Preface

Section 6

Section 7

Section 8

II

Control of Electrical Systems 95

Chapter 6 97

Fuzzy Logic Applications in Management 125

Chapter 7 127

Field-Programmable Gate Array for Fuzzy Controllers 157

Chapter 8 159

Determination of Optimal Transformation Ratios of Power System Transformers in Conditions of Incomplete Information Regarding

by Lezhniuk Petro Demianovych, Rubanenko Oleksandr Evgeniovich

The Fuzzy Logic Methodology for Evaluating the Causality of

Functional Safety of FPGA Fuzzy Logic Controller

by Mohammed Bsiss and Amami Benaissa

the Values of Diagnostic Parameters

and Rubanenko Olena Oleksandrivna

Factors in Organization Management by Nazarov Dmitry Mikhailovich

This book promotes new research results in the field of advanced fuzzy logic applications. Treating information using fuzzy logic has developed over the past 50 years, and this mathematical theory is an interesting tool for researchers to solve complex scientific and technical problems. Fuzzy logic has found applications in various sectors of human activity, such as: industry, business, finance, medicine, and more. The book includes new research results in scientific fields such as: fuzzy mathematics, adaptive neuro-fuzzy systems, inference methods, fuzzy control, expert systems, dynamic fuzzy neural networks, and others. The authors have published worked examples and case studies resulting from their research in the field. Readers will have access to new solutions and answers to questions related to emerging theoretical fuzzy logic applications and their implementation.

The book has eight sections: Introduction, Fuzzy Mathematics, Adaptive Neuro-Fuzzy Inference System, Inference Methods, Expert Systems, Control of Electrical Systems, Fuzzy Logic Applications in Management, and Field-Programmable Gate Array for Fuzzy Controllers.

The book includes an introductory chapter that presents basic properties of fuzzy relations and seven main chapters that illustrate research in the section domains.

The chapters were edited and published following a rigorous selection process, with only a small number of the proposed chapters being selected for publication.

The introductory chapter aims to recall algebraic relations that describe fuzzy rule bases and fuzzy blocks as algebraic applications. Also, the fuzzy block may be described graph-analytically with transfer functions and graphs.

The second chapter includes a study on the convergence of sequence spaces with respect to intuitionistic fuzzy norms and their topological and algebraic properties. The third chapter focuses mainly on building ANFIS and its application to identifying the online bearing fault. A traditional structure of ANFIS as a data-driven model is shown. A recurrent mechanism depicting the relation between the processes of filtering impulse noise and establishing ANFIS from a noisy measuring database is presented. One of the typical applications of ANFIS related to managing online bearing fault is presented. The fourth chapter presents methods of conditional inference for fuzzy control systems. The fifth chapter presents an application of fuzzy logic and fuzzy expert systems in material synthesis methods. The datadriven approach is used to construct a fuzzy model. Fuzzy C-means clustering is used to derive fuzzy if-then rules from material data that describe material composition. The sixth chapter presents an example of how to use fuzzy logic in control of electrical systems, in conditions of incomplete information regarding the values of diagnostic parameters. The seventh chapter includes a fuzzy logic methodology for evaluating the causality of factors in organization management. The chapter formulates the problem of causal relations in a broad sense and analyzes the methods for its solution with an emphasis on socioeconomic aspects. Systems approach, comparative experiment, economic and mathematical modeling, and other general

scientific methods are used. The eighth chapter includes a technical study on functional safety of an FPGA fuzzy logic controller. This chapter proposes to analyze the implementation of advanced safety architecture of fuzzy logic controllers with 1 out-of-2 controllers in FPGA using the reliability block diagram and the Markov model.

The editor thanks the authors for their excellent contributions in the field and their understanding during the process of editing. Also, the editor thanks all the editorial personnel involved in book publication. The publishing process provided a set of editorial standards, which ensured the quality of the scientific level of relevance of accepted chapters.

> Constantin Volosencu Professor, "Politehnica" University from Timisoara, Romania

> > Section 1

Introduction

Section 1 Introduction

**3**

following:

**Chapter 1**

*Constantin Volosencu*

subject matter of the calculation.

the problem addressed

tion solved.

of application

**1. General aspects**

Introductory Chapter: Basic

Properties of Fuzzy Relations

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

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

• Development of fuzzy controllers without a complex mathematical modeling of

• The possibility of implementing "human linguistic knowledge" on the applica-

• The possibility of using fuzzy logic for complex, nonlinear, and variable relations

• The possibility of performing exceptional treatments, that is, changing the cal-

• Possibility of using it when making decisions specific to artificial intelligence

• Interpolation among rules, usable in exceptional treatments to change the scope

culation strategies as a result of a change in the course of application

As advantages of fuzzy logic are useful in the calculations, we can list the

taken in consideration [2–5]. The author published also in the field [6].

