Presentation # 1 FL

8/9/2019 Presentation # 1 FL

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Amaury Caballero, Ph. D., P.EAmaury Caballero, Ph. D., P.E

Universidad Internacional deUniversidad Internacional dela Floridala Florida

Email: [email protected]: [email protected]

Logica Difusa: PrincipiosLogica Difusa: Principios

y Aplicacionesy Aplicaciones


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Principle of IncompatibilityPrinciple of Incompatibility

As the complexity of a system increases, our ability toAs the complexity of a system increases, our ability to

makemakeprecisepreciseand yetand yetsignificantsignificantstatements about itsstatements about its

behavior diminishes until a threshold is reached beyondbehavior diminishes until a threshold is reached beyondwhich precision and significance become almostwhich precision and significance become almost

mutually exclusive characteristics.mutually exclusive characteristics.


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Fuzzy LogicFuzzy Logic

Fuzzy logic may be viewed as a bridge between theFuzzy logic may be viewed as a bridge between theextensively wide gap between the precision crispextensively wide gap between the precision crisp

logic and the imprecision of both real world andlogic and the imprecision of both real world and

human interpretation.human interpretation. Prof. ZadehProf. Zadeh

As its name implies, the theory ofAs its name implies, the theory offuzzy setsfuzzy sets is,is,

basically, a theory ofbasically, a theory ofgradedgradedconcepts a theory inconcepts a theory inwhich everything is a matter of degree or, to put itwhich everything is a matter of degree or, to put it

figuratively, everything has elasticity.figuratively, everything has elasticity.

!.". Zimmermann!.". Zimmermann


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# A technology which enhances modelbased systemA technology which enhances modelbased system

designs using both intuition and engineeringdesigns using both intuition and engineering

heuristics.heuristics.

# $ecisions based on %degree of truth&$ecisions based on %degree of truth&

# A new paradigm of systems engineering whichA new paradigm of systems engineering which

helps achieve robust ' faulttolerant systemshelps achieve robust ' faulttolerant systems

# An efficient way of designing, optimizing 'An efficient way of designing, optimizing '

maintaining highly complex systems transparentlymaintaining highly complex systems transparently


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(he center of the fuzzy modeling is the idea of linguistic(he center of the fuzzy modeling is the idea of linguistic

variables.variables.

ExampleExample)) *f pro+ect duration is*f pro+ect duration is longlong, the completion risk, the completion risk

is increased.is increased. (he engine temperature is(he engine temperature is hothot.. A cruise missile has aA cruise missile has a longlong range at arange at a highhigh speed.speed. *f you are*f you are talltall, you are uite likely, you are uite likely heavyheavy..

(om is rather(om is rather talltallbut "udy isbut "udy isshortshort.. $isposable incomes in the$isposable incomes in the middlemiddletax payer istax payer is

adequateadequate.. (hat pro+ect reuires a(hat pro+ect reuires a largelargemanpowermanpower

commitment.commitment.


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ExampleExample::ForForthe fuzzy variable %the fuzzy variable %talltall& of men, the degrees& of men, the degrees

of membership depend on their heights.of membership depend on their heights.

!eight!ei

ght $egree of membership$e

gree of membership

-/&-/& /.//./-0&-0& /./1/./1

-1&-1& /.23/.234/&4/& /.-//.-/40&40& /.13/.1341&41& /.51/.516/&6/& 7.//7.//

(he variable %tall& is defined by the range of values for(he variable %tall& is defined by the range of values forheightsheights 8-/&, -0&, ..., 6/&98-/&, -0&, ..., 6/&9and the degrees of membershipand the degrees of membership8/.//, /./1, ..., 7.//98/.//, /./1, ..., 7.//9..

talltall)) heightsheights:/, 7;:/, 7;%%HeightsHeights& is the domain of %& is the domain of %talltall,& and :/, 7; the range.,& and :/, 7; the range.


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Types of UncertaintyTypes of Uncertainty

Probabilistic uncertaintyProbabilistic uncertainty example) rolling a diceexample) rolling a dice


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Fuzzy vs. ProbabilityFuzzy vs. Probability

Probabilistic =easoningProbabilistic =easoning %%(here is an 6-> chance that ?etty is old.&(here is an 6-> chance that ?etty is old.&

?etty is either old or not old 8the law of the excluded?etty is either old or not old 8the law of the excluded

middle9.middle9.

