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If
x
1
is
M
1
and
x
2
is
M
2
then
S
32
:
x
2
=
˙
8
.
0313
+
3
.
6645
x
1
−
0
.
0000
x
2
−
1
.
0903
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
33
:
x
2
=
˙
8
.
1236
+
3
.
8634
x
1
−
0
.
2363
x
2
−
1
.
0903
The resultant mean square error from this approximation is 0.0013. In this case, the
condition number of the extendedmatrix
X
a
becomes 1.4569e
+
004, thus improving
the reliability of results.
By using the identification with the classical minimum square method in an inter-
val close to the equilibrium point
x
1
∈[
−
4
,
4
]
x
2
∈[−
2
.
5
,
2
.
5
]
(1.104)
The resultant linear model of the system in this interval is:
x
2
=
˙
0
.
0092
+
15
.
2665
x
1
−
0
.
0000
x
2
−
1
.
4187
u
(1.105)
This approximation gives us an idea of the quantitative relation among various
variables. Comparing these parameters with those obtained in the previous fuzzy
model, it is evident that its values are, in general, far enough from its physical ones.
Moreover, if we compare the parameters of various adjacent subsystems, it can be
clearly verify that they are quite different.
By applying the tuning of parameters method, taking as a reference the parameters
obtained through minimum square method with a weighting factor
γ
=
0
.
001
∗
np
,
where np is the number of samples used in the identification, we obtain:
If
x
1
is
M
1
and
x
2
is
M
2
then
S
11
:
x
2
=
˙
0
.
1642
+
15
.
0162
x
1
−
0
.
3272
x
2
−
1
.
2458
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
12
:
x
2
=
˙
0
.
4847
+
14
.
6365
x
1
+
0
.
0000
x
2
−
1
.
1546
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
13
:
x
2
=
˙
0
.
1642
+
15
.
0162
x
1
+
0
.
3272
x
2
−
1
.
2458
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
21
:
x
2
=−
˙
0
.
0064
+
15
.
4291
x
1
+
0
.
0171
x
2
−
1
.
4291
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
22
:
x
2
=−
˙
0
.
0796
+
15
.
5791
x
1
−
0
.
0000
x
2
−
1
.
4536
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
23
:
x
2
=−
˙
0
.
0064
+
15
.
4291
x
1
−
0
.
0171
x
2
−
1
.
4291
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
31
:
x
2
=−
˙
0
.
0927
+
15
.
1524
x
1
+
0
.
2813
x
2
−
1
.
3232
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
32
:
x
2
=−
˙
0
.
2593
+
14
.
9994
x
1
+
0
.
0000
x
2
−
1
.
2568
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
33
:
x
2
=−
˙
0
.
0927
+
15
.
1524
x
1
−
0
.
2813
x
2
−
1
.
3232
u
which has a mean square error of 0.0332 greater than the previous one but with
more physical relationship between variables. Nevertheless, as it can be seen that
in this model
a
22
0
0796 which does not fulfil an important characteristic of
the existence of an instable equilibrium point which is also the objective of control
x
1
=−
0
.
=
x
2
=
u
=
0. In order that this point becomes an equilibrium in the fuzzy
model,
a
22
0
should be zero.
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