Abstract
Interval type-2 fuzzy controllers have received attention because of their potential in handling uncertainties better than type-1 (T1) fuzzy controllers. However, directly deriving the analytical structure of interval type-2 fuzzy controllers is a great challenge due to the iterative computation of Karnik–Mendel algorithm and similar ones. Analytical structure is necessary for the analysis and design of control systems (e.g. stability analysis). In this study, a novel technique is proposed for deriving the analytical structure of interval type-2 fuzzy controllers based on the configurations of the triangle interval type-2 input fuzzy sets, T1 output fuzzy singletons, product AND operators, Karnik–Mendel center-of-set type reducer, and centroid defuzzifier. Input space is partitioned into a group of input combinations (ICs), which are produced by superposing input ICs and case ICs, to obtain an analytical structure. Input ICs are evaluated by the upper/lower membership functions and case ICs using the equation of the terminating condition of Karnik–Mendel algorithm. An explicit analytical expression is obtained on any IC because each IC has the same membership functions and switch points. Finally, two examples are used to verify the validity of our method.
Highlights
Interval type-2 (IT2) fuzzy controllers are commonly used fuzzy system
Given that the upper/lower membership functions are piecewise linear functions, the input space 1⁄2ÀL À P, L + P 3 1⁄2ÀL À P, L + P needs be to partitioned into a set of sub-spaces called as input input combinations (ICs) to obtain the analytical structures of the IT2 fuzzy controllers
This article has discussed the analytical structure of IT2 fuzzy controllers with product AND operation
Summary
Interval type-2 (IT2) fuzzy controllers are commonly used fuzzy system. These controllers are popular because their secondary membership functions are all equal to 1, thereby simplifying the inference and computation of fuzzy systems.[1,2] In the last decade, IT2 fuzzy controllers have been developed with a sound momentum, and theoretical results and application examples have increased rapidly. Deriving the analytic structure of IT2 fuzzy controllers is difficult because it needs to calculate the switching point of the type-reducer algorithms using iterative calculation, where Karnik–Mendel (KM) and other algorithms are often applied. Studying the analytical structure of the IT2 fuzzy controllers based on product AND operation is important.
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