Fuzzy =easoningFuzzy =easoning %%?etty@s degree of membership within the set of old people?etty@s degree of membership within the set of old people

is /.6-.&is /.6-.&

?etty is like an old person, but could also have some?etty is like an old person, but could also have some

characteristics of a young person.characteristics of a young person.


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Fuzzy vs ProbabilityFuzzy vs Probability

risp Facts B distinct boundariesrisp Facts B distinct boundaries Fuzzy Facts B imprecise boundariesFuzzy Facts B imprecise boundaries Probability B incomplete factsProbability B incomplete facts

ExampleExample) Ccout reporting an enemy) Ccout reporting an enemy %%(wo tanks at grid DE -0& 8risp9(wo tanks at grid DE -0& 8risp9 %%A few tanks at grid DE -0& 8Fuzzy9A few tanks at grid DE -0& 8Fuzzy9

%%(here might be 3 tanks at grid DE -0&(here might be 3 tanks at grid DE -0&8Probabilistic98Probabilistic9


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Wat is Fuzzy Logic!Wat is Fuzzy Logic!

*t is a multivalued ?oolean logic.*t is a multivalued ?oolean logic.

*t uses rulebased control.*t uses rulebased control.

*t is based on a form of Artificial *ntelligence.*t is based on a form of Artificial *ntelligence.

*t uses human intuition for control.*t uses human intuition for control.


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Wat is Fuzzy Logic!Wat is Fuzzy Logic!

Fuzzy logic methodology is basically characterized byFuzzy logic methodology is basically characterized by

three traits)three traits)

879 *t does not consider whether something is true879 *t does not consider whether something is true

or false, but rather how true it is.or false, but rather how true it is.839 ?ecause its similar to human reasoning, its839 ?ecause its similar to human reasoning, its

implementation tends to be based on naturalimplementation tends to be based on natural

language.language.

829 *ts flexible and can model complex, nonlinear829 *ts flexible and can model complex, nonlinear

systems by using imprecise information.systems by using imprecise information.


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Fuzzy Logic " Wat is it #$T!Fuzzy Logic " Wat is it #$T!

Dot the solution to A


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Wen soul% Fuzzy Logic be use%!Wen soul% Fuzzy Logic be use%!

# *f no adeuate mathematical model for a given*f no adeuate mathematical model for a given

problem can be easily found.problem can be easily found.

# *f nonlinearity, timeconstraints or multiple*f nonlinearity, timeconstraints or multipleparameters exist.parameters exist.

*f engineering knowhow about the given problem*f engineering knowhow about the given problem

is available or can be acuired during the designis available or can be acuired during the designprocess.process.


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Wen soul% not Fuzzy Logic be use%!Wen soul% not Fuzzy Logic be use%!

# hen the problem can be easily solved usinghen the problem can be easily solved usingconventional control techniues, such as a P*$conventional control techniues, such as a P*$controller.controller.

# hen there is a simple, cleardefined and fasttohen there is a simple, cleardefined and fasttosolve mathematical model for the given problemsolve mathematical model for the given problemavailable.available.

# hen the problem cannot be solved at all. (herehen the problem cannot be solved at all. (hereare some problems with which even fuzzy logicare some problems with which even fuzzy logiccan not help you.can not help you.


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Wy Fuzzy Logic!Wy Fuzzy Logic!

Precision is not truth.Precision is not truth.G !enri HatisseG !enri Hatisse

Co far as the laws of mathematics refer to reality,Co far as the laws of mathematics refer to reality,they are not certain. And so far as they are certain,they are not certain. And so far as they are certain,they do not refer to reality.they do not refer to reality.

GG Albert IinsteinAlbert Iinstein

As complexity rises, precise statements lose meaningAs complexity rises, precise statements lose meaningand meaningful statements lose precision.and meaningful statements lose precision.

GG


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Wy Fuzzy Logic!Wy Fuzzy Logic!

!uman knowledge is fuzzy) expressed in %Fuzzy&!uman knowledge is fuzzy) expressed in %Fuzzy&


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Wy Use Fuzzy Logic!Wy Use Fuzzy Logic!

Do complex mathematical models for systemDo complex mathematical models for system

developmentdevelopment

Cimpler and more effective implementationCimpler and more effective implementation

Hore descriptiveHore descriptive

!igher faulttolerance and a better tradeoff!igher faulttolerance and a better tradeoff between system robustness and system sensitivitybetween system robustness and system sensitivity


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Wy &eluctance in Use of Fuzzy Logic!Wy &eluctance in Use of Fuzzy Logic!

