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6.7.3 Modelling and Prediction of Wang Data
This example deals with the modelling of a second-order nonlinear plant
y
(
k
)
g
(
y
(
k
1
),
y
(
k
2
))
u
(
k
)
(6.45a)
studied by Wang and Yen (1998, 1999a, and 1999b) and by Setnes and Roubos
(2000 and 2001), with
yk
(
)
yk
(
)( (
yk
)
. )
(6.45b)
gyk
((
1), (
yk
2))
2
2
1(
y
k
)
y
(
k
)
The goal is to approximate the nonlinear component
yg
of the
plant with a suitable fuzzy model. Wang and Yen (1999) generated 400 simulated
data points from the plant model (6.45a) and (6.45b). 200 samples of identification
data were obtained with a random input signal
u
(
k
) uniformly distributed in [-1.5,
1.5], followed by 200 samples of evaluation data obtained by using a sinusoid
input signal
(
(
k
1
),
y
(
k
2
))
as shown in Figure 6.9(a). This example was also
used by Setnes and Roubos (2000 and 2001) and a comparison with the results of
Wang and Yen (1998, 1999a, and 1999b) was made. Here, we also apply the
proposed Takagi-Sugeno-type neuro-fuzzy modelling scheme on the original
Wang-data and show the results for linear rules consequents and compare the
results with others described in the above references.
In order to apply the Takagi-Sugeno-type neuro-fuzzy modelling scheme the
original Wang data (which is available to us in the form of an XIO matrix of size
400 × 3 that contains the first two columns as inputs and the third column as the
desired output) was scaled and normalized down to the range [0, 1] for
convenience. In the following, since our objective is to approximate the nonlinear
component
uk
() sin2
S
25,
yg
of the plant, the same is treated as the desired
output from the neuro-fuzzy network, whereas
u
(
k
) and
y
(
k
) have been considered
as two inputs to the neuro-fuzzy network. The scaling and normalization were
performed separately on each column of the XIO matrix,
i.e.
(
(
k
1
),
y
(
k
2
))
>
@
, and
XIO
uyg
,
,
the
three
column
vectors
>
@
T
,
>
@
T
and
uuu
,
,
"
,
u
yyy
,
,
"
,
y
12
N
12
N
>
@
T
each contains
N
data points. The scaled and normalized vector
ggg
,
,
"
,
g
,
12
N
>
@
T
uKuu
(
), (
uu
),
"
,(
uu
)
u
,
1
min
2
min
N
min
nsc
0
lo
(6.46)
K
)
)
(
u
(
u
u
u
lo
min
0
hi
max
is then computed where
u
max
and
u
min
are the maximum and minimum values of the
u
vector, and
u
hi
= 1 and
u
lo
=
0 are the desired highest and lowest values of the
scaled or normalized
u
nsc
vector.
Once the scaling/normalization is performed, the scaled/normalized data are fed
to the neuro-fuzzy network with
n
= 2 inputs and
m
= 1 output for training. Once
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