Meaning of “fuzzy”

Dr.-Ing. Erwin SitompulPresident University

Lecture 7

Introduction to Neural Networksand Fuzzy Logic

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Meaning of “fuzzy”IntroductionFuzzy Logic

Covered with fuzz;Of or resembling fuzz;Not clear; indistinct

A fuzzy recollection of past events.Not coherent; confused

A fuzzy plan of action.Unclear, blurred, or distorted

Some fuzzy pictures from a Russian radar probe.

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4 Seasons

0

0.5

1

Time of the year

Mem

bers

hip

Spring Summer Autumn Winter

IntroductionFuzzy Logic

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Tall Persons

0 : A person is not tall

1 : A person is tall

IntroductionFuzzy Logic

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Incorporation of human’s perception

0 : room is not warm1 : room is warm

IntroductionFuzzy Logic

Room Temperature

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Characteristic function:

young

1, age( ) 20( )

0, age( ) 20

xx

x

young = { x P | age(x) ≤ 20 }

Set DefinitionFuzzy Logic

Classical Sets

A=“young”1

0

A ( )x

yearsx20x

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Classical Logic

Element x belongs to set Awith a certain “degree of membership”:

(x)[0,1]

Element x whether belongs to set A or not at all:

(x){0,1}

Fuzzy Logic

A=“young”1

0

A ( )x

yearsx21x

A=“young”1

0

21x

A ( )x

yearsx

Set DefinitionFuzzy Logic

Fuzzy Sets

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yearsx

A=“young”1

0

21x

A ( )x

0.4A

Fuzzy Set A = {(x,A(x)) | x X, A(x) [0,1]}is defined by a universe of discourse x where0 ≤ x ≤ 100 and a membership function A where A(x) [0,1]

Definition:

Fuzzy SetsSet DefinitionFuzzy Logic

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x

(x)

1

0

Set DefinitionFuzzy Logic

Some DefinitionsSupport of a fuzzy set A

supp(A) = { x X | A(x) > 0 }Core of a fuzzy set A

core(A) = { x X | A(x) = 1 }α-cut of a fuzzy set A

Aα = { x X | A(x) α}

α = 0.6

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Fuzzy Logic Control (FLC)Fuzzy Logic ControlFuzzy Logic

Fuzzy Logic Control (FLC) may be viewed as a branch of intelligent control which serves as an emulator of human decision-making behaviour which is approximate rather than exact.

FLC uses the IF-THEN rules, similar to binary control (Programmable Logic Controller, PLC).

Rule Format:Ri: IF x is Aj AND y is Bk THEN z is Cl

Ri: IF x is Aj OR y is Bk THEN z is Cl

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Logic Operators

A B A B A B

Fuzzy Logic OperatorsFuzzy Logic

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p q p q 0 0 0 0 1 1 1 0 1 1 1 1

p q p q 0 0 0 0 0.4 0.4 0 1 1

0.4 0 0.4 0.4 0.4 0.4 0.4 1 1 1 0 1 1 0.4 1 1 1 1

Boolean OR Fuzzy OR

Fuzzy Logic OperatorsFuzzy Logic

Boolean OR and Fuzzy OR

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p q p^q 0 0 0 0 1 0 1 0 0 1 1 1

p q p^q 0 0 0 0 0.4 0 0 1 0

0.4 0 0 0.4 0.4 0.16 0.4 1 0.4 1 0 0 1 0.4 0.4 1 1 1

Boolean AND Fuzzy AND

Boolean AND and Fuzzy ANDFuzzy Logic OperatorsFuzzy Logic

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Example: Air FanFuzzy Logic ControlFuzzy Logic

Conventional (On-Off) Control:IF temperature > X °C,

THEN run fan,ELSE stop fan.

Fuzzy Control: IF temperature is hot,

THEN run fan at full speed;

IF temperature is warm, THEN run fan at moderate speed;

IF temperature is comfortable, THEN maintain fan speed;

IF temperature is cool, THEN slow fan;

IF temperature is cold, THEN stop fan.

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Example: Stopping A Car

13600 N 0 N1500 kg

(0) 25 m(0) 10 m s

Fm

yy

Break force

Mass of the car

Initial position

Initial velocity

FF my y

m

Fuzzy Logic ControlFuzzy Logic

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Example: Stopping A Car

( )p dK e T ey

m

pF K e

, 0e w y w

p pK e K yy

m m

With Kp = –240, the car will stop at the traffic light after 10 s.

P-Control PD-Control

2

p

p d p

K my

w s K mT s K m

Choosing ζ = 1, Td = 1, Kp = 6000, the car will stop at the traffic light after 5 s.

Fuzzy Logic ControlFuzzy Logic

2

2 22n

n ns s

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Example: Stopping A CarFuzzy Logic ControlFuzzy Logic

Fuzzy Logic Control: IF distance is long AND approach is fast,

THEN brake zero; IF distance is long AND approach is slow,

THEN brake zero; IF distance is short AND approach is fast,

THEN brake hard; IF distance is short AND approach is slow,

THEN brake zero.

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Fuzzy Membership FunctionsFuzzy Logic ControlFuzzy Logic

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ResponseFuzzy Logic ControlFuzzy Logic