M.Yamakita

1
Engineering Education Methodologyon Intelligent
Control(Fuzzy Logic and Fuzzy Control)

• M.Yamakita
• Dept. of Mechanical and Control Systems Eng.
• Tokyo Inst. Of Tech.

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Natural Reasoning
IF he/she is Tall, THEN his/her foot is Big. IF
his/her foot is Big, THEN his/her shoes are
Expensive.
IF he is Tall, THEN his shoes are Expensive.
IF he/she is Tall, THEN his/her foot is Big. Mr.
Smith is Tall.

Mr. Smiths foot is Big.
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• Crisp Expert System

Inference (Reasoning)
Formal Logic
A ? B
A ? B B ? C
IF A THEN B A is true
IF A THEN B
A
IF B THEN C
A ? C
B
B is true
IF A THEN C
Modus Ponens
Hypothetical Syllogism
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Crisp Logic
Tall

Mr.A
181cm
(
170cm

(
Mrs.B
177cm
Short
(
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IF he is Tall, THEN his foot is Big.
Mrs. B is Tall.
Mrs. Bs foot is Big.

Mr. A is Very-Tall.
Mr.As foot is Very-Big.
Mr. A is Very-Tall.
Natural Reasoning
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Fuzzy Logic (Fuzzy Inference)
B
A
A ? B
B is true
A ? B A
A ? B B ? C
B
IF A THEN B
B ? C
IF B THEN C
B
IF A THEN C
A ? C
A ? C
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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
190
180
Height
Tall
Very Tall
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Representation of Fuzzy Set
1.Countable Set
2. Uncountable Set
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Example
1. Countable Case
Membership Function
1.0
0.5
x
170
190
180
Height
Tall
Very Tall
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1. Uncountable Case
Membership Function
1.0
0.5
x
170
190
180
Height
Tall
Very Tall
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Fuzzy Set Operations
1. Implication
2. Union
3. Intersection
4. Compliment
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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
fuzzy relation R of
is
In general,
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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
Z
Y
is
If A is a fuzzy set and R is a fuzzy relation,
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Fuzzy Inference
Direct Method (Mamdani)
(Max-Min Composition)
Caution! A and B are Fuzzy Sets.
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ATall BBig AVery 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.
ATall BBig A178 B is still Fuzzy Set
A is not fuzzy set or Defuzzy value
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Fuzzy Control
Rules
C
B
A


is

z
then

is
y

and


is
x
If

1

Rule
1
1
1
C
B
A


is

z
then

is
y

and


is
x
If

2

Rule
2
2
2
.




.



C
B
A

n
n
n
Input

is

x
Output
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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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Triangular Membership Function
Example
If x is NS, and y is PS, then z is PS If x is ZO,
and y is ZO, then z is ZO
R1
R2
NS
R1
PS
ZO
ZO
ZO
R2
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Simplification
NS
R1
PS
ZO
ZO
ZO
R2
Further Simplification (Height Method)
NS
PS
R1
PS
ZO
ZO
ZO
R2
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TS(Takegaki-Sugeno)Model

• Singleton Fuzzifier
• Product Inference
• Weighted Average Deffuzifier

PM
PS
R3
PS
PS
PS
R4
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References

• S.Murakami Fuzzy Control , Vol. 22, Computer and
  Applications 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)