fuzzy logic final

7/31/2019 Fuzzy Logic Final

http://slidepdf.com/reader/full/fuzzy-logic-final 1/54



• Overview of Fuzzy Logic1

• Fuzzy Sets and Operations2

• Fuzzy Variables3

• Fuzzy Example4

• Applications and Conclusion5

Today’s Contents



Overview of

Fuzzy Logic

- Amit Raj Satyal (Roll No: 01)



Origin

• Traces back to Ancient Greece

• Lotfi Asker Zadeh ( 1965 )

– First to publish ideas of fuzzy logic.

– Professor from UC, Berkeley to model the uncertainty of

Natural Language.

• Professor Toshire Terano ( 1972 )

–Organized the world's first working group on fuzzysystems.

• F.L. Smidth & Co. ( 1980 )

– First to market fuzzy expert systems.



Introduction

• What is Fuzzy

logic? – Fuzzy logic is a superset of conventional(Boolean) logic

that has been extended to handle the concept of partialtruth- truth values between "completely true" and

"completely false". – A way to represent variation or imprecision in logic

– A way to make use of natural language in logic

– Notions like rather warm or pretty cold can be formulatedmathematically and processed by computers a means tomodel the uncertainty of NL

• Fuzzy logic is an attempt to combine the twotechniques.



Characteristics

• In fuzzy logic, exact r

easoning is viewed as a limiting case of approximate reasoning.

• In fuzzy logic everything is a matter of degree.

• Any logical system can be fuzzified

• In fuzzy logic, knowledge is interpreted as a collection of

elastic or, equivalently , fuzzy constraint on a collection of

variables

• Inference is viewed as a process of propagation of elastic

constraints.



Voice from Expert - Rhinehart’s

He says“You take a bite out of the apple. Is it stillan apple? You take another bite out of the apple. Now, is it still an apple? Andanother bite. And then, another bite. At

some point, people will no longerperceive it as an apple. Fuzzy logic canrepresent this process, It becomes less of an apple as you move along.”

Fuzzy logic can assign percentages of

belongingness to the process. It’s not adigital sort of situation where it’s either aone (1) or a zero (0). It’s either an apple,or it’s not.'

Heads the chemical engineering school at Oklahoma State University.



Fuzzy Sets and

Operation

- Bigyan Sapkota (Roll No. 03)



Fuzzy setsConventional Sets

•A set is any collection of objects which can be treated

as a whole.

•Example: The set of non-integer

Fuzzy sets:•Following Zadeh many sets have more than an either-

or criterion for membership. Take example the set of

young people.

• A one year old baby clearly be a member of set, and a

100 years old person will not be a member of this set,

•But what about people at the age of 20,30, and 40

years?



Contd..

•Zadeh proposed a grade of membership, such that thetransition from membership to non-membership isgradual rather than abrupt.

•An item’s grade of membership is normally a realnumber between 0 and 1, denoted by µ.

•The grade of membership for all its members thusdescribes a fuzzy sets.

•Example: “Ram ate X eggs for breakfast” where XεU=

(1,2,3…8) U= [ 1 2 3 4 5 6 7 8]

µ=[1 1 1 1 .8 .6 .4 .2]



Fuzzy Logic 11

Membership Functions•

The function that ties a number to each element { of theuniverse is called the membership function

• Temp: {Freezing, Cool, Warm, Hot}

• Degree of Truth or "Membership"

50 70 90 1103010

Temp. (F°)

Freezing Cool Warm Hot

0

1



Fuzzy Logic 12

Example Membership Functions

• How cool is 36 F° ?

