzy sets (also called type-reduction) is a central block in GT2 FLSs. Recent studies prove the continuous Nie-Tan (CNT) algorithms to be actually an accurate approach to calculate the centroids of interval type-2 fuzzy sets (IT2 FSs). This paper compares the sum operation in discrete NT algorithms and the integral operation in CNT algorithms. According to the alpha-planes representation theory of general type-2 fuzzy sets (GT2 FSs), both the discrete and continuous NT algorithms can be extended to compute the centroids of GT2 FSs. Four computer simulation experiments indicate that, when the centroid type-reduced sets and defuzzified values of GT2 FSs are solved, to properly increase the number of sampling points of primary variable can make the results of discrete NT algorithms exactly approach to the accurate benchmark CNT algorithms. Furthermore, the computation efficiency of sampling based discrete NT algorithms is much higher than the CNT algorithms.
Highlights
As the computational complexity of interval type-2 fuzzy logic systems (IT2 FLSs [1]–[7]) is comparatively low, IT2 FLSs become the most commonly used T2 FLSs for handling fields with uncertainties [8], nonlinearities and timevarying
This paper considers the continuous NT (CNT) algorithms as the benchmark, and proves that the results of discrete NT algorithms can accurately approximate the CNT algorithms by varying the number of sampling of primary variable of general type-2 fuzzy sets (GT2 FSs)
The CNT algorithms are considered as the benchmark to compute the centroid type-reduced sets for
Summary
As the computational complexity of interval type-2 fuzzy logic systems (IT2 FLSs [1]–[7]) is comparatively low, IT2 FLSs become the most commonly used T2 FLSs for handling fields with uncertainties [8], nonlinearities and timevarying. Theorem 1 ([22]): While the number of sampling is infinity, the algorithms based on random sampling can calculate the accurate centroids of IT2 FSs. Proof: Firstly, consider the discrete IT2 FS RAα defined on the universe of discourse X. The CNT i=1 algorithms are equivalent to the exhaustive TR algorithms, which compute the centroid of IT2 FS by averaging the upper and lower bounds of FOU of IT2 FS. [23], [31], [32], the CNT algorithms can be extended to compute the centroids of GT2 FSs based on the α-planes representation theory. We can make conclusions between the CNT and NT algorithms for computing the centroids of GT2 FSs as: 1) NT algorithms compute the centroids in terms of the sum operation on sampling points xi(i = 1, · · · , N ). 3) NT algorithms perform the numerical computations according to the sum operation, whereas the CNT algorithms perform the computations symbolically by means of the integral operation
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