Type-1 Fuzzy Systems

This chapter explores many aspects of the type-1 fuzzy system that was introduced in Chap. 1 . It provides a very comprehensive and unified description of the two major kinds of type-1 fuzzy systems that are widely used in real-world applications—Mamdani and TSK fuzzy systems. Not only are derivations provided, but a lot of emphasis is also placed on understanding the potential benefits for using a type-1 fuzzy system. This chapter covers the following topics: The basic architecture of a type-1 fuzzy system; (IF-THEN) rules; singleton and non-singleton fuzzifiers; (derivation of) input-output formulas for the fuzzy inference engine; the effects of the two kind of fuzzifiers on the input-output formulas; combining or not combining fired-rule output sets on the way to defuzzification for Mamdani and TSK fuzzy systems; defuzzifiers (centroid, height, center-of-sets, and normalized and un-normalized TSK); a comprehensive numerical example that illustrates all of the computations for one kind of a Mamdani fuzzy system and two kinds of TSK fuzzy systems (this example is continued in later chapters); fuzzy basis functions that provide a mathematical description of a fuzzy system from its input to its output; course and fine sculpting of the state space used to explain the potential for improved performance of a type-1 fuzzy system over a non-fuzzy system; remarks and insights about a type-1 fuzzy system (including unique features of a type-1 fuzzy system, layered architecture interpretations for it, functional equivalence of it to other machine learning methods, universal approximation by it, continuity of it, rule explosion and some ways to control it, interpretability and explainability for it, and a top-down approach for obtaining it). There are three appendixes: Appendix 1 simplifies non-singleton fuzzification for a triangle fuzzy number and a trapezoidal antecedent membership function; Appendix 2 explains how to construct type-1 first-and second-order rule partitions for singleton and non-singleton fuzzification; and, Appendix 3 provides a procedure for determining the active rules in a first-order rule-partition (needed for explaining the output of a type-1 fuzzy system). Twenty-six examples are used to illustrate the important concepts. Chapter 9 builds upon the material that is in this chapter.