## **Chapter 1**

## Introductory Chapter: Basic Properties of Fuzzy Relations

*Constantin Volosencu*

## **1. General aspects**

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:


#### *Fuzzy Logic*

A fuzzy relation is a composed relation of defuzzification, inference, based on rules, and fuzzification. In the development of fuzzy relations, we have to answer the following questions. The lack of precise directives for conceiving a fuzzy relation. And in this case the following questions will be answered:


To answer these questions, a large number of fuzzy relations have been experimented, and calculus tests have been performed in a comparative analysis. The answers to these questions are reported by the values of the desired efficiency indicators and the values that can be provided by each fuzzy relation variant. By answering the questions posed, the empirical and unsystematic character of the operator's knowledge implementation and the synthesis of the fuzzy logic-based relation can be eliminated at a later design.

Next, for a better understanding of the phenomena occurring in the fuzzy relations, a brief presentation of their main basic properties will be made.

## **2. Properties**

#### **2.1 Fuzzy relation**

The basic fuzzy relation is a function of two variables:

$$\mathbf{y} = \mathbf{f}(\mathbf{x}\_1, \mathbf{x}\_2) \tag{1}$$

The variables are defined on universes of discourse, as real sets:

$$
\pi\_1 \in X\_1, \pi\_2 \in X\_2, \mathbf{y} \in Y \tag{2}
$$

A fuzzy relation may be described informationally by a structure as in **Figure 1**. It is composed relation, from defuzzification, inference based on rules, and fuzzification. The fuzzy values are defined and described with membership functions, defined on universes of discourse, with values on interval [0, 1]:

$$m(\mathfrak{x}) \colon X \to \{0, 1\} \tag{3}$$

The fuzzy set is defined as

$$A = \{\mathfrak{x}, m(\mathfrak{x})\}\tag{4}$$

**5**

**Figure 1.**

*The structure of a fuzzy relation.*

*Introductory Chapter: Basic Properties of Fuzzy Relations*

A fuzzy variable may have also five or seven fuzzy values.

An example of inference max-min is presented in **Figure 3**.

The fuzzy relations have the following algebraic properties.

The rule bases have also the same properties.

The fuzzy relation is characterized by some graphs [6]. First is the graph of function (1), represented in **Figure 4a**.

The membership functions are represented as graphs. A fuzzy variable with three fuzzy values NB, ZE, and PB and also three membership functions is repre-

The fuzzy relation is developed based on a rule base, for a fuzzy reasoning of the form [If *x*1 is … and *x*2 is … then *y* is …]. A primary rule base of 3 × 3 rules is

Several inference methods may be use, for example, max-min and sum-prod. Also there are some defuzzification methods: center of gravity, mean of maxima,

*f*(*x*1, *x*2) = *f*(*x*2, *x*1) (5)

*f*(*x*,−*x*) = *f*(−*x*,*x*) = 0 (6)

*f*(*x*,0) = *f*(0,*x*) = *x* (7)

But they do not have the associative property and nor the property of

The second graph is the graph of *y* with *x*1 as variable and x2 as parameter:

*y* = *f*(*x*1; *x*2) (8)

*y* = *f*(*xt*; *x*2) (9)

represented as a family of characteristics in **Figure 4b**. The third graph is a family

*DOI: http://dx.doi.org/10.5772/intechopen.88172*

sented in **Figure 2**.

presented in **Table 1**.

**2.2 Algebraic properties**

Inverse of *x* is −*x*:

Identity is 0 (ZE):

distributivity.

of characteristics:

**2.3 Graphs**

Commutative property

and others.

*Introductory Chapter: Basic Properties of Fuzzy Relations DOI: http://dx.doi.org/10.5772/intechopen.88172*

The membership functions are represented as graphs. A fuzzy variable with three fuzzy values NB, ZE, and PB and also three membership functions is represented in **Figure 2**.

A fuzzy variable may have also five or seven fuzzy values.

The fuzzy relation is developed based on a rule base, for a fuzzy reasoning of the form [If *x*1 is … and *x*2 is … then *y* is …]. A primary rule base of 3 × 3 rules is presented in **Table 1**.

Several inference methods may be use, for example, max-min and sum-prod. Also there are some defuzzification methods: center of gravity, mean of maxima, and others.

An example of inference max-min is presented in **Figure 3**.