Ingineers are trained using precise mathematicsIngineers are trained using precise mathematics

Host of us are more comfortable with theHost of us are more comfortable with theLawLaw

of the Excluded Middleof the Excluded Middle every proposition must either be (rue or Falseevery proposition must either be (rue or False

Dot enough software people are in charge ofDot enough software people are in charge of

engineering pro+ectsengineering pro+ects


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Fuzzy Logic ApplicationsFuzzy Logic Applications

'()'() AutomotiveAutomotive8a9 fuzzy engine control8a9 fuzzy engine control

8b9 fuzzy cruise control8b9 fuzzy cruise control

8c9 fuzzy antilock8c9 fuzzy antilock

braking systemsbraking systems

8d9 fuzzy transmission8d9 fuzzy transmission

systemssystems

'*)'*) AppliancesAppliances8a9 washing machines8a9 washing machines

8b9 air conditioners8b9 air conditioners

8c9 cameras8c9 cameras

8d9 E=s8d9 E=s

8e9 microwave ovens8e9 microwave ovens


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-loo% Pressure easurement /it Fuzzy Logic-loo% Pressure easurement /it Fuzzy Logic

=eliMn@s automatic wrist blood=eliMn@s automatic wrist blood

pressure monitor features bothpressure monitor features both

fuzzy logic inflation andfuzzy logic inflation and

controlled deflation to give youcontrolled deflation to give you

the most constant and exactthe most constant and exactrates of pressure.rates of pressure.

*t also features a large screen*t also features a large screen

(hat simultaneously displays(hat simultaneously displayssystolic, diastolic and pulsesystolic, diastolic and pulse

readings, an easy touse, onereadings, an easy touse, one

button design, and a memorybutton design, and a memory

that recalls up to 2/ readings.that recalls up to 2/ readings.


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&ice 0oo1er /it Fuzzy Logic&ice 0oo1er /it Fuzzy Logic

2o3irusi4 #euro"Fuzzy4 Logic &ice2o3irusi4 #euro"Fuzzy4 Logic &ice

0oo1er0oo1er make perfect rice with Zo+irushi@smake perfect rice with Zo+irushi@srice cooker using its advanced neurofuzzyrice cooker using its advanced neurofuzzy

logicN technology B brown rice, sushi rice,logicN technology B brown rice, sushi rice,

sweet rice and mixed rice are +ust a few ofsweet rice and mixed rice are +ust a few of

the options.the options.

A timer can be set up to 73 hours in advanceA timer can be set up to 73 hours in advance

and a start stop melody signal will let youand a start stop melody signal will let you

know when your rice is done.know when your rice is done.

A reheating and extended keepwarm cycleA reheating and extended keepwarm cycle

allow you to prepare up to -.- cups of yourallow you to prepare up to -.- cups of your

rice ahead of time and serve later.rice ahead of time and serve later.

=etractable cord for easy storage.=etractable cord for easy storage.


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Waser /it Fuzzy LogicWaser /it Fuzzy Logic


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Fuzzy 0ameraFuzzy 0amera


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5ro/ing Interest in Fuzzy 0ontrol!5ro/ing Interest in Fuzzy 0ontrol!

*t is a %new technology& and can be used to avoid*t is a %new technology& and can be used to avoidpatentclaims of similar solutions for technicalpatentclaims of similar solutions for technical

problems, which are based on a different techniue.problems, which are based on a different techniue.

*n "apan, fuzzy is %wanted& by consumers, since it*n "apan, fuzzy is %wanted& by consumers, since it

represents %hightech&. *n this case fuzzy techniuesrepresents %hightech&. *n this case fuzzy techniues

are mostly used as a marketing tool.are mostly used as a marketing tool.


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5ro/ing Interest in Fuzzy 0ontrol!5ro/ing Interest in Fuzzy 0ontrol!

(he development of fuzzy controllers is easier to(he development of fuzzy controllers is easier tolearn and reuires less skilled personnel than that oflearn and reuires less skilled personnel than that of

conventional controllers. (his results in cheaperconventional controllers. (his results in cheaper

productions.productions.

Fuzzy controllers provide more robustness thanFuzzy controllers provide more robustness than

conventional control.conventional control.

Fuzzy controllers are more appropriate to controlFuzzy controllers are more appropriate to control

nonlinear processes.nonlinear processes.


