New Methodology to Approximate Type-Reduction Based on a Continuous Root-Finding Karnik Mendel Algorithm

Interval Type-2 fuzzy systems allow the possibility of considering uncertainty in models based on fuzzy systems, and enable an increase of robustness in solutions to applications, but also increase the complexity of the fuzzy system design. Several attempts have been previously proposed to reduce the computational cost of the type-reduction stage, as this process requires a lot of computing time because it is basically a numerical approximation based on sampling, and the computational cost is proportional to the number of samples, but also the error is inversely proportional to the number of samples. Several works have focused on reducing the computational cost of type-reduction by developing strategies to reduce the number of operations. The first type-reduction method was proposed by Karnik and Mendel (KM), and then was followed by its enhanced version called EKM. Then continuous versions were called CKM and CEKM, and there were variations of this and also other types of variations that eliminate the type-reduction process reducing the computational cost to a Type-1 defuzzification, such as the Nie-Tan versions and similar enhancements. In this work we analyzed and proposed a variant of CEKM by viewing this process as solving a root-finding problem, in this way taking advantage of existing numerical methods to solve the type-reduction problem, the main objective being eliminating the type-reduction process and also providing a continuous solution of the defuzzification.