Maclaurin Series Expansion Complexity-Reduced Center of Sets Type-Reduction plus Defuzzification for Interval Type-2 Fuzzy Systems

This paper provides a mathematical analysis that shows how the crisp output of an IT2 FLS that is obtained by using the Begian-Melek-Mendel (BMM) formula compares to the one obtained by using center-of-sets type-reduction followed by defuzzification (COS TR + D). This is made possible by reformulating the structural solutions of the two optimization problems that are associated with COS TR, and then expanding each of them using a Maclaurin series expansion. As a result of doing this, we show that BMM is the zero-order approximation to COS TR + D. Additionally, by retaining the zero-order and first-order terms from the Maclaurin series expansions, we provide a new Enhanced BMM, one that is non-iterative, has a closed form and is much faster than using the EKM algorithms for COS TR. Although the Enhanced BMM formula is slower than BMM, we demonstrate, by means of extensive simulations, that it is from 5% to 50% more accurate than is BMM for achieving the same numerical solution that is obtained from COS TR + D; and, it is at least 94% faster than when EKM is used for COS TR + D, which makes the Extended BMM a very strong candidate for use in real time applications of IT2 FLSs.