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is calculated in Layer 2. With the use of the Gaussian membership function, the
operations performed in this layer are
(
2
)
2
(
u
m
)
ij
ij
,
(8.10)
2
V
(
2
)
a
e
ij
where m ij and V ij are, respectively, the center (or mean) and the width (or variance)
of the Gaussian membership function of the j th term of the i th input variable x i .
Unlike other clustering-based partitioning methods, where each input variable has
the same number of fuzzy sets, the number of fuzzy sets of each input variable is
not necessarily identical in the SONFIN.
Layer 3: A node in this layer represents one fuzzy logic rule and performs pre-
condition matching of a rule. Here, we use the following AND operation for each
Layer-3 node,
–
()
3
()
3
a
u i
,
(8.11)
i
where n is the number of Layer-2 nodes participating in the IF part of the rule.
Layer 4: This layer is called the consequent layer. Two types of nodes are used
in this layer, denoted blank and shaded circles in Fig. 8.9, respectively. The node
denoted by a blank circle (blank node) is the essential node representing a fuzzy
set (described by a Gaussian membership function) of the output variable. Only
the center of each Gaussian membership function is delivered to the next layer for
the LMOM (local mean of maximum) defuzzification operation [15], and the
width is used for output clustering only. Different nodes in Layer 3 may be con-
nected to the same blank node in Layer 4, meaning that the same consequent fuzzy
set is specified for different rules. The function of the blank node is
,
(
4
)
(
4
)
¦
a
u
a
(8.12)
0
i
j
j
where a 0i = m 0i , the center of a Gaussian membership function. As to the shaded
node, it is generated only when necessary. Each node in Layer 3 has its own cor-
responding shaded node in Layer 4. One of the inputs to a shaded node is the out-
put delivered from Layer 3, and the other possible inputs (terms) are the input
variables from Layer 1. The shaded node function is
,
(
4
)
¦
(
4
)
a
a
x
u
(8.13)
ji
j
i
j
where the summation is over all the inputs and a ji is the corresponding parameter.
Combining these two types of nodes in Layer 5, we obtain the whole function
performed by this layer for each rule as
¦
a
(
4
)
(
a
x
a
)
u
(
4
)
.
(8.14)
ji
j
0
i
i
j
Layer 5 : Each node in this layer corresponds to one output variable. The node
integrates all the actions recommended by Layers 3 and 4 and acts as a defuzzifier
with
(
5
)
¦
(
4
)
¦
(
3
a
a
a
.
(8.14)
i
i
i
i
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