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6.4.1 Neural Network Representation of Fuzzy Logic Systems
The fuzzy logic system considered here for constructing neuro-fuzzy structures is
based on a Takagi-Sugeno-type fuzzy model with Gaussian membership functions.
It uses product inference rules and a weighted-average defuzzifier defined as
M
M
p
l
f
()
x
¦¦
y
z
l
l
,
z
(6.9a)
j
j
l
1
l
1
where
j
= 1, 2, 3, ...,
m
;
l
= 1, 2, 3, ...,
M
;
n
l
y
l
l
,with
i
1,2,3,
"
, ;
n
(6.9b)
T
¦
T
x
0
j
ij
i
j
i
1
and
j
= 1, 2, 3, ...,
m
;
l
= 1, 2, 3, ...,
M
;
`
^
n
2
2
l
l
l
;
exp
(6.9c)
P
P
z
x
x
x
c
V
i
l
l
i
G
i
G
i
i
i
i
i
1
with
i
= 1, 2, 3, ...,
n
.
l
j
, where
i
V
i
Here, we assume that
i
and, are the
input and output universes of discourse respectively. The corresponding
l
th rule
from the above fuzzy logic system can be written as
U
,
!
0,
and
U
y
c
V
V
j
i
j
R
l
: If x
1
is G
l
1
and x
2
is G
l
2
and … and x
n
is G
l
n
l
l
l
l
l
Then
y
T
"
(6.10)
T
x
T
x
T
x
1
2
n
j
0
j
2
j
n
j
1
j
where,
x
i
with
i=
1, 2, …,
n
; are the
n
system inputs, whereas
f
j
, with
j=
1, 2, …,
m
; are its
m
system outputs, and
G
i
l
,
with
i=
1, 2, …,
n
; and
l=
1, 2, …,
M
; are the
Gaussian membership functions of the form (6.9c) with the corresponding mean
and variance parameters
l
y
as the output
l
and
V
respectively and with
l
c
i
i
consequent of the
l
th rule.
It is to be noted that the Gaussian membership functions (
G
i
l
) actually represent
linguistic terms such as
low
,
medium
,
high
,
etc
. The rules (6.10), as specified
above, are known as Takagi-Sugeno rules.
In the fuzzy logic system (6.9a) - (6.9c) the Gaussian membership function is
deliberately chosen because the same membership function is continuously
differentiable at all points. This is an essential requirement to apply the gradient-
method-based training algorithm. Furthermore, it is also important to note that the
fuzzy logic system (6.9a) - (6.9c) is capable of uniformly approximating any
nonlinear function to any degree of accuracy over a universe of discourse
n
U
\
(Wang, 1994).
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