#### **2.2 Algebraic properties**

*Fuzzy Logic*

A fuzzy relation is a composed relation of defuzzification, inference, based on rules, and fuzzification. In the development of fuzzy relations, we have to answer the following questions. The lack of precise directives for conceiving a fuzzy rela-

• What real mathematical variables have to be chosen for fuzzy processing?

• How many fuzzy values and what membership functions are chosen for fuzzy

To answer these questions, a large number of fuzzy relations have been experimented, and calculus tests have been performed in a comparative analysis. The answers to these questions are reported by the values of the desired efficiency indicators and the values that can be provided by each fuzzy relation variant. By answering the questions posed, the empirical and unsystematic character of the operator's knowledge implementation and the synthesis of the fuzzy logic-based

Next, for a better understanding of the phenomena occurring in the fuzzy rela-

*y* = *f*(*x*1, *x*2) (1)

A fuzzy relation may be described informationally by a structure as in **Figure 1**. It is composed relation, from defuzzification, inference based on rules, and fuzzification. The fuzzy values are defined and described with membership functions,

*m*(*x*):*X* → [0, 1] (3)

*A* = {*x*,*m*(*x*)} (4)

*x*<sup>1</sup> ∈ *X*1, *x*<sup>2</sup> ∈ *X*2, *y* ∈ *Y* (2)

tions, a brief presentation of their main basic properties will be made.

The variables are defined on universes of discourse, as real sets:

defined on universes of discourse, with values on interval [0, 1]:

The basic fuzzy relation is a function of two variables:

tion. And in this case the following questions will be answered:

• How to choose the universes of discourse for fuzzy variables?

• What is the structure of the fuzzy relation?

• Which is the rule base of the fuzzy relation?

• Which defuzzification method is better?

relation can be eliminated at a later design.

**2. Properties**

**2.1 Fuzzy relation**

The fuzzy set is defined as

• Which method of inference should be chosen?

relationship variables?

**4**

The fuzzy relations have the following algebraic properties. Commutative property

$$f(\mathbf{x}\_1, \mathbf{x}\_2) = f(\mathbf{x}\_2, \mathbf{x}\_1) \tag{5}$$

Inverse of *x* is −*x*:

$$f(\infty, -\infty) = f(-\infty, \infty) = \mathbf{0} \tag{6}$$

Identity is 0 (ZE):

$$f(\mathbf{x}, \mathbf{O}) = f(\mathbf{O}, \mathbf{x}) = \mathbf{x} \tag{7}$$

The rule bases have also the same properties.

But they do not have the associative property and nor the property of distributivity.

#### **2.3 Graphs**

The fuzzy relation is characterized by some graphs [6]. First is the graph of function (1), represented in **Figure 4a**. The second graph is the graph of *y* with *x*1 as variable and x2 as parameter:

$$\mathbf{y} = \mathbf{f}(\mathbf{x}\_1; \mathbf{x}\_2) \tag{8}$$

represented as a family of characteristics in **Figure 4b**. The third graph is a family of characteristics:

$$\mathcal{Y} = f(\mathbf{x}\_t; \mathbf{x}\_2) \tag{9}$$

**Figure 1.** *The structure of a fuzzy relation.*

## **Figure 2.**

*Membership function.*


#### **Table 1.**

*Primary 3 × 3 rule base.*

**Figure 3.** *Example of inference max-min.*

represented in **Figure 4c**, where

$$\mathcal{X}\_t = \mathcal{X}\_1 + \mathcal{X}\_2 \tag{10}$$

is a compound variable. This graph is situated in the first and third quadrants and it has a sector property.

And the fourth graph is the variable gain:

$$K(\mathbf{x}\_i; \mathbf{x}\_2) = \frac{y}{\overline{\mathbf{x}\_t}} \tag{11}$$

represented, as a family of characteristics, in **Figure 4d**, with the value in origin:

$$K\_0 = \lim\_{\overline{\mathcal{X}\_t} \ni \overline{\mathcal{X}\_t}} \frac{\mathcal{Y}}{\mathcal{X}\_t} \tag{12}$$

**7**

**3. Conclusion**

*Graphs of a fuzzy relation.*

**Figure 4.**

for example.

The fuzzy relations may be classified according the rule base, membership functions, number of fuzzy variables, inference, and defuzzification. They have transfer characteristic graphs which may be numerical calculated. The graphs may be used for grapho-analytical analysis of fuzzy relations and their applications, because only the analytical description of the fuzzy systems is difficult because of the complexity of operations made inside: fuzzification, inference, and defuzzification. The rule bases and the fuzzy relations may have algebraic properties, the commutative property, inverse, and identity, but not the associative property, so no kind of algebraic structures may be developed. The fuzzy relations are nonlinear functions. They have applications in many domain, like fuzzy controllers with variable gain,

*Introductory Chapter: Basic Properties of Fuzzy Relations*

*DOI: http://dx.doi.org/10.5772/intechopen.88172*

The graphs are obtained for a fuzzy relation with three fuzzy values, membership function from **Figure 2**, the primary 3 × 3 rule base, max-min inference, and defuzzification with center of gravity.

