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Table 1 Estimated parameters
True values
Chiu (n
q
¼
20)
DBSCAN (n
q
¼
21)
k-means (n
q
¼
7)
2
4
3
5
2
4
3
5
2
4
3
5
2
4
3
5
h
1
0
:
4
0
:
5
0
0
:
4046
0
:
5138
0
0
:
4054
0
:
4903
0
0
:
4064
0
:
5464
0
:
3
:
2919
:
2992
:
2598
2
4
3
5
2
4
3
5
2
4
3
5
2
4
3
5
h
2
0
:
7
6179
0
:
5336
0
:
4740
0
:
7369
0
:
6675
0
:
5239
0
:
6955
0
:
5903
0
:
4939
0
:
0
:
6
0
:
5
2
4
3
5
2
4
3
5
2
4
3
5
2
4
3
5
h
3
0
:
4
0
4015
:
0
4679
:
0
:
4792
0
:
2
0
:
2071
0
:
1977
0
:
2101
0
:
2
0
:
2042
0
:
2298
0
:
2406
After obtaining the estimated parameter vectors, we apply the SVM algorithm in
order to estimate the regions. We can then attribute each parameter vector to the
corresponding region where it is de
ned. The estimated outputs obtained with three
algorithms are presented in Fig.
2
.
Table
2
presents the quality measures (
18
), (
19
) and (
20
) of the two proposed
methods and the k-means method. The obtained results prove the ef
ciently of the
proposed methods compared with the existing method (k-means).
5.3 Identi
cation Results of a Nonlinear Model
Consider the nonlinear system described by the following equation (Lai et al.
2010
):
1
:
5y
ð
k
1
Þ
y
ð
k
2
Þ
ð
Þ
¼
ð
k
2
Þ
þ
ð
ð
Þþ
ð
Þ
Þ
y
k
sin y
k
1
y
k
2
1
þ
y
2
ð
k
1
Þþ
y
2
ð
Þ
24
þ
u
ð
k
1
Þþ
0
:
8u
ð
k
2
Þ
This nonlinear system can be modeled by a PWARXmodel of the form (Lai
2011
):
8
<
T
1
h
uð
k
Þ
if
u 2
H
1
.
h
ð
Þ
¼
ð
Þ
y
k
25
:
T
s
uð
k
Þ
if
u 2
H
s
where
T
uð
k
Þ
¼
½
y
ð
k
1
Þ;
y
ð
k
2
Þ;
u
ð
k
1
Þ;
u
ð
k
2
Þ
ð
26
Þ
T
u
¼
u
T
:
ð
27
Þ
1
h
i
are the parameter vectors and s is the number of submodels to be determined. u
(k) is a random input in the range of [
2, 2].
−
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