Graphics Reference
In-Depth Information
In each universe of discourse,
U
i
and
Y
i
and
u
i
and
y
i
take on values with correspond-
ing linguistic variables
u
i
and
y
i
that describe the characteristics of the variables.
denotes the
j
th linguistic value of the
u
i
linguistic variable defined over
the universe of discourse
U
i
. If we assume that many linguistic values are defined
in
U
i
, the linguistic variable
u
i
Suppose
A
i
j
that takes on the elements from the set of linguistic
values may be denoted by Equation (4.2).
{
}
j
AAj
=
:
=
12
,, ,
…
N
(4.2)
i
i
i
In the same manner, we can consider that
B
i
j
to denote the
j
th value of the linguistic
may be represented by ele-
ments taken from the set of linguistic values denoted by the following equation.
defined over the universe of discourse
Y
i
.
y
i
variable
y
i
{
}
p
BBp
=
:
=
12
,, ,
…
M
(4.3)
i
i
i
Given a condition where all the premise terms are used in every rule and a rule is
formed for each possible combination of premise elements, we have rule set with
N
i
number of rules that can be expressed as:
n
∏
NNNN
=⋅
2
…
⋅
⋅
(4.4)
i
1
n
i
=
1
Based on the membership functions, the conversion of a crisp input value into its
corresponding fuzzy value is known as fuzzification. The defuzzification of the
resultant fuzzy set from the inference system to a quantifiable value may be done
using the centroid (centre of gravity) method [43]. The principle is to select the value
in the resultant fuzzy set such that it would lead to the smallest error on average
given any criterion. To determine
y
*
, the least square method can be used and the
square of the error is accompanied by the weight of the grade of the membership
function
µ
B
()
. Therefore, the defuzzified output
y
*
may be obtained by finding a
solution to the following equation.
(
)
∫
µ
2
()
*
*
y
=
argmin
*
y
yydu
−
(4.5)
B
y
Differentiating with respect to
y
*
and equating the derivative to zero yields:
∫
∫
()
yydy
µ
B
*
=
y
Y
(4.6)
()
µ
ydy
B
Y
which gives the value of the abscissa of the centre of gravity of the area below the
membership function
µ
B
()
.
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