A Tour of Type-1 and Interval Type-2 Fuzzy Sets Theory

A succinct review of type-1 basic theory is described in this chapter and should be enough for the reader, who is not familiar with fuzzy set theory, to be able to understand these concepts (Jafelice et al. Teoria dos Conjuntos Fuzzy. Springer Briefs in Mathematics SBMAC (in Portuguese), vol 17, 2nd edn. Springer, Berlin, 2012). The next topic is the description of a fuzzy technique that expands the relationship between the function’s variations and the respective state variables, to obtain through a FRBS the trajectory of the state variables as a function of time. In the sequence, an introduction to a new fuzzy system identification with multiple inputs and single output is reviewed. These two aforesaid methodologies are used in the applications of the type-1 and interval type-2 fuzzy sets in Chap. 3. Type-2 fuzzy set concept and its components are defined as being: the primary membership, the secondary membership function, the domain of uncertainty, and the upper and lower membership functions. Examples are shown and exercises are proposed for illustration and to reinforce the subject matter. Next, a more comprehensive and didactic explanation of interval type-2 fuzzy sets, describing recent theoretical adjustments made by Mendel, are developed (Mendel et al. Inf Sci 340–341:337–345, 2016; Mendel, Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions, 2nd edn. Prentice-Hall, Upper Saddle River, 2017). Three methods of inference for the interval type-2 FRBS are presented: Wu, Takagi–Sugeno–Kang, and Mamdani.