(英文PPT5)模糊控制讲义第二章(2.7 2.8)

2.8 Fuzzy Logic and Fuzzy Reasoning
2.8.1 Fuzzy logic 2.8.2 Fuzzy linguistic operator 2.8.3 Fuzzy linguistic variable 2.8.4 Fuzzy reasoning
2.8.1 Fuzzy logic
3 Determinant operator (判定化算子 判定化算子) 判定化算子
∆ ~ ~ (P A)(u) = dα [ A(u)] α
Where Pα is the determinant operator, dα is a real function in [0，1].
x ≤α 0 1 1 d α ( x) = α < x ≤ 1 − α (0 < α ≤ ) 2 2 x > 1−α 1
0 u > 30 [倾向young](u) = P1 [ young](u) = d 1 ([ young](u)) = 1 u ≤ 30 2 2
2.8.3 Fuzzy linguistic variable
Fuzzy linguistic variable is first brought up by Zadeh. Linguistic variable is the word of sentence in natural language, and not the value in math. Definition: A linguistic variable could be written as:
λ > 1 , H λ is
called concentrated operator. (集中化算子)
H
5 4
为相当，H 2 为很， H 4 为极.
2.8.2 Fuzzy linguistic operator
Example 2.8.1: [相当老]（u ）＝
~ ~ ( H 5 A )( u ) = [ A ( u )] =
P : He is fond of English. Example: Q : He is fond of French.
The conjunction “or”, “and”, “not”, “if … then …” are often used to union two propositions into one.
is called the transpose of
v a.
2.7.1 Definition of fuzzy vector
~ We can express the fuzzy set A in finite space U after introducing the fuzzy vector. ∆ ~ a i = µ A ( u i ), i = 1, 2 , L , n
CHAPTER 2 Fuzzy Mathematics
2.7 Fuzzy Vector 2.8 Fuzzy Logic and Fuzzy Reasoning
2.7 Fuzzy Vector 2.7.1 Definition of fuzzy vector 2.7.2 Cartesian product of fuzzy vector 2.7.3 Dot product of fuzzy vector
2.8.2 Fuzzy linguistic operator
1 Tone operator (语气算子 语气算子) 语气算子 Definition:
~ A
∆ ~ ~ λ ( H λ A)(u ) =[ A(u )]
is a fuzzy set in U
,
Hλ
is tone operator,
λ is a positive real number.
~ ~ Definition： ( FA )( u ) = ( E o A )( u ) = ∨ ( E (u , v ) ∧ A ( v )) ：
v∈U ∆
~ Where F is the fuzzy operator. E is a relation in U . ~ E is usually a normal school. −(u−v)2 e ~ | u − v |< δ E(u, v) = 0 | u − v |≥ δ
2.8.1 Fuzzy logic
The above examples are single proposition. It is often to unite two single propositions into one, which is called as a composite proposition. A proposition is usually written as a capital.
. ,
v a = (0.2,0.4,0.6,0.8)
v b = (0.7,0.5,0.3,0.2)
Note: The Cartesian product of fuzzy vector is a fuzzy matrix. And the dot product of fuzzy vector is a value.
Where δ is a parameter that reflects the degree of fuzziness.
2.8.2 Fuzzy linguistic operator
Example 2.8.2:
A(u ) is a certain number 4.
1 u = 4 A(u ) = 0 u ≠ 4
−(u−4)2
e ~ ~ (FA)(u) = (E o A)(u) = ∨ (E(u, v) ∧ A(v)) = v∈U 0
| u − 4 |< δ | u − 4 |≥ δ
( FA)(u ) is a fuzzy number that means round about 4.
2.8.2 Fuzzy linguistic operator
2.7.2 Cartesian product of fuzzy vector
v then the Cartesian product of fuzzy vectors a v and b is v v v∆ v
v Definition: a ∈ M 1×n
,
v b ∈ M 1×m
a×b =a ob
Some words are used in language in order to emphasize the affirmative tone, such as, very, quite, best. There are some others are often used to make a word fuzzy, such as, maybe, probably. And correspondingly, some words like “incline” are used to make something positive. All these words are called as fuzzy linguistic operators.
=
2.8.2 Fuzzy linguistic operator
λ < 1 , H λ is
H
1 4
called loose operator. (散漫化算子) 为微， H 1 为略， H 3 为比较.
2
4
2.8.2 Fuzzy linguistic operator
2 Fuzzy operator（模糊化算子） （模糊化算子）
4
5 4
0 5 u − 50 − 2 − 4 [1 + ( 5 ) ]
0 ≤ u ≤ 50 50 < u ≤ 200
~ ~ u ）＝ ( H 4 A)(u ) = [ A(u )] 4 [极老]（
0 [1 + ( u − 50 ) − 2 ] − 4 5 0 ≤ u ≤ 50 50 < u ≤ 200
2.8.1 Fuzzy logic
The above example (3) is discussed again here in order to derivate the fuzzy proposition. Here “high temperature” is a fuzzy conception apparently. So the sentence include fuzzy conception is called fuzzy proposition. Such sentence is very common in our life. For example:
~ P : He runs very quickly.
We use a value in [0,1] to describe the degree of how true or false a fuzzy proposition is.
2.8.2 Fuzzy linguistic operator
“1” is usually used to express that a proposition is true and “0” means false. So it is called the dualistic logic. And math studying on dualistic logic is called Boolean algebra or Logic algebra.
2.8.2 Fuzzy linguistic operator
Example 2.8.3: The membership function of fuzzy set “young” is