Fuzzy Logic: Fundamentals and Applications

By Fouad Sabry

What Is Fuzzy Logic

One type of many-valued logic is known as fuzzy logic. In this type of logic, the truth value of variables can be any real number that falls between 0 and 1. It is utilized for the purpose of managing the idea of partial truth, in which the truth value may fluctuate between being entirely true and being entirely false. In contrast, the truth values of variables in Boolean logic can only ever be the integer values 0 or 1, as these are the only two possible outcomes.

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Fuzzy Logic

Chapter 2: Many-valued Logic

Chapter 3: Fuzzy Control System

Chapter 4: Fuzzy Set

Chapter 5: Lotfi A. Zadeh

Chapter 6: Fuzzy Mathematics

Chapter 7: Fuzzy Rule

Chapter 8: Type-2 Fuzzy Sets and Systems

Chapter 9: Adaptive Neuro Fuzzy Inference System

Chapter 10: Soft Computing

(II) Answering the public top questions about fuzzy logic.

(III) Real world examples for the usage of fuzzy logic in many fields.

(IV) 17 appendices to explain, briefly, 266 emerging technologies in each industry to have 360-degree full understanding of fuzzy logic' technologies.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of fuzzy logic.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jun 25, 2023

Book preview

Chapter 1: Fuzzy logic

One kind of many-valued logic is known as fuzzy logic. In this type of logic, the truth value of variables may be any real integer that falls between 0 and 1. It is used to manage the idea of partial truth, in which the truth value may vary between being totally true and being completely false. This concept is used to handle. In contrast, the truth values of variables in Boolean logic can only ever be the integer values 0 or 1, and not any other value.

In 1965, Iranian-Azerbaijani mathematician Lotfi Zadeh proposed the fuzzy set theory, which is often credited as being the birthplace of the term fuzzy logic.

There have been several successful applications of fuzzy logic, ranging from control theory to artificial intelligence.

In traditional reasoning, one can only reach conclusions that are either correct or incorrect. However, there are other propositions that might have a variety of responses, such as the responses you could get from a group of individuals when you ask them to name a color. In situations like these, the truth is revealed as the consequence of reasoning based on inaccurate or incomplete information, in which the responses that were tested are mapped out on a spectrum.

An example of a fundamental application would be one that describes the numerous sub-ranges of a continuous variable. For instance, a temperature measurement for anti-lock brakes might have many distinct membership functions, each establishing a certain temperature range that is necessary for effective operation of the brakes. Every function converts the given temperature measurement to a truth value that falls somewhere in the range of 0 to 1. Once these truth values have been determined, they may be utilized to figure out how the brakes should be adjusted. The fuzzy set theory offers a method for accurately describing ambiguity.

In applications of fuzzy logic, non-numerical values are often used as a means of facilitating the presentation of rules and facts.

The rule-based Mamdani system is the one that is most well recognized. It operates according to the regulations listed below:

All input values will be fuzzy-checked using fuzzy membership functions.

To calculate the fuzzy output functions, carry out all of the rules in the rulebase that are appropriate.

In order to get crisp output values, the fuzzy output functions must first be defuzzed.

The act of allocating the numerical input of a system to fuzzy sets that include a certain degree of membership is referred to as fuzzification. This level of membership may take on any value within the range of 0 to 1, inclusive. If it is zero, then the value does not fit into the fuzzy set that has been provided, but if it is one, then the value fits in perfectly with the fuzzy set that has been provided. The degree of uncertainty that a given value belongs in the set may be represented by any number between 0 and 1, inclusive. Because these fuzzy sets are often characterized by words, assigning the system input to fuzzy sets enables us to reason with it in a way that is linguistically natural.

For instance, the meanings of the terms cold, warm, and hot are represented by functions mapping a temperature scale in the figure that can be seen further down on this page. One truth value corresponds to each of the three functions, therefore each point on that scale has a total of three truth values. The three arrows, which indicate the truth values, are used to measure a certain temperature, which is shown by the vertical line in the picture. The fact that the red arrow is pointing to zero indicates that this temperature should be read as not hot. Another way to say this is that this temperature does not belong to the fuzzy set hot. The orange arrow, which points to 0.2, may say that the temperature is somewhat warm, while the blue arrow, which points to 0.8, would say that the temperature is quite chilly. As a result, this temperature has a membership of 0.2 in the fuzzy set referred to as warm, and a membership of 0.8 in the fuzzy set referred to as cold. Fuzzification is the process that determines the degree of membership that is given to each fuzzy collection.

Each value in a fuzzy set will have a slope when the value is growing, a peak where the value is equal to 1 (which might have a length of 0 or longer), and a slope while the value is falling. This is because fuzzy sets are sometimes specified as being in the form of a triangle or a trapezoid. One example of this is the typical logistic function, which can be expressed as

{\displaystyle S(x)={\frac {1}{1+e^{-x}}}} , characterized by the symmetry properties listed below

{\displaystyle S(x)+S(-x)=1.}

Because of this, it stands to reason that

{\displaystyle (S(x)+S(-x))\cdot (S(y)+S(-y))\cdot (S(z)+S(-z))=1}

The relationship between membership values and fuzzy logic may be compared to that of Boolean logic. In order to do this, suitable alternatives for the fundamental operators AND, OR, and NOT need to be accessible. There are a few different approaches to take here. The Zadeh operators are an example of a frequent kind of replacement:

The results that the fuzzy expressions provide for TRUE/1 and FALSE/0 are the same as the results that the Boolean expressions produce.

There are also additional operators, which are of a more linguistic character and are referred to as hedges, which may be employed. In most cases, these are adjectives like extremely or somewhat that are used to change the meaning of a set by the application of a mathematical formula. A criterion has been developed to determine whether or not a particular choice table defines a fuzzy logic function. Additionally, a straightforward algorithm for the synthesis of fuzzy logic functions has been proposed, and it is based on the previously introduced concepts of minimum and maximum constituents. A fuzzy logic function is the representation of a disjunction between components of minimum, where a constituent of minimum is the conjunction of variables of the