![Page 1: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/1.jpg)
Fuzzy Logic
Dr. Chi-Tai, Cheng
![Page 2: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/2.jpg)
Outline
• What is Fuzzy Logic?
• History and applications of Fuzzy Logic
• Fuzzy Control– Fuzzification– Fuzz Inference Mechanism– Fuzzy Knowledge Base
– Defuzzification
• Hand calculation of Fuzzy Control
• Paper of Fuzzy Control
![Page 3: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/3.jpg)
What is Fuzzy Logic?
• Traditional logic system: (Crisp set)– Item does or doesn’t belong to a group
(1, 0)• Human• Sports
Not Human
Human Hockey Basketball
![Page 4: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/4.jpg)
What is Fuzzy Logic?
• Fuzzy logic system:– Degree of a set [1, 0]
• Cold/Hot• Rich/Poor• Teenager/
![Page 5: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/5.jpg)
What is Fuzzy Logic?
Traditional logic system
Characteristic function
1
0
1
0
Fuzzy logic systemMembership function
![Page 6: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/6.jpg)
What is Fuzzy Logic?
• Example: • Input: Angle of Shoot, Distance of
Shoot• Output: Chance of Success
Small Normal Big
Close Normal Good Chance
Good Chance
Normal No Chance Normal Good Chance
Far No Chance No Chance Normal
![Page 7: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/7.jpg)
History and applications of Fuzzy Control
• Fuzzy logic was first proposed by Lotfi A. Zadeh of the University of California at Berkeley in a 1965 paper.
• Mid-1980, “ inverted pendulum” experiment.
![Page 8: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/8.jpg)
History and applications of Fuzzy Control
![Page 9: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/9.jpg)
History and applications of Fuzzy Control
ZE PS PBNB
NS
![Page 10: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/10.jpg)
History and applications of Fuzzy Control
• When can we use Fuzzy Control?
![Page 11: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/11.jpg)
History and applications of Fuzzy Control
• Fuzzy and Probability
![Page 12: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/12.jpg)
History and applications of Fuzzy Control
• For continuous values [0, 1]• For big and complicate problem• For no explicit define math problem• For human language - linguistic
variables
![Page 13: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/13.jpg)
Jacky’s Assignment
![Page 14: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/14.jpg)
Fuzzy set
AA offunction pmemebershi ],1 ,0[X: →µ
setfuzzy ,}X|))( ,{( ∈= xxxA Aµ
membership of degree ],1 ,0[)( ∈xAµ1
0
Teenager :A
13 1911 21
![Page 15: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/15.jpg)
Fuzzy set
• Support set
• Core
• Crossover point
• Height 0
13 1911 21
}X ,0)(|{Supp ∈∀>= xxxA Aµ
}X ,1)(|{Core ∈∀== xxxA Aµ
5.0)( =xµ
)(max)(HeightX
xA Ax
µ∈
=
1Teenager :A
![Page 16: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/16.jpg)
α-cut sets
• α-cut sets1] [0,, ,set Universal:X ∈= α F(X)A
}X ,)(|{ ∈∀≥= xxxA A αµα
0
1
<∉≥∈
=))(( 0
))(( 1)(
αµαµ
χα
αα xAxif
xAxifx
A
AA
0.5 =α
![Page 17: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/17.jpg)
Membership function
• Discrete membership function• Continuous membership function
– Triangular shape– Trapezoid– S function– Z function– Pi function– Bell shape
![Page 18: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/18.jpg)
Discrete membership function
