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Advance Topics in Mathematical Methods ME71

Fuzzy sets,System andModelling

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Fuzzy sets were introduced by Zadeh in 1965 to represent/manipulate

data and information possessing nonstatistical uncertainties.L.A.Zadeh, Fuzzy Sets, Information and Control, 8(1965) 338-353.

There are two main characteristics of fuzzy systems that give them

better performance for specific applications.

Fuzzy systems are suitable for uncertain or approximate reasoning,

especially for the system with a mathematical model that is difficult

to derive.

Fuzzy logic allows decision making with estimated values under

incomplete or uncertain information.

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Classical sets or crisp set

A = {12, 24, 36, 48, }

Notation: A = {x | x = 12n, n is a natural number}

A = {cities adjoining Hyderabad}

A = {Mahbubnagar, Medak, Nalgonda, Rangareddy}

x

xxA if0

if1)(

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Classical sets vs Fuzzy set

A = {cities near from Hyderabad}

A = {Hyderabad, Adilabad, Khammam, Karimnagar, Mahbubnagar,

Medak, Nalgonda, Nizamabad, Rangareddy, Warangal, Bidar,

Gulbarga,.., Mumbai, Pune, Bangalore,., Delhi,Islamabad, Kabul, Kathmandu, Singapore. Rome, London,

Paris.}

Is the above information precise? : No it is Fuzzynot clear,

distinct, or precise; blurred

Definition of fuzzy logic : A form of knowledge representation suitable for

notions that cannot be defined precisely, but which depend upon their

contexts.

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Fuzzy set : example 1

A = {cities near from Hyderabad}

What is near:A distance less the D kilometer

D = 100; if we are interested in adjoining cities

D = 200/300/400; if we are interested in cities in APD = 300/400/500..; if we are interested in cities in AP or Karnataka

D= 3000/4000/if we are interested in cities in India AP or Karnataka

Information is not precise and is context based

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Fuzzy set :: example 1

A = {cities near from Hyderabad}What is not near (far): A distance more the D kilometer

Say, D = 200; if we are interested in cities in AP

Lets make some rules:

1. If 0 D 100; very near ( depending upon road condition and traffic one can reach within 2 hr)

2. If 50< D 150; near ( depending upon road condition and traffic one can reach within 3hrs)

3. If 100< D 200; far ( depending upon road condition and traffic u can reach within 6hrs)

4. If D> 200; very far

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1. If 0 D 100; very near ( depending upon road condition and traffic one can reach within 2 hr)

100 200 300

0

1

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100 200 300

0

1

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Lets make some rules:1. If 0 D 100; very near ( depending upon road condition and traffic one can reach within 2 hr)




100 200 300

0

1

100 200 300

0

1

100 200 300

0

1

100 200 300

0

1

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Fuzzy set :

100 200 300

0

1

100 200 300

0

1

100 200 300

0

1

100 200 300

0

1

Distance (x)

Mem

bership(x)

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Fuzzy set :

100 200 300

0

1

100 200 300

0

1

100 200 300

0

1

100 200 300

0

1

Distance (x)

Mem

bership(x)

Imp. Note: + sign stands for the union of membership grades; /

stands for a marker and does not imply division.

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Fuzzy set :

100 200 300

0.25

1

100 200 300

1

100 200 300

0.75

1

X=75

0

1

Distance (x)

Mem

bership(x)

A(x=75) = 0.25/very near + 0.75/near + 0.0 far + 0.0/very far

A(x=300) = 0.00/very near + 0.00/near + 0.0 far + 1.0/very far

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Fuzzy set : Example 2

150 210170 180 190 200160

Height, cmDegreeofMembership

150 210180 190 200

1.0

0.0

0.2

0.4

0.6

0.8

160

Degreeof

Membership

Short Average Tall

170

1.0

0.0

0.2

0.4

0.6

0.8

Fuzzy Sets

CrispSets

Short Average Tall

Negnevitsky, Pearson Education, 2005

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Fuzzy set :

A fuzzy set A can be denoted as

If x is discrete

If x is continuous

Example 2. The membership function of the fuzzy set of real numbers close

to 1, is can be defined as

A x xA

x X

i i

i

( ) /

A x xA

X

( ) /

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Triangular Fuzzy Number :

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Trapezoidal Fuzzy Number :

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Operations on crisp sets

Intersection Union

Complement

Not A

A

Containment

AA

B

BA AA B


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Operations on fuzzy sets

Intersection:The intersection of A and B is defined

as

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Intersection:The union of A and B is defined as

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Complement

0x

1

(x)

0x

1

Containment

0x

1

0x

1

AB

Not A

A

Intersection

0x

1

0x

AB

Union

0

1

AB

AB

0x

1

0x

1

B

A

B

A

(x)

(x) (x)

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Complement

0x

1

(x)

0x

1

Containment

0x

1

0x

1

AB

Not A

A

Intersection

0x

1

0x

AB

Union

0

1

AB

AB

0x

1

0x

1

B

A

B

A

(x)

