Information Technology Reference
In-Depth Information
2.2
Antecedent Unit
The antecedent unit is used for finding the degrees of truth for the antecedents. To
find the degree of truth, it is necessary to store the membership functions for the fuzzy
sets. The membership functions need not have linear edges and have to be approx-
imated for storing them. They are approximated with piece-wise linear segments as
shown in Fig. 3. The approximation has been done in such a way that the ordinates of
the end-points of the linear-segments are multiples of
Y
and need not be stored. The
abscissas of the end-points of the linear segments are stored. For each membership
function there are 8 abscissas.
Fig. 3.
Membership Function
Fig. 4.
Intersecting membership functions
For a particular truth value
ʱ
, it is necessary to calculate the
ʱ
-cut. It is possible to
know the segments that
ʱ
intersects with, depending on the ordinates of the segment
end-points between which
ʱ
lies. The left and right end-points of the
ʱ
-cut can then be
calculated as
X
L
(
ʱ
) =
X
Lni
+
p
Lni
(
ʱ
-
Y
ni
) .
(1)
X
R
(
ʱ
) =
X
Rni
-
p
Rni
(
ʱ
-
Y
ni
) .
(2)
Where
p
Lni
=
-
.
(3)
p
Rni
=
|
-
|
.
(4)
The truth space of the antecedents is represented by 8 bits. The degree of truth for
an antecedent is calculated by finding the maximum value of
ʱ
for which the
ʱ
-cut of
the antecedent and input has a non-null intersection. This is done by first identifying
the segments of the membership functions which intersect followed by finding the
intersection point of these segments. In Fig. 4, are given 2 membership functions A
and B, whose intersection point needs to be found out. To identify the intersecting
segments, the first step is to find out the membership function nearer to the origin.
Depending on that, the upper end-points of the intersecting segments will satisfy one
of the equations
X
RA
X
LB
(A is nearer to the origin) or
X
RB
X
LA
(B is nearer to the origin) .
(5)