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67ample: Fuzzy Temperature 0ontrol67ample: Fuzzy Temperature 0ontrol


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*f =oom (emperature is*f =oom (emperature is 0ol%0ol%then Fan Cpeed isthen Fan Cpeed is 8lo/8lo/

*f =oom (emperature is*f =oom (emperature is WarmWarmthen Fan Cpeed isthen Fan Cpeed is

e%iume%ium

*f =oom (emperature is*f =oom (emperature is 9ot9otthen Fan Cpeed isthen Fan Cpeed is FastFast

Fuzzy Temperature 0ontrolFuzzy Temperature 0ontrol


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Dote the

overlapping

of fuzzy

subsets again it leads

to smooth

approximation

of the functionbetween the

Fan Cpeed '

(emperature


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0alculation of te output:0alculation of te output:

After the fuzzy modeling is done there is an operation phase)

alculate the Fan Cpeed when =oom (emperature L 33 / .

DM(IO 33/

belongs to the Cubsets arm and !ot


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Fuzzification an% InferenceFuzzification an% Inference

*f =oom (emperature is Warm(hen Fan Cpeed is e%ium


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Fuzzification an% InferenceFuzzification an% Inference

*f =oom (emperature is 9otthen Fan Cpeed is Fast

Te uestion no/ is: Wat is te output value!


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Example of Crisp OperationsExample of Crisp Operations


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Fuzzy Sets NotationFuzzy Sets Notation

A notation convention or u!!y setsA notation convention or u!!y sets

"hen the universe o discourse, #, is"hen the universe o discourse, #, is

discretediscrete

and fnite, is as ollo"s or a u!!y set A:and fnite, is as ollo"s or a u!!y set A: Discrete:Discrete:

Analo$ousAnalo$ous


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Fuzzy Sets OperationsFuzzy Sets Operations


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Fuzzy Intersection (Conjunction)Fuzzy Intersection (Conjunction)


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Fuzzy ComplementFuzzy Complement

Fuzzy max min CompositionFuzzy max min Composition


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Fuzzy max-min CompositionFuzzy max-min Composition

%u&&ose ' is a u!!y relation on the%u&&ose ' is a u!!y relation on the

Cartesian s&ace # ( ), % is a u!!yCartesian s&ace # ( ), % is a u!!yrelation on ) ( *, and + is a u!!yrelation on ) ( *, and + is a u!!y

relation on # ( * then u!!y ma(-relation on # ( * then u!!y ma(-

min com&osition is defned in termsmin com&osition is defned in terms

o the settheoretic notation ando the settheoretic notation and

membershi& unctiontheoreticmembershi& unctiontheoretic

notation in the ollo"in$ manner:notation in the ollo"in$ manner:


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Fuzzy max-product CompositionFuzzy max-product Composition


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Oter Forms of te CompositionOter Forms of te Composition

OperationOperation


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Fuzzy 8etsFuzzy 8ets

Fuzzy set theory is a generalization of classical setFuzzy set theory is a generalization of classical set

theory B a generalization that deals excellently withtheory B a generalization that deals excellently with

imprecision.imprecision.

(he power of fuzzy logic is that it enables you to(he power of fuzzy logic is that it enables you to

accurately describe a process or behavior withoutaccurately describe a process or behavior without

using mathematics.using mathematics.


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lassical set theory) An ob+ect is either in or not inlassical set theory) An ob+ect is either in or not in the set.the set.

ant talk about nonsharp distinctionsant talk about nonsharp distinctions

Fuzzy set theoryFuzzy set theory An ob+ect is in a set by matter of degreeAn ob+ect is in a set by matter of degree

membershipmembership hh L 7./L 7./ in the set in the set membershipmembership hh L /./L /./ not in the set not in the set

/./ Q membership/./ Q membership

hh

Q 7./Q 7./

partially in the set partially in the set

Fuzzy sets have a smooth boundaryFuzzy sets have a smooth boundary Dot completely in or outDot completely in or out


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0risp 8ets an% Fuzzy 8ets0risp 8ets an% Fuzzy 8ets


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0lassical 8ets0lassical 8ets

a set of patients with %highfever&

universe of discourse

25.7o

0/.4o

07.-o

a set of patients without

%high fever&

24.0o

26./o

21.1o

26.2o


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a fuzzy set of patients with

%high fever&

universe of discourse

25.7o

0/.4o

07.-o

a fuzzy set of patients

without %high fever&

24.0o

26./o

26.2o


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7 /.1 /.6 /.4 /.- /.3 /

8a%e of 5ray

Degree of being

roun% 8roundness9

/.- .4 /.4- /.- /.4- /.0

/.6 /.-- /.0 /.1 /.5 7./


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(he basic idea of thefuzzy set theoryis that an element

belongs to afuzzy set with a certain degree ofmembership, with fuzzy boundaries

Aproposition is neither true nor false, but may be partly

true 8or partly false9 to any degree. (his degree is

usually taken as a real number in the interval :/,7;

Fuzzy logic is an extension of classic twovalued logic Bthe truth value of a sentenceisnot restricted to true or

false.