![Page 19: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/19.jpg)
Triangular shape
≤≤≤
≤≤≤
=−−−−
xbfor
bxafor
axafor
axfor
xabxb
aaax
A
1
1
1
1
0
0
)(
1
1
1
1
µ
0
1
a1 a b1
![Page 20: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/20.jpg)
Trapezoid
≤≤≤≤≤≤≤
≤
=
−−
−−
xbfor
bxbfor
bxafor
axafor
axfor
x
bbxb
aaax
A
1
1
1
1
0
1
0
)(
1
1
1
1
µ
0
1
a1 a b b1
![Page 21: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/21.jpg)
S function
• monotonical increasing
≤≤≤−≤≤
≤
=−−
−−
xfor
xfor
xfor
xfor
xSx
x
γγββα
α
γβααγγ
αγα
1
)(21
)(2
0
),,;(2
2
![Page 22: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/22.jpg)
Z function
≤≤≤≤≤−
≤
=−−
−−
xfor
xfor
xfor
xfor
xZx
x
γγββα
α
γβααγγ
αγα
0
)(2
)(21
1
),,;(2
2
![Page 23: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/23.jpg)
Pi function
≥++−≤−−
=Πγβγγγγγγβγ
γβ β
β
xforxS
xforxSx
) , , ;(1
) , , ;(),;(
2
2
![Page 24: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/24.jpg)
Bell shape
bA
acx
x 2
1
1)(
−+=µ
c
1
![Page 25: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/25.jpg)
Relation
• Binary relational matrix• Example: Standard weight
60 70 80 90
150 1 0 0 0
160 0 1 0 0
170 0 0 1 0
180 0 0 0 1
60 70 80 90
150 1 0.6 0.2 0
160 0.6 1 0.6 0.2
170 0.2 0.6 1 0.6
180 0 0.2 0.6 1
Traditional logic Fuzzy logic
![Page 26: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/26.jpg)
Traditional Logic Operate
• Brothers & Classmates
Brother Classmates
}David Carl, Bill, Albert,{=A}Grand ,Foster ,Eason{=B
Eason
Foster
Grand
Albert
0 1 0
Bill 0 0 1
Carl 0 0 0
David
1 0 0
Eason
Foster
Grand
Albert
0 1 1
Bill 0 1 0
Carl 1 0 1
David
1 1 0
![Page 27: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/27.jpg)
Traditional Logic Operate - Union
• Relationship for who are brothers OR classmates (Union) => max(A, B)
Eason
Foster
Grand
Albert 0 1 0Bill 0 0 1Carl 0 0 0David 1 0 0
Eason
Foster
Grand
Albert
0 1 1Bill 0 1 0Carl 1 0 1David 1 1 0
∪ =Eason Foster Grand
Albert 0 1 1
Bill 0 1 1
Carl 1 0 1
David 1 1 0
![Page 28: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/28.jpg)
Traditional Logic Operate - Intersection
• Relationship for who are brothers AND classmates (Intersection) => min(A, B)
Eason
Foster
Grand
Albert 0 1 0Bill 0 0 1Carl 0 0 0David 1 0 0
Eason
Foster
Grand
Albert
0 1 1Bill 0 1 0Carl 1 0 1David 1 1 0
∩ =Eason Foster Grand
Albert 0 1 0
Bill 0 0 0
Carl 0 0 0
David 1 0 0
![Page 29: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/29.jpg)
Fuzzy Logic Operate
• B Higher & Heavier than A
Higher Heavier
}David Carl, Bill, Albert,{=A
}Grand ,Foster ,Eason{=B
Eason
Foster
Grand
Albert
0.8 0 0.9
Bill 1 0.8 1
Carl 0.1 0 0.7
David
0.7 0 0.8
Eason
Foster
Grand
Albert
0.4 0.9 0.3
Bill 0 0.4 0
Carl 0.9 0.5 0.8
David
0.6 0.7 0.5
![Page 30: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/30.jpg)
Fuzzy Logic Operate - Union
• Relationship for B are higher OR heavier (Union) => max(A, B)
Eason
Foster
Grand
Albert 0.8
0 0.9
Bill 1 0.8
1
Carl 0.1
0 0.7
David 0.7
0 0.8
Eason
Foster
Grand
Albert 0.
40.9
0.3
Bill 0 0.4
0
Carl 0.9
0.5
0.8
David 0.
60.7
0.5
∪ =Eason Foster Grand
Albert 0.8 0.9 0.9
Bill 1 0.8 1
Carl 0.9 0.5 0.8
David 0.7 0.7 0.8
![Page 31: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/31.jpg)
Fuzzy Logic Operate - Intersection
• Relationship for B are higher AND heavier (Union) => min(A, B)
Eason
Foster
Grand
Albert 0.8
0 0.9
Bill 1 0.8
1
Carl 0.1
0 0.7
David 0.7
0 0.8
Eason
Foster
Grand
Albert 0.
40.9
0.3
Bill 0 0.4
0
Carl 0.9
0.5
0.8
David 0.
