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Gpstews م۳( رس
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© The Architecture of Fuzzy Inference Systems
° Fuzzy Models:
Mamdani Fuzzy models
Sugeno Fuzzy Models
Tsukamoto Fuzzy models
© Partition Styles for Fuzzy Models
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Gpstews م۳( رس
۱۳ ل
روص" IePereue Gystews
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Converts thecrispinput toa linguistic
variableusing themembership functions
storedin the fuzzy knowledge base.
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Converts thecrispinput toa linguistic
variableusing themembership functions
storedin the fuzzy knowledge base.
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Converts thefuzzy output of theinference
engine tocrispusing membership functions
analogous to the ones used by the fuzzifier.
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مدب
ال رامع را ره ,مرن عک دموا رما موم ۲6۵ 111
implementsa nonlinear mapping fromitsinput space to
output space.
۳ =
aggregator| defuzzifier
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Gpstews م۳( رس
اكد دوه(
(Puzgp wodels
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® Original Goal:Controfasteamengine &
boiler combination by a set of Linguistic
control rules obtainedfromexperienced
human operators.
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Max-Min Composition isused.
The Reusvuny Grokewe
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0ع اک که موه Max-Product
The Reusvuny Grokewe
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© Converts thefuzzy output of theinferenceengine to
crispusing membership functions analogous tothe
ones used by thefuzzifier.
© Fivecommonly used defuzzifying methods:
Centroidofarea (COA)
Bisector ofarea (BOA)
Mean of maximum (MOM)
SmalCest of maximum (SOM)
Largest of maximum (LOM)
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smallest of max. centroid of area
bisecter of area
largest of max.
mean of max.
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2 هار 2)< 2 ۲82
2
۲۳۳۳۳ >
مش وير سمو 5-5 bisecter of area
\— mean of max.
= -
fuade= fund,
= Zo04
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Ra if Xis small then Vis small
R2iIf Xismediumthen Yismed
Railf XisCargethen Vislarge
Gxavple ع ی
Y= output € [0, 10]
Max-min compositionandcentroiddefuzzification wereused.
Overall input-output curve
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‘Ra1:if Xissmall & Vissmall then Zis negative!
‘RaiIf Xissmall & YisCarge then Zis negatives
Ratlf Xislarge& Yis small then Zis positive sm
تنم Xislarge& Vislargethen Zispositivela
X,Y, Ze[-5, 5]
Max-min compositionandcentroiddefuzzification wereused.
Overall input-output curve
1
موس
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Gpstews م۳( رس
Guyew
(Puzay Oodels
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© Alsoknownas TSK fuzzy model
~ Takagi, Sugeno & Kang, 7985
* Goal: Generation of fuzzy rules froma
giveninput-output dataset.
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Ques oP TGC Oodet رد۳
If xis dyis nz = f(x, y)
Fuzzy Sets Crisp Function
fx, yisvery oftena
polynomial functionw.r.t. x
andy,
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2: )ام که ]ام و2 ۲0۵02 < -x +y +1
R2:if Xissmalland Yislargethen z = -y +3
R3:if XisCargeand Yissmall then Z = -x +3
R4:if Xislargeand Yislargethenz=x+t+yt2
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+ لاو + رم > و2
۲ + لا + لايم - و2
weighted average|
W424+WeZ2
۷,
ze
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RaiIfXissmall then Y= 0.1X+ 6.4
22:11 Xismediumthen Y= -0.5X + 4
Railf Xislargethen Y = X- 2
X=input €[-10, 10]
(0) Overall ¥O Curve far Crisp Rules
(a) Antecedent MFs for Crisp Rules
5 medium اقا
08 {
£08 {
eo ۱
2 اوه |
0 1
2 3 80 6 10 10 3 a 3 18
2 x
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RaiIfXissmall then Y= 0.1X+ 6.4
22:11 Xismediumthen Y= -0.5X + 4
Railf Xislargethen Y = X- 2
X=input > ]-10, 10[
(0) Overall VO Curve for Fuzzy Rules
(c) Antecedent MFs for Fuzzy Rules
small medium large
5 مصاع 3
8 2
3
4> 208
بو 2
2
02 7
ip aes 8 oh
Ifwe have smoothmembership functions (fuzzy rules) the overall
input-output curve becomes a smoother one.
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‘Xis small and Yissmall then z
‘Raxif Xissmalland YisCarge then
‘Rarif XisCargeand Yissmall then
رام Xislargeand Yislargethen
X, YeL5, 5]
“x+y 41
“y +3
=-x43
xty+2
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Gpstews م۳( رس
Tsuboi
(Puzay wodels
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The consequent of eachfuzzy if-then-
ruleis represented by a fuzzy set witha
monotonical MF.
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weighted average
Wy * We
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Ra: If Xissmall then Yis C,
R2:If Xismediumthen Vis C,
R3:if XisCargethen Yis C;
سم اسهم
پم اجه osm تست و
oo ۰ ۲۳۰
Gos, ۲ 1 doe
Bos \ Fos
soe ۱
202 ذخام 3
5 5
صب
۳
0
1
ا
‘
2
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Gpstews م۳( رس
(Puntitiocr Otptes Por
Puzay Oodels
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If <antecedence> then <consequence>.
——— ye
Thesamestylefor Different stylesfor
Mamdani Fuzzy models Mamdani Fuzzy models
Sugeno Fuzzy Models * Sugeno Fuzzy Models
* Tsukamoto Fuzzy models * Tsukamoto Fuzzy models
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Tree Scatter
Partition Partition
1 2
Grid
Partition
(a)