Fuzzy 8ets


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Fuzzy Cubset A

uzziness

7

/Arisp Cubset A uzziness

8x9

# (ypical functionsthat can be used to represent a fuzzy set aresigmoid, gaussianand pi. !owever, these functions increase

the time of computation. (herefore, in practice, most

applications represent fuzzy subsetsrepresent fuzzy subsetsby linear fit functions!

Fuzzy 8ets


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#

A linguistic variableis a fuzzy variable.For example, the statement %"ohn is tall& implies that the

linguistic variable"ohntakes the linguistic value tall.

#

(he range ofpossible valuesof a linguistic variablerepresents the universe of discourseof that variable.

For example, the universe of discourse of the linguisticvariablespeedmight have the range between / and 33/

kmRh and may include such fuzzy subsets as very slow,slow, medium,fast, and very fast.

Fuzzy 8ets

F 8 tF 8ets


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A fuzzy set has a membership function that allowsA fuzzy set has a membership function that allowsvarious degrees of membership for the elements of avarious degrees of membership for the elements of a

given set.given set.

(he membership function may be defined in terms(he membership function may be defined in terms

of discrete values, or more commonly by a graph.of discrete values, or more commonly by a graph.

hen membership function is described by anhen membership function is described by ananalytic expression, we can +ust use the membershipanalytic expression, we can +ust use the membership

function to describe the fuzzy subset.function to describe the fuzzy subset.

8


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Classical Logic

Ilementxbelongs to setA

or it does not)

h8x9S/,7T

AL%young&

x:years;

hA8x9

7

/

Fuzzy Logic

Ilementxbelongs to setA

with a certain degree of

membership:

h8x9:/,7;

AL%young&

x:years;

hA8x9

7

/


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67ample: 0risp set Tall67ample: 0risp set Tall

Fuzzy sets and concepts are commonly used in naturalFuzzy sets and concepts are commonly used in naturallanguagelanguage

"ohn is"ohn is talltall

#an is#an is smartsmartAlex isAlex is happyhappy$he class is$he class ishothot

(he crisp set(he crisp set TallTallcan be defined as Scan be defined as Sxx U heightU heightxx V 7.1V 7.1metersT. ?ut what about a person with a height L 7.65metersT. ?ut what about a person with a height L 7.65meters hat about 7.61 meters W hat about 7.-3meters hat about 7.61 meters W hat about 7.-3metersmeters


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F 8


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# (hexaxisrepresents the universe of discoursetherange of all possible valuesapplicable to a chosen

variable.

#

(heyaxisrepresents the membership value of thefuzzy set.

Fuzzy 8ets

F C


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#crisp and fuzzy subsets defined on the universe

7-/ 37/76/ 71/ 75/ 3//74/

Height% cm#egree ofMem&ership

(all Hen

7-/ 37/71/ 75/ 3//

7./

/./

/.3

/.0

/.4

/.1

74/

#egree ofMem&ership

Chort Average Chort(all

76/

7./

/./

/.3

/.0

/.4

/.1

Fuzzy Cets

risp Cets

Chort Average

(all

(all

Fuzzy Cets


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Fuzzy 8etFuzzy 8et

Fuzzy sets ) young, middle aged and oldFuzzy sets ) young, middle aged and old

Xoung A7

Hid

AgeA3 MldA2

AgeH

emb

ers

hip

/ 1/2-3/ 4/0-

8F 8


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Eariables whose states are defined by linguisticEariables whose states are defined by linguisticconcepts likeconcepts like lowlow,, mediummedium,, highhigh..

(hese linguistic concepts are fuzzy sets themselves.(hese linguistic concepts are fuzzy sets themselves.

Eery


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. . . Yhot. . .

Ytemperature

Yagreeable


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A fuzzy subset has a membership function thatA fuzzy subset has a membership function that allows various degrees of membership for theallows various degrees of membership for the

elements of a given set.elements of a given set.

(he membership function may be defined in terms(he membership function may be defined in terms

of discrete values, or more commonly by a graph.of discrete values, or more commonly by a graph.