60.7
0.5
=Eason Foster Grand
Albert 0.4 0 0.3
Bill 0 0.4 0
Carl 0.1 0 0.7
David 0.6 0 0.5
∩
![Page 32: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/32.jpg)
Composition operationComposition operation
• ExampleJerry} Ister, {Harry,=A
Mickey} Luffy, {Kalman,=B
}Lab {Hockey,C =
Harry
Ister Jerry
Kalman
0.8 0 0.3
Luffy 0.6 0.9 0.9
Mickey
0 0.2 1
Hockey
Lab
Harry 0.9 0
Ister 0.2 0.4
Jerry 0.7 0.9
![Page 33: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/33.jpg)
Composition operationComposition operation
• Where will we meet Kalman, Luffy and Mickey tomorrow?
Harry
Ister Jerry
Kalman
0.8 0 0.3
Luffy 0.6 0.9 0.9
Mickey
0 0.2 1
Hockey
Lab
Harry 0.9 0
Ister 0.2 0.4
Jerry 0.7 0.9
Hockey
Lab
Kalman
0.8 0.3
Luffy 0.7 0.9
Mickey 0.7 0.9
![Page 34: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/34.jpg)
Fuzzy Control
Fuzzy Inference Mechanism
Fuzzification Defuzzification
Fuzzy Knowledge Base
Controlled system
Crisp
Process Output Actual Control
CrispFuzzyTerms
FuzzyTerms
![Page 35: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/35.jpg)
Fuzzy Control
• Fuzzification• Fuzzy Knowledge Base• Fuzzy Inference Mechanism• Defuzzification
![Page 36: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/36.jpg)
Fuzzification
• Fuzzification transforms crisp values into grades of membership for linguistic terms of fuzzy sets.– Example: Shooting Success Rate : input
1->Angle
![Page 37: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/37.jpg)
Fuzzy Knowledge Base
• Use linguistic variables to build the IF-THEN rules in Fuzzy Knowledge Base.
• Example: Shooting Success Rate Knowledge
Base
Distance
Close Normal Far
Angle
Small NormalNo
Chance
NoChanc
e
Normal
GoodChance
NormalNorma
l
BigGood
ChanceGood
Chance
GoodChanc
e
![Page 38: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/38.jpg)
Fuzzy Inference Mechanism
• Combines the fuzzy terms obtained from the fuzzification with the rule base and data base
• Min-Max2222121
21212111
1
Bis THEN Y A is X AND A is X IF:R Bis THEN Y A is X AND A is X IF:R
1X
1X 2X
2X
Y
Y
Y
111 A is x
121 A is x
212 A is x
222 A is x
1 Bis y
2 Bis y*B
![Page 39: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/39.jpg)
Defuzzification – Crisp Value
• Weighted Average Defuzzification
∑∑ ⋅
=
ii
ii
i
w
wmu*
![Page 40: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/40.jpg)
Defuzzification – Fuzzy Set
• Center of Gravity DefuzzificationDiscrete Continuous
∑∑ ⋅
=
ii
iii
x
xxu
)(
)(*
µ
µ
∫∫ ⋅
=
x
x
dxx
xdxx
u)(
)(
*µ
µx
*µ
x*
µ
![Page 41: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/41.jpg)
Defuzzification
0.80.50.4
1 2 3 Output
1
94.14.08.05.0
4.038.025.01 =++
×+×+×
![Page 42: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/42.jpg)
Defuzzification
0.80.50.4
1 2 3 Output
1
94.14.08.05.0
4.038.025.01 =++
×+×+×
![Page 43: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/43.jpg)
Center of Gravity Defuzzification
∑∑
∑∑∑∑
∑∑∑∑
∑
⋅=
=⋅
=−⋅
=⋅−⋅
=⋅−
=⋅
)(
)(
)()(
0)()(
0)()(
0)()(
0)(CoG)-(
x
xxCoG
xCoGxx
xCoGxx
xCoGxx
xCoGx
xx
µµ
µµ
µµ
µµ
µ
µ
![Page 44: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/44.jpg)
Example
KnowledgeBase
Distance
Close Normal Far
Angle
Small NormalNo
Chance
NoChanc
e
Normal
GoodChance
NormalNorma
l
BigGood
ChanceGood
Chance
GoodChanc
e
NormalGoodChanc
e
NoChanc
e
0.5 10
Input 1: Shooting Angle
Input 2: Shooting Distance
Output:Shooting Success Rate
• Use Shooting Angle and Shooting Distance to calculate the Shooting Success Rate for Ice Hockey
![Page 45: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/45.jpg)
Example
Input 1: Shooting Angle
• Input 1: Shooting Angle
![Page 46: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/46.jpg)
Example
Input 2: Shooting Distance
• Input 2: Shooting Distance
