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Fuzzy Logic: Membership Functions

Sachin Saxena

Assistant Professor,

CSE DEPARTMENT,

ABES, Engineering College, Ghaziabad, Uttar Pradesh

sachin.saxena@abes.ac.in

+91-8909603708

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Access this PPT at: https://sachinplacement.blogspot.com

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ABES Engineering College, Ghaziabad

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Overview

  • Outline to the left in green
  • Current topic in yellow

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Variables
  • Fuzzy Logic Operators
  • Fuzzy Control
  • Case Study

Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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References

  • L. Zadah, “Fuzzy sets as a basis of possibility” Fuzzy Sets Systems, Vol. 1, pp3-28, 1978.
  • T. J. Ross, “Fuzzy Logic with Engineering Applications”, McGraw-Hill, 1995.
  • K. M. Passino, S. Yurkovich, "Fuzzy Control" Addison Wesley, 1998.

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Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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L. Zadah

Fuzzy Logic

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Introduction

  • Fuzzy logic:
    • A way to represent variation or imprecision in logic
    • A way to make use of natural language in logic
    • Approximate reasoning
  • Humans say things like "If it is sunny and warm today, I will drive fast"
  • Linguistic variables:
    • Temp: {freezing, cool, warm, hot}
    • Cloud Cover: {overcast, partly cloudy, sunny}
    • Speed: {slow, fast}

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Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Crisp (Traditional) Variables

  • Crisp variables represent precise quantities:
    • x = 3.1415296
    • A ∈{0,1}
  • A proposition is either True or False
    • A ∧ B ⇒ C
  • King(Richard) ∧ Greedy(Richard) ⇒ Evil(Richard)
  • Richard is either greedy or he isn't:
    • Greedy(Richard) ∈{0,1}

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Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Fuzzy Sets

  • What if Richard is only somewhat greedy?
  • Fuzzy Sets can represent the degree to which a quality is possessed.
  • Fuzzy Sets (Simple Fuzzy Variables) have values in the range of [0,1]
  • Greedy(Richard) = 0.7
  • Question: How evil is Richard?

Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Fuzzy Linguistic Variables

Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Fuzzy Linguistic Variables

  • Fuzzy Linguistic Variables are used to represent qualities spanning a particular spectrum
  • Temp: {Freezing, Cool, Warm, Hot}
  • Membership Function
  • Question: What is the temperature?
  • Answer: It is warm.
  • Question: How warm is it?

Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Membership Functions

  • Temp: {Freezing, Cool, Warm, Hot}
  • Degree of Truth or "Membership"

Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Membership Functions

  • How cool is 36 F° ?

Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Membership Functions

  • How cool is 36 F° ?
  • It is 30% Cool and 70% Freezing

Fuzzy Logic

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0.7

0.3

  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Fuzzy Logic

  • How do we use fuzzy membership functions in predicate logic?
  • Fuzzy logic Connectives:
    • Fuzzy Conjunction, ∧
    • Fuzzy Disjunction, ∨
  • Operate on degrees of membership in fuzzy sets

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Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Fuzzy Disjunction

  • A∨B max(A, B)
  • A∨B = C "Quality C is the disjunction of Quality A and B"

Fuzzy Logic

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  • (A∨B = C) ⇒ (C = 0.75)
  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Fuzzy Conjunction

  • A∧B min(A, B)
  • A∧B = C "Quality C is the conjunction of Quality A and B"

Fuzzy Logic

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  • (A∧B = C) ⇒ (C = 0.375)
  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Example: Fuzzy Conjunction

Calculate A∧B given that A is .4 and B is 20

Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Example: Fuzzy Conjunction

Calculate A∧B given that A is .4 and B is 20

Fuzzy Logic

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    • Determine degrees of membership:

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Example: Fuzzy Conjunction

Calculate A∧B given that A is .4 and B is 20

Fuzzy Logic

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    • Determine degrees of membership:
      • A = 0.7

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Example: Fuzzy Conjunction

Calculate A∧B given that A is .4 and B is 20

Fuzzy Logic

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    • Determine degrees of membership:
      • A = 0.7 B = 0.9

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Example: Fuzzy Conjunction

Calculate A∧B given that A is .4 and B is 20

Fuzzy Logic

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    • Determine degrees of membership:
      • A = 0.7 B = 0.9
    • Apply Fuzzy AND
      • A∧B = min(A, B) = 0.7

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0.7

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Fuzzy Control

  • Fuzzy Control combines the use of fuzzy linguistic variables with fuzzy logic
  • Example: Speed Control
  • How fast am I going to drive today?
  • It depends on the weather.
  • Disjunction of Conjunctions

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Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Inputs: Temperature

  • Temp: {Freezing, Cool, Warm, Hot}

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Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Inputs: Temperature, Cloud Cover

  • Temp: {Freezing, Cool, Warm, Hot}

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  • Cover: {Sunny, Partly, Overcast}

Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Output: Speed

  • Speed: {Slow, Fast}

Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Rules

  • If it's Sunny and Warm, drive Fast

Sunny(Cover)∧Warm(Temp)⇒ Fast(Speed)

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  • If it's Cloudy and Cool, drive Slow

Cloudy(Cover)∧Cool(Temp)⇒ Slow(Speed)

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  • Driving Speed is the combination of output of these rules...

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Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Example Speed Calculation

  • How fast will I go if it is
    • 65 F°
    • 25 % Cloud Cover ?

Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Fuzzification:�Calculate Input Membership Levels

  • 65 F° ⇒ Cool = 0.4, Warm= 0.7

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Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Membership Functions

Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary

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Ref: https://researchhubs.com/post/engineering/fuzzy-system/fuzzy-membership-function.html

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Summary

  • Fuzzy Logic provides way to calculate with imprecision and vagueness
  • Fuzzy Logic can be used to represent some kinds of human expertise
  • Fuzzy Membership Sets
  • Fuzzy Linguistic Variables
  • Fuzzy AND and OR
  • Fuzzy Control

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Fuzzy Logic

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  • References
  • Introduction
  • Crisp Variables
  • Fuzzy Sets
  • Linguistic Variables
  • Membership Functions
  • Fuzzy Logic
    • Fuzzy OR
    • Fuzzy AND
    • Example
  • Fuzzy Control
    • Variables
    • Rules
    • Fuzzification
    • Defuzzification
  • Summary