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*f*fis theis the universe of discourseuniverse of discoursewith elementwith elementxx, then a, then afuzzy su&setfuzzy su&set, denoted by, denoted byAA, on, on is a set of ordered pairsis a set of ordered pairs

such thatsuch that

AAS8S8xx,, hhAA88xx99U99UxxTTwherewhere hhAA88xx9 is the9 is the mem&ership functionmem&ership functionofofxxininAAandand

is defined byis defined by hhAA)):/, 7;:/, 7;

Ilements with zero membership are usually not listed.Ilements with zero membership are usually not listed.



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*f the universe of discourse*f the universe of discourseis discrete and finite,is discrete and finite,

L SL Sxx77,,xx33, W,, W,xxnnT, there are three different ways toT, there are three different ways to

describe the fuzzy subsetdescribe the fuzzy subsetAA..

8a98a9 AALL hhAA88xx779R9Rxx77 hhAA88xx339R9Rxx33 W W hhAA88xxnn9R9Rxxnn

8b98b9AAL S8L S8xx77,, hhAA88xx7799, 899, 8xx33,, hhAA88xx3399, W , 899, W , 8xxnn,, hhAA88xxnn99T99T

8c98c9AAL SL ShhAA88xx779,9, hhAA88xx339, W ,9, W , hhAA88xxnn9T9T

*f the universe of discourse*f the universe of discourseis continuous, we canis continuous, we can

represent the fuzzy subsetrepresent the fuzzy subsetAAbyby

AAxx

::hhAA

88xx9R9Rxx;;



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Cince all information contained in a fuzzy subset can beCince all information contained in a fuzzy subset can bedescribed by its membership function, it is useful todescribed by its membership function, it is useful to

develop a lexicon of terms to describe various specialdevelop a lexicon of terms to describe various special

features of this function.features of this function.



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(he(he corecore of a fuzzy subsetof a fuzzy subset AAis defined as that region ofis defined as that region of

the universe that isthe universe that is

characterized by fullcharacterized by full

membership inmembership inAA..

'ore'ore88AA9 L S9 L SxxUU hhAA88xx9 L 7,9 L 7,xxTT

?oundary

h8x9ore

Cupport

7./



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(he(hesupportsupport8or domain9 of8or domain9 of

a fuzzy subseta fuzzy subsetAA,, (upp(upp88AA9,9,

is a crisp set of allis a crisp set of allxx

such thatsuch that hhAA

88xx9V/.9V/.

(upp(upp88AA9 L S9 L SxxUU hhAA88xx9V/,9V/,xxTT?oundary

h8x9ore

Cupport

7./

F 8 tF 8 t


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ExampleExample::


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yy

(he(he &oundary&oundary of a fuzzy subsetof a fuzzy subset AAis defined as that region ofis defined as that region of

the universe containingthe universe containing

elements that have a nonzeroelements that have a nonzero

membership but not completemembership but not complete

membership.membership.

)oundary)oundary88AA9 L S9 L SxxU / QU / QhhAA88xx9Q 7,9Q 7,xxTT

?oundary

h8x9ore

Cupport

7./



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(he(heheightheightof a fuzzy subsetof a fuzzy subsetAAis the maximum valueis the maximum value of the membership function, i.e.,of the membership function, i.e.,

heightheight88AA9 L9 L maxmaxSShhAA88xx9T.9T.

A fuzzy subsetA fuzzy subsetAAisis normalnormalif and only ifif and only if

supsupxx::hhAA88xx9; L 7,9; L 7,

that is, the supreme ofthat is, the supreme of hhAA88xx9 over9 overis unity.is unity.

A fuzzy subset isA fuzzy subset issu&normalsu&normalif it is not normal.if it is not normal.



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Hembership functions can beHembership functions can besymmetricalsymmetrical ororasymmetricalasymmetrical..

(hey are typically defined on one dimensional universe,(hey are typically defined on one dimensional universe,but they certainly can be described on multibut they certainly can be described on multi

dimensional universe.dimensional universe.

For high dimensions, the membership functions becomeFor high dimensions, the membership functions become

hypersurfaces.hypersurfaces.


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ExampleExample::


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8tan%ar% embersip Functions8tan%ar% embersip Functions

*n*nfuzzy controlfuzzy control mainly usemainly usedd)) LL,, andand functions, defined by straight line segments.functions, defined by straight line segments.

7

u

) : , ; 8 9

8 [ , 9 8 9 R 8 9

*

u u

u

u

u

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Presentation # 1 FL