General Type-2 Fuzzy Systems

This chapter explores many aspects of the general type-2 (GT2) fuzzy system that was introduced in Chap. 1 . It provides a very comprehensive and unified description of two major kinds of aggregated horizontal-slice GT2 fuzzy systems (due to Christian Wagner and Hani Hagras, and called herein a WH GT2 fuzzy system) that may be used in real-world applications—WH GT2 Mamdani and WH GT2 TSK fuzzy systems. Importantly, it also distinguishes between WH GT2 fuzzy systems that include type-reduction followed by defuzzification and those that bypass type-reduction and use direct defuzzification. It also focuses on what exactly “design of a WH GT2 fuzzy system” means, provides a tabular way for making the choices that are needed in order to fully specify a WH GT2 fuzzy system, introduces two approaches to design—the partially dependent approach and the totally independent approach, and describes one design method and one extensive case study. Not only are derivations provided, but a lot of emphasis is also placed on understanding the potential benefits of using a WH GT2 fuzzy system over an IT2 fuzzy system. The coverage of this chapter focuses on singleton fuzzification, the use of the horizontal-slice representation of a GT2 FS, and a horizontal-slice architecture for a GT2 fuzzy system—the WH GT2 fuzzy system, and includes: GT2 (IF-THEN) rules; what a GT2 non-singleton fuzzifier is; derivation of horizontal-slice formulas for firing sets and fired-rules output sets; combining fired-rule output sets on the way to defuzzification for WH GT2 Mamdani fuzzy systems; horizontal-slice type-reduction for horizontal-slice GT2 Mamdani and TSK fuzzy systems; defuzzification (this is where horizontal slices are aggregated); a summary of the computational steps for two WH GT2 Mamdani and two WH GT2 TSK GT2 fuzzy systems; a comprehensive numerical example that illustrates all of the computations for a WH GT2 Mamdani fuzzy system that uses COS type-reduction + average of end points defuzzification; proposed WH NT and WH BMM direct defuzzification methods; the continuation of the comprehensive example to illustrate all of the computations for the proposed WH NT and WH BMM IT2 fuzzy systems; GT2 fuzzy basis functions which provide a mathematical description of a GT2 fuzzy system from its input to its output; course and fine sculpting of the state space as well as novelty partitions used to explain the potential for improved performance of a WH GT2 fuzzy system over an IT2 fuzzy system; a penultimate summary table for explaining the potential for improved performance as one goes from crisp to T1 to IT2 to WH GT2 fuzzy systems; remarks and insights about a WH GT2 fuzzy system; a detailed overview of what exactly “design of a WH GT2 fuzzy system” means; one design method; requirements that need to be met in the study of real-world applications of WH GT2 fuzzy systems; and a case study of WH GT2 fuzzy logic control. 13 examples are used to illustrate the chapter’s important concepts.