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By carefully observing the functional forms (6.9a) - (6.9c), it can be seen that
the above fuzzy logic system can be represented as a three-layer multi-input, multi-
output feedforward network as shown in Figure 6.6. Because of the neuro
implementation of the Takagi-Sugeno type fuzzy logic system, Figure 6.6 actually
represents a Takagi-Sugeno-type of multi-input, multi-output neuro-fuzzy network,
where, instead of the connection weights and the biases as in backpropagation
neural networks, we have the mean (
c
i
) and the variance (
V
i
) parameters of
TT
),
i.e
.
y
l
j
from the rules
consequent, as the equivalent adjustable parameters of the network.
If the adjustable parameters of the neuro-fuzzy network are suitably selected,
then the above fuzzy logic system can correctly approximate any nonlinear system
based on given input-output data pairs.
Gaussian membership functions, along with (
l
0
,
l
j
ij
6.4.2 Training Algorithm for Neuro-fuzzy Network
The fuzzy logic system, once represented as the equivalent multi-input, multi-
output feedforward network (Figure 6.6), can generally be trained using any
suitable training algorithm, such as the standard backpropagation algorithm (Palit
et al.
, 2002) that is generally used for neural networks training. However, because
of its relatively slow speed of convergence, this algorithm needs to be further
improved. Alternatively, a more efficient second-order training algorithm, such as
the Levenberg-Marquardt algorithm described in the Section 6.4.2.3, can also be
used.
6.4.2.1 Backpropagation Training of Takagi-Sugeno-type Neuro-fuzzy Network
Let a set of
N
input-output data pairs
p
p
d
,
, with
p
= 1, 2, 3, ...,
N
; and
x
j
p
p
d
p
{
p
,
x
,
"
,
p
n
U
\ , and
n
\ is given. The objective is to
m
x
x
x
j
1
2
j
determine a fuzzy logic system
p
f
in the form of (6.9a) - (6.9c), such that the
x
j
performance function
S
, defined as
2
N
j
S
0.5
¦
0.5
T
j
(6.11a)
e
E E
j
j
p
1
m
and
S
¦
S
"
,
(6.11b)
S
S
S
j
1
2
m
j
1
`
is minimized, where
E
j
is the column vector of errors
^
p
p
d
f
()
p
, and
p =
e
x
j
j
j
1, 2, ..., N
; for the
j
th output from the fuzzy logic system. In addition, we also
assume that the number of fuzzy rules and also the number of membership
functions (to be implemented)
M
are given. In this way the problem is reduced to
the adjustment of
y
l
j
,
i.e.
the parameters (
l
ij
j
,
) from the rules consequent and
T
T
0
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