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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)
Fuzzy Inference Systems
主講人 : 虞台文
Content
The Architecture of Fuzzy Inference Systems
Fuzzy Models:
–
–
–
Mamdani Fuzzy models
Sugeno Fuzzy Models
Tsukamoto Fuzzy models
Partition Styles for Fuzzy Models
Fuzzy Inference Systems
The Architecture of
Fuzzy Inference Systems
Fuzzy Systems
Input
Fuzzifier
Inference
Engine
Fuzzy
Knowledge base
Defuzzifier
Output
Fuzzy Control Systems
Input
Fuzzifier
Inference
Engine
Fuzzy
Knowledge base
Defuzzifier
Plant Output
I nput
Fuzzifier
Fuzzif ier
I nf erence
Engine
Def uzzif ier
Fuzzy
Knowledge bas e
Converts the crisp input to a linguistic
variable using the membership functions
stored in the fuzzy knowledge base.
O utput
I nput
Fuzzifier
Fuzzif ier
I nf erence
Engine
Def uzzif ier
Fuzzy
Knowledge bas e
Converts the crisp input to a linguistic
variable using the membership functions
stored in the fuzzy knowledge base.
O utput
I nput
Fuzzif ier
I nf erence
Engine
Def uzzif ier
Inference Engine
Fuzzy
Knowledge bas e
Using If-Then type fuzzy rules converts the
fuzzy input to the fuzzy output.
O utput
I nput
Defuzzifier
Fuzzif ier
I nf erence
Engine
Def uzzif ier
Fuzzy
Knowledge bas e
Converts the fuzzy output of the inference
engine to crisp using membership functions
analogous to the ones used by the fuzzifier.
O utput
Nonlinearity
In the case of crisp inputs & outputs, a fuzzy inference system
implements a nonlinear mapping from its input space to
output space.
Fuzzy Inference Systems
Mamdani
Fuzzy models
Mamdani Fuzzy models
Original Goal: Control a steam engine &
boiler combination by a set of linguistic
control rules obtained from experienced
human operators.
Max-Min Composition is used.
The Reasoning Scheme
Max-Product Composition is used.
The Reasoning Scheme
I nput
Defuzzifier
Fuzzif ier
I nf erence
Engine
Fuzzy
Knowledge bas e
Def uzzif ier
O utput
Converts the fuzzy output of the inference engine to
crisp using membership functions analogous to the
ones used by the fuzzifier.
Five commonly used defuzzifying methods:
–
Centroid of area (COA)
–
Mean of maximum (MOM)
–
–
–
Bisector of area (BOA)
Smallest of maximum (SOM)
Largest of maximum (LOM)
I nput
Defuzzifier
Fuzzif ier
I nf erence
Engine
Fuzzy
Knowledge bas e
Def uzzif ier
O utput
I nput
Fuzzif ier
Defuzzifier
I nf erence
Engine
Def uzzif ier
O utput
Fuzzy
Knowledge bas e
zzCOA
COA
zdz
((zz))zdz
,,
dz
((zz))dz
ZZ
ZZ
zBOA
zBOA
dz
dz,,
((zz))dz
((zz))dz
zBOA
zBOA
AA
zdz
zdz
ZZ
zzMOM
,,
MOM
dz
dz
Z
AA
AA
AA
Z
**
where
Z
{
z
;
(
z
)
where Z {z; AA (z) }}
Example
R1 : If X is small then Y is small
R2 : If X is medium then Y is medi
R3 : If X is large then Y is large
X = input [10, 10]
Y = output [0, 10]
Max-min composition and centroid defuzzification were used.
Overall input-output curve
Example
R1: If X is small & Y is small then Z is negative l
R2: If X is small & Y is large then Z is negative sm
R3: If X is large & Y is small then Z is positive sm
R4: If X is large & Y is large then Z is positive lar
X, Y, Z [5, 5]
Max-min composition and centroid defuzzification were used.
Overall input-output curve
Fuzzy Inference Systems
Sugeno
Fuzzy Models
Sugeno Fuzzy Models
Also known as TSK fuzzy model
–
Takagi, Sugeno & Kang, 1985
Goal: Generation of fuzzy rules from a
given input-output data set.
Fuzzy Rules of TSK Model
If x is A and y is B then z = f(x, y)
Fuzzy Sets
Crisp Function
f(x, y) is very often a
polynomial function w.r.t. x
and y.
Examples
R1: if X is small and Y is small then z = x +y +1
R2: if X is small and Y is large then z = y +3
R3: if X is large and Y is small then z = x +3
R4: if X is large and Y is large then z = x + y + 2
The Reasoning Scheme
Example
R1: If X is small then Y = 0.1X + 6.4
R2: If X is medium then Y = 0.5X + 4
R3: If X is large then Y = X – 2
X = input [10, 10]
oth
o
m
s
n
u
Example
R1: If X is small then Y = 0.1X + 6.4
R2: If X is medium then Y = 0.5X + 4
R3: If X is large then Y = X – 2
X = input [10, 10]
If we have smooth membership functions (fuzzy rules) the overall
input-output curve becomes a smoother one.
Example
R1: if X is small and Y is small then z = x +y +1
R2: if X is small and Y is large then z = y +3
R3: if X is large and Y is small then z = x +3
R4: if X is large and Y is large then z = x + y + 2
X, Y [5, 5]
Fuzzy Inference Systems
Tsukamoto
Fuzzy models
Tsukamoto Fuzzy models
The consequent of each fuzzy if-then-
rule is represented by a fuzzy set with a
monotonical MF.
Tsukamoto Fuzzy models
Example
R1: If X is small then Y is C1
R2: If X is medium then Y is C2
R3: if X is large then Y is C3
Fuzzy Inference Systems
Partition Styles for
Fuzzy Models
Review Fuzzy Models
If <antecedence> then <consequence>.
The same style for
Mamdani Fuzzy models
• Sugeno Fuzzy Models
• Tsukamoto Fuzzy models
•
Different styles for
Mamdani Fuzzy models
• Sugeno Fuzzy Models
• Tsukamoto Fuzzy models
•
Partition Styles for Input Space
Grid
Partition
Tree
Partition
Scatter
Partition