![Page 47: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/47.jpg)
Example
• Knowledge Base
KnowledgeBase
Distance
Close Normal Far
Angle
Small Normal NoChance
NoChanc
e
Normal
GoodChance Normal Norma
l
Big GoodChance
GoodChance
GoodChanc
e
![Page 48: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/48.jpg)
Knowledge Base
If Angle is Small and Distance is Close then output = NormalIf Angle is Small and Distance is Normal then output = No ChanceIf Angle is Small and Distance is Far then output = No ChanceIf Angle is Normaland Distance is Close then output = Good ChanceIf Angle is Normal and Distance is Normal then output = NormalIf Angle is Normal and Distance is Far then output = NormalIf Angle is Big and Distance is Close then output = Good ChanceIf Angle is Big and Distance is Normal then output = Good ChanceIf Angle is Big and Distance is Far then output = Good Chance
![Page 49: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/49.jpg)
Example
• Angle = 20, Distance = 90. Output is ?
• Angle = 20, Distance = 81. Output is ?
– (Use Weighted Average Defuzzification)
![Page 50: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/50.jpg)
Angle = 20, Distance = 90
If Angle is Small and Distance is Close then output = NormalIf Angle is Small and Distance is Normal then output = No ChanceIf Angle is Small and Distance is Far then output = No ChanceIf Angle is Normaland Distance is Close then output = Good ChanceIf Angle is Normal and Distance is Normal then output = NormalIf Angle is Normal and Distance is Far then output = NormalIf Angle is Big and Distance is Close then output = Good ChanceIf Angle is Big and Distance is Normal then output = Good ChanceIf Angle is Big and Distance is Far then output = Good Chance
![Page 51: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/51.jpg)
Angle = 20, Distance = 90
Angle “20” is 20% degrees of Small Set and 80% degrees of Normal Set
Distance “90” is 100% degrees of Normal Set
![Page 52: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/52.jpg)
Angle = 20, Distance = 90
4.08.02.0
5.08.002.0 =+
×+×=output
![Page 53: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/53.jpg)
Angle = 20, Distance = 81
458.01.09.01.02.0
11.05.09.05.01.002.0 =+++
×+×+×+×=output
![Page 54: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/54.jpg)
Quiz
• Angle = 15, Distance = 99
![Page 55: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/55.jpg)
A Fuzzy Logic Controller for Car-like Mobile Robots
• Input:– curvature, position error and orientation
error
• Output:– speed and direction
![Page 56: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/56.jpg)
Input - Curvature
![Page 57: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/57.jpg)
Input - Position error
![Page 58: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/58.jpg)
Input - Orientation error
![Page 59: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/59.jpg)
Output - Speed
![Page 60: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/60.jpg)
Output - Direction
![Page 61: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/61.jpg)
Rules
• straightLine & zeroDisp & zeroAngle– Fast & straight
• straightLine & zeroDisp & posHighAngle– Medium speed & sharpLeft
• leftCircle & posLowDisp &negLowAngle– Medium speed & lowLeft
![Page 62: Fuzzy Logic - cs.umanitoba.ca · –Fuzz Inference Mechanism –Fuzzy Knowledge Base –Defuzzification •Hand calculation of Fuzzy Control •Paper of Fuzzy Control. What is Fuzzy](https://reader035.vdocuments.mx/reader035/viewer/2022062509/60f8494697908474373d7463/html5/thumbnails/62.jpg)
Reference
• http://www4.cs.umanitoba.ca/~jacky/Teaching/Courses/COMP_4180-IntelligentMobileRobotics/current/index.php• Ching-Chang Wong, 2010 Fuzzy Control class, TKU, Taiwan.• http://en.wikipedia.org/wiki/Fuzzy_control_system