Introductory Chapter: Basic Properties of Fuzzy Relations

Abstract

Treating information using fuzzy logic has developed over the past 50 years, this mathematical theory being an interesting tool for researchers to solve complex scientific and technical problems. In these years, research has always yielded new results in the field of advanced fuzzy logic applications. Fuzzy logic has found applications in various sectors of human activity, such as, industry, business, finance, medicine, and in many scientific fields such as, machine learning, big data technologies, fuzzy control, expert systems, dynamic fuzzy neural networks, and others. Fuzzy logic provides a different way of dealing with mathematical calculus problems. In the case of fuzzy logic, conventional algorithms are replaced by a series of linguistic rules of the If (then) condition (conclusion). Thus, a heuristic algorithm is obtained, and human experience can be taken into account in the subject matter of the calculation. This introductory chapter aims to recall some basic notions, main properties of fuzzy relations. Fuzzy rule bases and fuzzy blocks may be seen as relations between fuzzy sets and, respectively, between real sets, with algebraic properties as commutative property, inverse and identity. The fuzzy relations are developed with different rule bases, fuzzy values, membership functions, inference, and defuzzification methods, and they may be characterized with transfer characteristic graphs. The book [1] can be considered a reference in the field. Other references may be taken in consideration [2-5]. The author published also in the field [6]. As advantages of fuzzy logic are useful in the calculations, we can list the following:

Stability Analysis of Systems with Fuzzy PI Controllers Applied to Electric Drives

“…The block diagram of the speed controller based on fuzzy logic had a structure as shown in Figure 2 [17][18][19].…”

“…The block diagram of the speed controller based on fuzzy logic had a structure as shown in Figure 2 [17][18][19]. The speed controller is a PI fuzzy controller with integration (summation) at the output.…”

“…The fuzzy block BF from Figure 2 may be developed using Mamdani's structure, with different membership functions, fuzzyfication methods, rule bases, inference methods and defuzzification methods [17][18][19]. In this paper, an example of fuzzy block with membership functions from Figure 3a, a rule base with nine rules, min-max inference and defuzzification with center of gravity is taken in consideration.…”

“…It has the highest non-linear character. Its transfer characteristics are presented in Figure 3b-d, where x t1 = e + de [17][18][19].…”

“…The values of KM may be determined from Figure 3d [17][18][19]. The new structure obtained from Figure 5 after the transformation w tions ( 16)-( 18) is presented in Figure 6.…”