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To achieve this target, we can either fix the value of this parameter or assign it a
very large weighting factor as explained in Sect.
1.4.2
. In this case, a weighting factor
of 100
γ
has been assigned to this parameter and finally the fuzzy model becomes:
If
x
1
is
M
1
and
x
2
is
M
2
then
S
11
:
x
2
=
˙
0
.
1642
+
15
.
0164
x
1
−
0
.
3271
x
2
−
1
.
2458
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
12
:
x
2
=
˙
0
.
4848
+
14
.
6366
x
1
+
0
.
0002
x
2
−
1
.
1546
u
:
If
x
1
is
M
1
and
x
2
is
M
2
then
x
2
=
0
.
1642
+
15
.
0162
x
1
+
0
.
3272
x
2
−
1
.
2458
u
S
13
If
x
1
is
M
1
and
x
2
is
M
2
then
S
21
:
x
2
=−
.
+
.
4272
x
1
+
.
0172
x
2
−
.
0
0073
15
0
1
4291
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
22
:
x
2
=
15
.
5778
x
1
−
0
.
0003
x
2
−
1
.
4536
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
23
:
x
2
=−
0
.
0072
+
15
.
4287
x
1
−
0
.
0170
x
2
−
1
.
4291
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
31
:
x
2
=−
˙
0
.
0001
+
15
.
1478
x
1
+
0
.
3000
x
2
−
1
.
3232
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
32
:
x
2
=−
˙
0
.
2646
+
14
.
9965
x
1
+
0
.
0080
x
2
−
1
.
2568
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
33
:
x
2
=−
˙
0
.
0942
+
15
.
1516
x
1
−
0
.
2821
x
2
−
1
.
3232
u
The minimum square error is 0.0333 which is slightly greater than that of the
previous case but it represents the equilibrium point of the physical system, i.e.,
a
22
0
0. It can be clearly noticed in this model that the distance between adjacent
subsystem is relatively small.
=
1.7 Conclusions
New efficient approaches have been presented to improve the local and global
approximation of TS fuzzy model. The main problem of TS identification method is
that it can not be applied when the membership functions are overlapped by pairs.
This restricts the use of the TS method because this type of membership function has
been widely used during the last two decades in the stability, controller design and
are popular in industrial control applications. The approaches developed here can
be considered as a generalized version of TS method with optimized performance
in approximating nonlinear functions. We propose a non iterative method through a
weighting of parameters approach and an iterative algorithmby applying the extended
Kalman filter, based on the same idea of parameters' weighting. We show that the
Kalman filter is an effective tool in the identification of TS fuzzy model. An Illus-
trative example of an inverted pendulum has been chosen to evaluate the robustness
and remarkable performance of the proposed method and the high accuracy obtained
in approximating nonlinear and unstable systems locally and globally in comparison
with the original TS model. Simulation results have shown the potential, simplicity
and generality of the algorithm.
Acknowledgments
The authors would like to thank the Spanish Ministry of Economy and
Competitiveness for its support to this work through projects DPI2010-21247-C02-01 andDPI2010-
17123, the Regional Government of Andalusia (Spain) for supporting TEP-6124 project, as well as
the European Union Regional Development for funding the last two projects.
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