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Computational Method

by Abhishek ApoorvAbhishek Apoorv
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Abhishek Apoorv
Abhishek Apoorv
Introduction to fuzzy logic Franck Dernoncourt franck.dernoncourt@gmail.com MIT, January 2013
Contents Contents i List of Figures ii 1 Introduction 1.1 Set theory refresher . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 2 Fuzzy logic 2.1 Fuzzy sets . . . . . . . . 2.2 The linguistic variables . 2.3 The fuzzy operators . . 2.4 Reasoning in fuzzy logic 2.5 The defuzzification . . . 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 8 10 11 14 15 3 Training fuzzy inference systems 18 3.1 Neuro-fuzzy systems . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Evolutionary computation . . . . . . . . . . . . . . . . . . . . . . . 19 4 Acknowledgments 20 Bibliography 21 i
List of Figures of the set {1, 5, 6, 7, 10} . . . . . . . . . . . . . . . . . . . . . . . . . . . . of sets . . . . . . . . . . 1.1 1.2 1.3 1.4 Graphical representation Union of two sets . . . . Intersection of two sets . Graphical representation . . . . . . . . 2 3 3 4 2.1 2.2 . 5 . . 6 7 2.11 2.12 2.13 2.14 2.15 2.16 ”The classical set theory is a subset of the theory of fuzzy sets” . . . Membership function characterizing the subset of ’good’ quality of service . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical representation of a conventional set and a fuzzy set . . . . Comparison between a identity function of a conventional set and a membership function of fuzzy set . . . . . . . . . . . . . . . . . . . . A membership function with properties displayed . . . . . . . . . . . Linguistic variable ’quality of service’ . . . . . . . . . . . . . . . . . . Linguistic variable ’quality of food’ . . . . . . . . . . . . . . . . . . . Linguistic variable ’tip amount’ . . . . . . . . . . . . . . . . . . . . . Example of fuzzy implication . . . . . . . . . . . . . . . . . . . . . . Example of fuzzy implication with conjunction OR translated into a MAX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of fuzzy implication using the decision matrix . . . . . . . Defuzzification with the method of the mean of maxima (MeOM) . . Defuzzification with the method of center of gravity (COG) . . . . . Overview diagram of a fuzzy system: . . . . . . . . . . . . . . . . . . Decisions of a system based on fuzzy system . . . . . . . . . . . . . . Decisions of a system based on classical logic . . . . . . . . . . . . . 3.1 3.2 Example of a feedforward neural network . . . . . . . . . . . . . . . . 18 Structure of a neuro-fuzzy system . . . . . . . . . . . . . . . . . . . . . 19 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 . 8 . 9 . 9 . 10 . 12 . . . . . . . 13 13 14 15 16 16 17
Chapter 1 Introduction Fuzzy logic is an extension of Boolean logic by Lotfi Zadeh in 1965 based on the mathematical theory of fuzzy sets, which is a generalization of the classical set theory. By introducing the notion of degree in the verification of a condition, thus enabling a condition to be in a state other than true or false, fuzzy logic provides a very valuable flexibility for reasoning, which makes it possible to take into account inaccuracies and uncertainties. One advantage of fuzzy logic in order to formalize human reasoning is that the rules are set in natural language. For example, here are some rules of conduct that a driver follows, assuming that he does not want to lose his driver’s licence: If the light is red... if my speed is high... If the light is red... if my speed is low... If the light is orange... If the light is green... if my speed is average... if my speed is low... and if close... and if far... and if far... and if close... the light is then I brake hard. the light is the light is then I maintain my speed. then I brake gently. the light is then I accelerate. Intuitively, it thus seems that the input variables like in this example are approximately appreciated by the brain, such as the degree of verification of a condition in fuzzy logic. To exemplify each definition of fuzzy logic, we develop throughout this introductory course a fuzzy inference system whose specific objective is to decide the amount of 1

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