Uncertain Rule-Based Fuzzy Systems pp 449-527 | Cite as

# Interval Type-2 Fuzzy Systems

## Abstract

This chapter explores many aspects of the interval type-2 fuzzy system that was introduced in Chap. 1. As was done for type-1 fuzzy systems, it provides a very comprehensive and unified description of the two major kinds of interval type-2 fuzzy systems that are widely used in real-world applications—IT2 Mamdani and TSK fuzzy systems. Importantly, it also distinguishes between IT2 fuzzy systems that include type-reduction followed by defuzzification and those that bypass type-reduction and use direct defuzzification. The coverage of this chapter includes IT2 rules, three kinds of fuzzifiers (singleton, type-1 non-singleton, and IT2 non-singleton), input–output formulas for the fuzzy inference engine (also valid for GT2 fuzzy systems), the effects of the three kind of fuzzifiers on the input–output formulas (valid for IT2 fuzzy systems), IT2 first-and second-order rule partitions, combining or not combining fired-rule output sets on the way to defuzzification, type-reduction (centroid, height, and center-of-sets) + defuzzification for an IT2 Mamdani fuzzy system, type-reduction + defuzzification for four kinds of IT2 TSK fuzzy systems, novelty partitions, approximate type-reduction and defuzzification (the Wu–Mendel Uncertainty Bounds), direct defuzzification (Nie–Tan and Biglarbegian–Melek–Mendel), IT2 fuzzy basis functions which provide a mathematical description of an IT2 fuzzy system from its input to its output, remarks, and insights about an IT2 fuzzy system (including layered architecture interpretations for it, fundamental differences between type-1 and IT2 fuzzy systems, universal approximation by it, continuity of it, rule explosion and some ways to control it, and rule interpretability for it), and historical notes. Seventeen examples are used to illustrate the important concepts and there is also a comprehensive numerical example in Sects. 9.7 and 9.11.

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