50 70 90 1103010

Temp. (F°)


0

1



Fuzzy Logic 13

Membership Functions

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

50 70 90 1103010

Temp. (F°)


0

1

0.7

0.3



Fuzzy Logic 14

Fuzzy logic connectives

–Fuzzy Conjunction,

– Fuzzy Disjunction,

– Fuzzy negation, p’

–if then

– if and only if

•Operate on degrees of membership in fuzzysets



Fuzzy Logic 15

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)



Fuzzy Logic 16

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)



Fuzzy Logic 17

Example: Fuzzy Conjunction

Calculate A

B 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



Fuzzy Logic 18


Calculate A


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:



Fuzzy Logic 19


Calculate A


0

1

A

0

1

B

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


A = 0.7

0.7



Fuzzy Logic 20


Calculate A


0

1

A

0

1

B

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


A = 0.7 B = 0.9

0.7

0.9



Fuzzy Logic 21


Calculate A


0

1

A

0

1

B

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


A = 0.7 B = 0.9Apply Fuzzy AND

AB = min(A, B) = 0.7

0.7

0.9



Fuzzy Logic 22

It means student are happy most of the time

It means teacher are happy slightly less than half of their time.

Proposition Truth value

Student of NCIT are happy 0.8

Teachers of NCIT are happy 0.4

For the instance



Fuzzy Logic 23

Truth value of Negation of propositon in Fuzzy logic:1 minus the truth value of propostion

Example : what is the truth value of statement

Negation of Proposition Truth value

Student of NCIT are not happy 1- 0.8 = 0.2

Teachers of NCIT are happy 1-0.4 = 0.6



Fuzzy

Variables

- Krishna Prasad Paudel (Roll No. 11)



Linguistic variable

• Just like an algebraic variable takes numbers as

values, a linguistic variable takes words or

sentences as values (Zadeh in Zimmermann,

1991).

• set of values that it can take is called its term set.

• Each value in the term set is a Fuzzy variable

defined over a base variable



Contd… • Linguistic variable->Fuzzy variable->base

variable• Example: Tank example If the level is low

• Here, Low is a fuzzy variable, and defined with

degree of membership [0,1]• a value of the linguistic variable is level

{high,low)

C d



Contd..• Fuzzy linguistic terms often consist of two parts:

1.Fuzzy predicate (primary term)• If the set defining the predicates of individual is a fuzzy

set, the predicate is called a fuzzy Predicate

• Example

– “z is expensive.” – “w is young.”

• Terms “expensive” and “young” are fuzzy terms. So, thesets “expensive(z)” and “young(w)” are fuzzy sets

• When a fuzzy predicate “ x is P” is given, we can interpret

it “P( x ) is a fuzzy set”. The membership degree of x in theset P is defined by the membership function µP( x )

Example: expensive, old, rare, dangerous, good, etc.



2. Modifiers

• It is an operation that modifies the meaning

of a term

• Example: The sentence ‘‘very close to 0’’, the

word very modifies Close to 0 which is a fuzzy

set.

• very, likely, almost impossible, extremely

unlikely, etc

• A new term can be obtained when we add a modifier“very” to a primary term

– µvery young(u) = (µyoung(u))2



Logic Process

n



Fuzzy Example

- Simon Shrestha (Roll No. 26)



Fuzzy Logic 31

Fuzzy Control

• Fuzzy Control combines the use of fuzzylinguistic variables with fuzzy logic

• Example: Speed Control

• How fast am I going to drive today?

• It depends on the weather.

• Disjunction of Conjunctions



Fuzzy Logic 32

Inputs: Temperature


50 70 90 1103010

Temp. (F°)


0

1



Fuzzy Logic 33

Inputs: Temperature, Cloud Cover


• Cover: {Sunny, Partly, Overcast}

50 70 90 1103010

Temp. (F°)


0

1

40 60 80 100200

Cloud Cover (%)

OvercastPartly CloudySunny

0

1



Fuzzy Logic 34

Output: Speed

• Speed: {Slow, Fast}

50 75 100250

Speed (mph)

Slow Fast

0

1



Fuzzy Logic 35

Rules

• If it's Sunny and Warm, drive Fast

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

• If it's Cloudy and Cool, drive SlowCloudy(Cover)Cool(Temp) Slow(Speed)

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



Speed Calculation

• How fast will I go if it is – 65 F°

– 25 % Cloud Cover ?