(x) (x)

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Important: Operations on fuzzy sets are also fuzzy, i.e., union

of intersection may be treated different ( different operators)

Eg. min-max operator

Union ( fuzzy OR)

)]x(),x([max)x( BABA

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Eg. Min and max operator

Intersection ( fuzzy AND)

)]x(),x([min)x( BABA

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Eg. min-max operator

Min-max :intersection distributive over union

)()( )()()( xx CABACBA

min[ A, max(B,C) ]=min[ max(A,B), max(A,C) ]

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Eg. Min, max and min-max operator

Min-max :union distributive over intersection

)()( )()()( xx CABACBA

max[ A,min(B,C) ]= max[ min(A,B), min(A,C) ]

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Eg. sum-product operator

)()()( A xxx BBA

]1),()([min)( A xxx BBA

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Fuzzy System

Input FuzzifierInference

EngineDefuzzifier Output

FuzzyKnowledge base

Fuzzier : Crisp input to a linguistic variable usingthe membership functionsEg: If 0 D 100; very near

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Fuzzy SystemInput Fuzzifier

Inference


FuzzyKnowledge base

Fuzzy Rule: There are many Fuzzy rules

e.g. Zadeh Mamdani rule (( commonly known as Mamdani Rule)If xisAand yis Bthen z

ifDis near andT is moderate thencustomers= more

ifDis very near and Tis highthen customers= reasonable

ifDis near and T is high then customers= reasonable

ifDis V near and Tis high then customers= less

ifDis far or very far then customers = less

If D =75 and T =21 how many customers will be there ???????

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Inference


FuzzyKnowledge base

Fuzzy Rule: There are many Fuzzy rules

e.g. Takagi, Sugeno & Kang( commonly known as Sugeno Rule)If xiisAiand yiis Bithen zi=f(xi,yi)

(f(xi,yi)is a polynomial of x and y)

ifDis near and Tis moderate z= 3D+2T+1

ifXis very small and Yis very large then z= xy

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Inference


FuzzyKnowledge base

Fuzzy Rule: Sugeno Rule

ifDis near and Tis moderate z= 3D+2T+1

ifXis very small and Yis very large then z= xy

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Inference


FuzzyKnowledge base

Inference:converts the fuzzy input to the fuzzy output, i.e., computation

of output fuzzy membership grade and their aggregation

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Inference


FuzzyKnowledge base

Defuzzifier:converts the fuzzy output to crisp output

Centroid of area (COA) Bisector of area (BOA) Mean of maximum (MOM) Smallest of maximum (SOM) Largest of maximum (LOM)

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Inference


FuzzyKnowledge base


Centroid of area (COA) Bisector of area (BOA) Mean of maximum (MOM) Smallest of maximum (SOM) Largest of maximum (LOM)

( )

,( )

A

Z

COA

A

Z

z zdz

z z dz

*

,

{ ; ( ) }

Z

MOM

Z

A

zdz

zdz

where Z z z

( ) ( ) ,

BOA

BOA

z

A A

z

z dz z dz

Used in Mamdani

FIS

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Inference


FuzzyKnowledge base


Weighted sum Weighted Average

Used in Mamdani

SugenoFIS

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Fuzzy System

Mamdani FIS with Max-min inference

R1:If x =A1, y=B1 then Z=C1

R2: If x =A2, y=B2 then Z=C2

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Fuzzy System

Mamdani FIS with Max-Product inference

R1:If x =A1, y=B1 then Z=C1

R2: If x =A2, y=B2 then Z=C2

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Advantages of the Sugeno Method

It is computationally efficient.

It works well with linear techniques (e.g., PID control).

It works well with optimization and adaptive techniques.

It has guaranteed continuity of the output surface.

It is well suited to mathematical analysis.

Advantages of the Mamdani Method

It is intuitive.

It has widespread acceptance.

It is well suited to human input.

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Fuzzy system example (Mamdani)

(1) fuzzification

Fuzzy Distance Fuzzy temperature Fuzzy Visitor

If D =75 and temperature =25 how many customers are accepted?

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(2) Rule

R1 : If Distance is near and temperature is pleasant then sufficient visitors

R2 : If Distance is very far or temperature is harsh then no visitorsR3 : If Distance is near and temperature is bad then less visitors

R4 : If Distance is far and temperature is reasonable then less visitorsR5 : If Distance is very near and temperature is pleasant then more visitors

R1

R2

R2

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(2) Rule




R3

R4

R5

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(2) Rule

Max- min operator

R1

R2

R2

min

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(2) Rule




R3

R4

R5

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(2) Rule

Max- min operator

R1

min

( )

,( )

A

Z

COA

A

z zdz

z

z dz

( ) ( ) ,BOA

BOA

z

A A

z

z dz z dz

*

,

{ ; ( ) }

Z

MOM

Z

A

zdz

zdz

where Z z z

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