1. Dept. of Mechanical and Control Systems Eng.
Engineering　Education　Methodology on Intelligent Control (Fuzzy Logic and Fuzzy Control)
M.Yamakita
Dept. of Mechanical and Control Systems Eng.
Tokyo Inst. Of Tech.
2019/2/24

2. Natural Reasoning IF he/she is Tall, THEN his/her foot is Big.
IF his/her foot is Big, THEN his/her shoe’s are Expensive.
IF he is Tall, THEN his shoe’s are Expensive.
IF he/she is Tall, THEN his/her foot is Big.
Mr. Smith is Tall.
Mr. Smith’s foot is Big.
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3. Crisp Expert System Inference (Reasoning) Formal Logic A → B A → B
Condition part
(Antecedent part)
Implication operator
Conclusion
(Operation part)
A　→　B
B　→　C
IF　A　THEN　B
A　→　B
IF　A　THEN B
A　is true
IF　B　THEN　C A
A　→　C
IF　A　THEN　C B
B is true
Hypothetical　Syllogism
Modus　Ponens
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4. Crisp Logic （ （ Mr.A Mrs.B Tall ] 181ｃｍ １７０ｃｍ ] 177ｃｍ Short (
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5. IF he is Tall, THEN his foot is Big.
Mrs. B is Tall.
Mrs. B’s foot is Big.
Mr. A is Very-Tall. ×
Mr. A is Very-Tall.
Mr.A’s foot is Very-Big.
Natural Reasoning
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6. Fuzzy Logic （Fuzzy Inference)A’ B’
A　→　B
A　→　B A’ B’
IF　A　THEN B
A’　is true
B’ is true
A　→　B A’
A　→　B
B　→　C
IF　A　THEN　B B’
IF　B　THEN　C
B 　→　C B’
A’　→　C’
IF　A’　THEN　C’
A’　→　C’
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7. How To Realize Fuzzy Inference ?
Introduction of membership function !
We consider a member of a set as well as the degree of the membership.
Degree of
property
100%
50%
30% ) x
170
180
190
Height
Tall
Very Tall
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8. Representation of Fuzzy Set
１．Countable Set
2. Uncountable Set
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9. Example 1.0 0.5 x 170 180 190 Height Tall Very Tall 1. Countable Case
Membership
Function
1.0
0.5 x
170
180
190
Height
Tall
Very Tall
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10. 1.0 0.5 x 170 180 190 Height Tall Very Tall 1. Uncountable Case
Membership
Function
1.0
0.5 x
170
180
190
Height
Tall
Very Tall
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11. Fuzzy Set Operations 1. Implication 2. Union 3. Intersection
4. Compliment
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12. Fuzzy Relation Definition [Fuzzy Relation]
Let　assume that X and Y are sets. Fuzzy relation R of X and Y is
a fuzzy subset of X x Y as
In general,
fuzzy relation R of is
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13. Composition of Relations
Definition [Composition of Fuzzy Relations]
Let R and S are fuzzy relations, i.e.,
Composition of fuzzy relations, R and S, is a fuzzy set defined by R S X Y Z
If A is a fuzzy set and R is a fuzzy relation, is
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14. Fuzzy Inference A → B A’ B’ Caution! A’ and B’ are Fuzzy Sets.
Direct Method (Mamdani)
A　→　B A’ B’
IF　A　THEN B
A’　is true
B’ is true
（Max－Min　Composition)
Caution! A’ and B’ are Fuzzy Sets.
2019/2/24

15. If he/she is tall then his/her foot is big. He is very tall.
A=Tall
B=Big
A’=Very Tall
If he/she is tall then his/her foot is big.
He is very tall.
If he/she is tall then his/her foot is big.
He is 178cm tall.
A=Tall
B=Big
A’=178
B’ is still Fuzzy Set
A’ is not fuzzy set or Defuzzy value
2019/2/24

16. Fuzzy Control B A A B C A B C A B C Rules Input Output x is is and y1 : If x is A
and y is = B
then z is C 1 1 1
Rule 2 : If x is A
and y is = B
then z is C 2 2 2 . . is z
then y
and x If : n
Rule = A B C n n n
Input x is B A is
and o y
Output
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17. Defuzzication Control Input is Number Defuzzication
If x and y are defuzzy values,
This operation is sometimes replaced by x (multiplication)
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18. Triangular Membership Function-1 1 NM NS NB ZO PS PM PB
Example
R1:
If x is NS, and y is PS, then z is PS
If x is ZO, and y is ZO, then z is ZO
R2: NS PS PS R1 PS ZO ZO ZO R2
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19. Further Simplification (Height Method)NS PS PS R1 PS ZO ZO ZO R2
Further Simplification (Height Method) NS PS PS R1 PS ZO ZO ZO R2
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20. TS（Takegaki－Sugeno）Model
Singleton Fuzzifier
Product　Inference
Weighted Average Deffuzifier PM PS PS R3 PS PS PS R4
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21. References
S.Murakami: Fuzzy Control , Vol. 22, Computer and Application’s Mook, Corona Pub.(1988) in Japanese
K.Hirota: Fuzzy !?, Inter AI (Aug,88-June,90) in Japanese
S.S.Farinwata et. Ed.: Fuzzy Control, Wiley (2000)
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