• Steps Involved

– Fuzzification

–Calculation

– Defuzzification



Fuzzification

• 65 F° Cool = 0.4, Warm= 0.7

50 70 90 1103010

Temp. (F°)


0

1



Fuzzification

• 25% CoverSunny = 0.8, Cloudy = 0.2

40 60 80 100200

Cloud Cover (%)

OvercastPartly CloudySunny

0

1



Calculation

• If it's Sunny and Warm, drive FastSunny(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



50 75 100250Speed (mph)

Slow Fast

0

1

Defuzzification

•Speed is 20% Slow and 70% Fast

• Find centroids where membership is 100%

50 75 100250

Speed (mph)

Slow Fast

0

1



50 75 100250Speed (mph)

Slow Fast

0

1

Defuzzification

• Speed = weighted mean = (2*25+...

• Speed = (2*25+7*75)/(9) = 63.8 mph

50 75 100250

Speed (mph)

Slow Fast

0

1



Fuzzy

Applications

- Subash Paudyal



Applications

• Industrial control• Quality control

• Elevator control and scheduling

• Train control• Traffic control

• Loading crane control

•Reactor control

• Automobile transmissions

• Automobile climate control



Applications

• Automobile body painting control• Automobile engine control

• Paper manufacturing

• Steel manufacturing• Power distribution control

• Software engineering

•Expert systems

• Operation research

• Decision analysis



Applications

• Financial engineering• Assessment of credit-worthiness

• Fraud detection22.Mine detection

• Pattern classification

• Oil exploration• Geology

• Civil Engineering

• Chemistry

• Mathematics• Medicine

• Biomedical instrumentation

• Health-care products



Rule Base

• Air Temperature – Set cold {50, 0, 0}

– Set cool {65, 55, 45}

– Set just right {70, 65, 60}

– Set warm {85, 75, 65}

– Set hot {, 90, 80}• Fan Speed

– Set stop {30, 0, 0}

– Set slow {50, 30, 10}

– Set medium {60, 50, 40}

– Set fast {90, 70, 50}

– Set blast {, 100, 80}



Rules

• Air Conditioning Controller Example:

– IF Cold then Stop

– If Cool then Slow

– If OK then Medium

– If Warm then Fast

– IF Hot then Blast



Fuzzy Air Conditioner

S t o p

S l o w

M e d i u m

F a s t

B l a s t

0

10

20

30

40

50

60

70

80

90

100

0

1

45 50 55 60 65 70 75 80

0

C o l d

C o o l

85 90

J u s t

R i g h t

W a r m

H o t

if Coldthen Stop

IF CoolthenSlow

If Just Rightthen

Medium

If WarmthenFast

If HotthenBlast



Mapping Inputs to Outputs1

S t o p

S l o w

M e d i u m

F a s t

B l a s t

0

10

20

30

40

50

60

70

80

90

100

0

1

45 50 55 60 65 70 75 80

0

C o l d

C o o l

85 90

J u s t

R i g h

t

W a r m

H o t

t



Ongoing Research

• Hybrid IntelligentSystems.



Drawbacks

• Requires tuning of membership functions

• Fuzzy Logic control may not scale well to

large or complex problems

• Deals with imprecision, and vagueness,

but not uncertainty



Summary

• Provides way to calculate with imprecision andvagueness

• Used to represent some kinds of human

expertise• Provides an alternative way to represent

linguistic and subjective attributes of the realworld in computing.

• Able to be applied to control systems and otherapplications in order to improve the efficiencyand simplicity of the design process.



• Fuzzy Logic - Making Inferences with Imprecise

Concepts

http://aaai.org/AITopics/FuzzyLogic

• Fuzzy Logic – From wikipedia

http://en.wikipedia.org/wiki/Fuzzy_logic • This slide deck and related resources:

<hyperlink here>

REFERENCES


http://en.wikipedia.org/wiki/Fuzzy_logic

http://en.wikipedia.org/wiki/Fuzzy_logic




Q UESTIONS?

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