(英文PPT4)模糊控制讲义第二章(2.5 2.6)

2.6.2 Operation of fuzzy relation
Example 2.6.2
1 0.2 0 R 0.2 1 0.1 0 0.1 1
1 0.2 0 1 0.2 0 1 0.2 0.1 R R 0.2 1 0.1 0.2 1 0.1 0.2 1 0.1 R 0 0.1 1 0 0.1 1 0.1 0.1 1
1 rij rij 0 rij
R
is called the support matrix of
R
.
Notes: The support matrix R is a Boolean matrix.
2.5.2 Support of fuzzy matrix
Example 2.5.4:
2.6.2 Operation of fuzzy relation
Characteristics of fuzzy relation:
1 自反性
x X
Every element in diagonal
R ( x, x) 1
of fuzzy matrix is 1.
RT R
2 对称性
2.5.4 Transpose of fuzzy matrix
Rows change into lines and lines change into rows.
0.5 0.7 Q 0.2 0.8
0.5 0.2 Q 0.7 0.8
T
2.6 Fuzzy Relation
height and weight is in table.
2.6.1 Definition of fuzzy relation
Fuzzy relation between height and weight
~ R
140
Y
40 1 50 0.8 60 0.2 70 0.1 80 0
150
0.8
0.2 0.1
sik (qij r jk ),1 i n,1 k l
j 1
m
The composition of fuzzy matrix is also called as multiplication of fuzzy matrix.
2.5.3 Composition of fuzzy matrix
We can find from the above example that how does fuzzy mathematics describe the fuzzy conception in real world in the form of math. Using this method, we can handle the fuzzy conception in computer.
3 Contain: R S R ( x, y) S ( x, y),( x, y) X Y 4 Equal:
R S R ( x, y) S ( x, y),( x, y) X Y
c
T
5 Complement: R Rc ( x, y) 1 R ( x, y),( x, y) X Y 6 Transpose: R R ( y, x) R ( x, y),( x, y) X Y
( x, y) X X
R ( x, y) R ( y, x)
3 传递性
RR R
R ( x, z ) [ R ( x, y ) R ( y, z )]
y
( x, y), ( y, z ), ( x, z ) X X
2.6.2 Operation of fuzzy relation
1
0.8 0.2
0.8
1 0.8
0.2
0.8 1
0.1
0.2 0.8
X
160 170
180
0
0.1
0.2
0.8
1
2.6.1 Definition of fuzzy relation
1 0 .8 R 0 .2 0 .1 0 0 1 0.8 0.2 0.1 0 .8 1 0 .8 0 . 2 0 .2 0 .8 1 0 .8 0 .1 0 .2 0 .8 1 0 .8 0 .2 0.1
Y y1 , y2 ,, ym ~ An element rij in fuzzy matrix R means the relation between the ith element x i in X and the jth element y j
in Y .
~ R ( xi , y j ) rij
2.6.2 Operation of fuzzy relation
R and S are two fuzzy relations in X Y .
1 Union:
R S [ R ( x, y), S ( x, y)],( x, y) X Y
2 Intersection: R S [ R ( x, y), S ( x, y)],( x, y) X Y
2.6.1 Definition of fuzzy relation
Fuzzy relation can described by fuzzy matrix when X and Y are finite set. When X Y , it is called the fuzzy relation in X .
2.6.1 Definition of fuzzy relation
2.6.2 Operation of fuzzy relation 2.6.3 Composition of fuzzy relation
2.6.1 Definition of fuzzy relation
Relation: A subset R of a Cartesian product of X and Y , is called a dualistic relation from X to Y (or relation for short).
RU ( S T ) ?
0.4 0.3 S 0.6 0.8
(R S ) (R T ) ?
0.7 0.6 T 0.5 0.7
2.5.2 Support of fuzzy matrix
Definition:
R nm [0,1]
R (rij )
X x1 , x2 ,, xn
2.6.1 Definition of fuzzy relation
Example 2.5.6
X is the space of height in an area and Y is the space
of weight.
X ,150,160,170,180 ,(cm), 140 Y 40,50,60,70,80 ,(kg), the relation between
( x, y) R
xRy
( x, y) R
xR y
~ Fuzzy relation: X and Y are two sets, R is a fuzzy ~ subset in X Y, R is also called a fuzzy relation from X to Y .
~ The degree of membership of ( x, y ) is R ( x, y )
2.5.4 Transpose of fuzzy matrix
2.5.1 Basic definitions of fuzzy matrix
If ( i n , j m ), there exist rij [0,1], then means R (rij ) nm is a fuzzy matrix. Usually, nm m all the fuzzy matrixes ( rows, lines). n
2.5.3 Composition of fuzzy matrix
Definition: Q , R , Q (qij ) nm , R (r jk ) ml , the composition matrix S of these two matrixes is a fuzzy matrix with n rows and l lines, written as Q R whose element is
R

nm
0.5,0.7
0 .5 0 . 3 0 .9 R 0 .8 1 0 .2 0 .7 0 0 .4
R 0. 5
1 0 1 1 1 0 1 0 0
R 0 .7
0 0 1 1 1 0 1 0 0
R, S nm
R (rij ) nm S (sij ) nm
Hale Waihona Puke Baidu
1 Union:
R S (rij sij ) nm
2 Intersection: R S (rij sij ) nm
2.5.1 Basic definitions of fuzzy matrix
Example 2.5.5:
0.2 0.5 1 Q 0.7 0.1 0.8
0 .6 0.5 R 0 .4 1 0 .1 0 .9
Q R?
Notes: (popularly)
Q R RQ
(Q R) S (Q S ) ( R S )
3 Complement:
R (1 rij ) nm
c
4 Contain: rij sij (i 1,2,, n; j 1,2,, m) RS 5 Equal:
rij sij (i 1,2,, n; j 1,2,, m)
RS
2.5.1 Basic definitions of fuzzy matrix
CHAPTER 2 Fuzzy Mathematics
2.5 Fuzzy Matrix
2.6 Fuzzy Relation
2.5 Fuzzy Matrix
2.5.1 Basic definitions of fuzzy matrix
2.5.2 Support of fuzzy matrix 2.5.3 Composition of fuzzy matrix
Example 2.6.1 Two people have the fuzzy relations:
“similitude” 自反性（Y）对称性（Y）传递性（N） “enemy” “love”
自反性（N）对称性（Y）传递性（N）
自反性（Y）对称性（?）传递性（?）
“younger” 自反性（N）对称性（N）传递性（Y）
Example 2.5.2:
O, E, R M nm
0.7 0.5 R 0.9 0.2
R R c , R RC ?
2.5.1 Basic definitions of fuzzy matrix
Example 2.5.3:
R, S , T nm
0.7 0.5 R 0.9 0.2
Example 2.5.1:
0.7 0.5 R 0.9 0.2
0.4 0.3 S 0.6 0.8
R S, R S, R ?
C
Notes: A fuzzy matrix and its complement are not complementary events similar with fuzzy sets.