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M
where
l
l
b
,with
b
¦
and
l
h
is the normalized degree of fulfilment
h
z
z
l
1
(firing strength) of
l
th rule.
Equations (6.11a) - (6.16d) represent the
backpropagation training algorithm
(BPA) for Takagi-Sugeno-type multi-input multi-output neuro-fuzzy networks or
the equivalent fuzzy logic system of form (6.9a) - (6.9c ) with linear fuzzy rules
consequent part as
l
l
l
l
l
l
y
T
"
.
T
x
T
x
T
x
T
x
j
0
j
1
2
j
2
3
j
3
nj
n
1
j
In the above Takagi-Sugeno-type fuzzy rules (linear) consequent, if the
coefficients
i
T
for
i
= 1, 2, 3, ...,
n
;
l
=1, 2, 3, ...,
M
; and
m
= 1, then the
equivalent neuro-fuzzy network is identical with the multi-input, single-output
neuro-fuzzy network described by Wang and Mendel (1992b) and Palit and
Popovic (1999 and 2000a). The resulting fuzzy logic system can be seen as a
special case of both the Mamdani- and -Takagi-Sugeno-type systems, where the
rule consequent is a singleton (constant number) fuzzy set. However, if
l
0,
T
ij
for
i =
1, 2, 3, ...,
n
;
l=
1, 2, 3,
…
,
M
; and for
m =
1, then the resulting fuzzy logic
system is identical with Takagi-Sugeno type multi-input and single-output neuro-
fuzzy network, as described by Palit and Babuška (2001).
It is generally known that the backpropagation algorithm based on steepest
descent rule, in order to avoid the possible oscillations in the final phase of the
training, uses a relatively low learning rate
z
0
K . Therefore, the
backpropagation training usually requires a large number of recursive steps or
epochs. The acceleration of the training process with classical backpropagation,
however, is achievable if the adaptive version of the learning rate or the
momentum version of the steepest descent rule is used:
1
`
^
l
k
1
l
k
K
1
mo
f
l
b
T
T
d
z
0
j
0
j
j
j
(6.17a)
'
mo
l
(
k
)
T
0
j
`
^
l
k
1
l
k
K
1
mo
f
l
b
T
T
d
z
x
ij
ij
j
i
j
(6.17b)
'
mo
l
ij
(
k
)
T
`
^
2
i
l
l
l
k
1
k
K
1
mo
A
2
l
c
c
z
x
c
V
i
i
i
i
(6.17c)
'
mo
i
k
1
c
`
^
3
2
i
l
l
l
k
1
k
K
1
mo
A
2
l
V
V
z
xc
V
i
i
i
i
(6.17d)
'
mo
i
k
1
V
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