*Introductory Chapter: Basic Properties of Fuzzy Relations DOI: http://dx.doi.org/10.5772/intechopen.88172*

*Fuzzy Logic*

**Figure 2.**

**Table 1.**

*Primary 3 × 3 rule base.*

*Membership function.*

**y x1**

x2 NB NB NB ZE

ZE NB ZE PB PB ZE PB PB

**6**

**Figure 3.**

*Example of inference max-min.*

and it has a sector property.

represented in **Figure 4c**, where

defuzzification with center of gravity.

And the fourth graph is the variable gain:

*<sup>K</sup>*(*xt*; *x*2) = *<sup>y</sup>*

*xt* = *x*<sup>1</sup> + *x*<sup>2</sup> (10)

**NB ZE PB**

(11)

(12)

is a compound variable. This graph is situated in the first and third quadrants

represented, as a family of characteristics, in **Figure 4d**, with the value in origin:

*<sup>K</sup>*0 = lim*xt*→0

The graphs are obtained for a fuzzy relation with three fuzzy values, membership function from **Figure 2**, the primary 3 × 3 rule base, max-min inference, and

\_ *xt*

*y* \_ *xt*

**Figure 4.** *Graphs of a fuzzy relation.*

## **3. Conclusion**

The fuzzy relations may be classified according the rule base, membership functions, number of fuzzy variables, inference, and defuzzification. They have transfer characteristic graphs which may be numerical calculated. The graphs may be used for grapho-analytical analysis of fuzzy relations and their applications, because only the analytical description of the fuzzy systems is difficult because of the complexity of operations made inside: fuzzification, inference, and defuzzification. The rule bases and the fuzzy relations may have algebraic properties, the commutative property, inverse, and identity, but not the associative property, so no kind of algebraic structures may be developed. The fuzzy relations are nonlinear functions. They have applications in many domain, like fuzzy controllers with variable gain, for example.

*Fuzzy Logic*

## **Author details**

Constantin Volosencu "Politehnica" University, Timisoara, Romania

\*Address all correspondence to: constantin.volosencu@aut.upt.ro

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**9**

*Introductory Chapter: Basic Properties of Fuzzy Relations*

*DOI: http://dx.doi.org/10.5772/intechopen.88172*

[2] Cox E. Adaptive fuzzy systems. IEEE

[3] Mendel JM. Fuzzy logic systems for engineering: A tutorial. Proceedings of

[4] Thomas DE, Armstrong-Helouvry B. Fuzzy logic control, a taxonomy of demonstrated benefits. Proceedings of

[5] Zimmerman HJ. Fuzzy Sets Theory and its Applications. Boston: Kluwer-

[6] Volosencu C. Properties of fuzzy systems. WSEAS Transactions on Systems. 2009;**8**(2):210-228

[1] Buhler H. Reglage par Logique Floue. Lausanne: Press Polytechnique et

Universitaires Romands; 1994

**References**

Spectrum. 1993

the IEEE. 1995

the IEEE. 1995

Nijhoff Pub; 1985

*Introductory Chapter: Basic Properties of Fuzzy Relations DOI: http://dx.doi.org/10.5772/intechopen.88172*

## **References**

*Fuzzy Logic*

**8**

**Author details**

Constantin Volosencu

"Politehnica" University, Timisoara, Romania

provided the original work is properly cited.

\*Address all correspondence to: constantin.volosencu@aut.upt.ro

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

[1] Buhler H. Reglage par Logique Floue. Lausanne: Press Polytechnique et Universitaires Romands; 1994

[2] Cox E. Adaptive fuzzy systems. IEEE Spectrum. 1993

[3] Mendel JM. Fuzzy logic systems for engineering: A tutorial. Proceedings of the IEEE. 1995

[4] Thomas DE, Armstrong-Helouvry B. Fuzzy logic control, a taxonomy of demonstrated benefits. Proceedings of the IEEE. 1995

[5] Zimmerman HJ. Fuzzy Sets Theory and its Applications. Boston: Kluwer-Nijhoff Pub; 1985

[6] Volosencu C. Properties of fuzzy systems. WSEAS Transactions on Systems. 2009;**8**(2):210-228

Section 2

Fuzzy Mathematics

11
