By: GAJANAND KUMAWAT

07EUDEC0226th April 2011 1

6th April 2011 Fuzzy Logic 2

Introduction

• Lotfi A. Zadeh introduced the theory of Fuzzy

Logic in 1965.

• 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"

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 3

Fuzzy Sets

• 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]

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 4

Fuzzy Linguistic Variables

• Fuzzy Linguistic Variables are used to represent

natural language in logic.

• Linguistic variables:

• Temp: {Freezing, Cool, Warm, Hot}

• Cloud Cover: {overcast, partly cloudy, sunny}

• Speed: {slow, fast}

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 5

Membership Functions

• Temp: {Freezing, Cool, Warm, Hot}

• Degree of Truth or "Membership"

50 70 90 1103010

Temp. (F°)

Freezing Cool Warm Hot

0

1

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 6

Membership Functions

• How cool is 36 F° ?

50 70 90 1103010

Temp. (F°)

Freezing Cool Warm Hot

0

1

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 7

Membership Functions

• How cool is 36 F° ?

• It is 30% Cool and 70% Freezing

50 70 90 1103010

Temp. (F°)

Freezing Cool Warm Hot

0

1

0..7

0..3

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 8

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

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 96th April 2011 Fuzzy Logic 9

Fuzzy Disjunction

• AB max(A, B)

• AB = C "Quality C is the disjunction of

Quality A and B"

0

1

0.375

A

0

1

0.75

B

• (AB = C) (C = 0.75)

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 10

Fuzzy Conjunction

• AB min(A, B)

• AB = C "Quality C is the conjunction of

Quality A and B"

0

1

0.375

A

0

1

0.75

B

• (AB = C) (C = 0.375)

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 11

Example: Fuzzy Conjunction

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

0

1

A

0

1

B

.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 12

Example: Fuzzy Conjunction

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

0

1

A

0

1

B

.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40

• Determine degrees of membership:

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 13

Example: Fuzzy Conjunction

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

0

1

A

0

1

B

.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40

• Determine degrees of membership:

• A = 0.7

0.7

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 14

Example: Fuzzy Conjunction

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

0

1

A

0

1

B

.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40

• Determine degrees of membership:

• A = 0.7 B = 0.9

0.7

0.9

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 15

Example: Fuzzy Conjunction

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

0

1

A

0

1

B

.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40

• Determine degrees of membership:

• A = 0.7 B = 0.9

• Apply Fuzzy AND

• AB = min(A, B) = 0.7

0.7

0.9

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 16

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

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control• Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 17

Inputs: Temperature

• Temp: {Freezing, Cool, Warm, Hot}

50 70 90 1103010

Temp. (F°)

Freezing Cool Warm Hot

0

1

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control• Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 18

Inputs: Temperature, Cloud Cover

• Temp: {Freezing, Cool, Warm, Hot}

• Cover: {Sunny, Partly Cloudy, Overcast}

50 70 90 1103010

Temp. (F°)

Freezing Cool Warm Hot

0

1

40 60 80 100200

Cloud Cover (%)

OvercastPartly CloudySunny

0

1

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control• Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 19

Output: Speed

• Speed: {Slow, Fast}

50 75 100250

Speed (mph)

Slow Fast

0

1

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control• Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 20

Rules

• If it's Sunny and Warm, drive Fast

Sunny(Cover)Warm(Temp) Fast(Speed)

• If it's Cloudy and Cool, drive Slow

Cloudy(Cover)Cool(Temp) Slow(Speed)

• Driving Speed is the combination of output of

these rules...

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control• Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 21

Example Speed Calculation

• How fast will I go if it is

• 65 F°

• 25 % Cloud Cover ?

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control• Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 22

Fuzzification:

Calculate Input Membership Levels

• 65 F° Cool = 0.4, Warm= 0.7

50 70 90 1103010

Temp. (F°)

Freezing Cool Warm Hot

0

1

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control• Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 23

Fuzzification:

Calculate Input Membership Levels

• 65 F° Cool = 0.4, Warm= 0.7

• 25% Cover Sunny = 0.8, Cloudy = 0.2

50 70 90 1103010

Temp. (F°)

Freezing Cool Warm Hot

0

1

40 60 80 100200

Cloud Cover (%)

OvercastPartly CloudySunny

0

1

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control• Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 24

...Calculating...

• If it's Sunny and Warm, drive Fast

Sunny(Cover)Warm(Temp)Fast(Speed)

0.8 0.7 = 0.7

Fast = 0.7

• If it's Cloudy and Cool, drive Slow

Cloudy(Cover)Cool(Temp)Slow(Speed)

0.2 0.4 = 0.2

Slow = 0.2

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 25

Defuzzification:

Constructing the Output• Speed is 20% Slow and 70% Fast

• Find centroids: Location where membership is

100%

50 75 100250

Speed (mph)

Slow Fast

0

1

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control• Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 26

Defuzzification:

Constructing the Output• Speed is 20% Slow and 70% Fast

• Find centroids: Location where membership is

100%

50 75 100250

Speed (mph)

Slow Fast

0

1

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control• Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 27

Defuzzification:

Constructing the Output• Speed is 20% Slow and 70% Fast

• Speed = weighted mean

= (2*25+...

50 75 100250

Speed (mph)

Slow Fast

0

1

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control• Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 28

Defuzzification:

Constructing the Output• Speed is 20% Slow and 70% Fast

• Speed = weighted mean

= (.2*25+.7*75)/(.9)

= 63.8 mph

50 75 100250

Speed (mph)

Slow Fast

0

1

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control• Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 29

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 & OR

• Fuzzy Control

• Introduction

• Fuzzy Sets

• Linguistic Variables

• Membership Functions

• Fuzzy Logic• Fuzzy OR

• Fuzzy AND

• Example

• Fuzzy Control • Variables

• Rules

• Fuzzification

• Defuzzification

• Summary

6th April 2011 Fuzzy Logic 30

THANK You...