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Fig. 1.3
Membership functions for the angle
x
of the inverted pendulum
Fig. 1.4
Membership functions for the angular velocity of the inverted pendulum
x
1
=
θ
(1.100)
x
1
=
˙
x
2
(1.101)
x
2
=
θ
(1.102)
mlx
2
sen
u
+
(
x
1
)
g
sen
(
x
1
)
−
cos
(
x
1
)(
)
M
+
m
x
2
=
˙
(1.103)
m
cos
2
(
x
1
)
4
(
3
−
)
l
M
+
m
Firstly, the model of the inverted pendulum is estimated in three operation points
for both the angle and its derivative. The universe of discourse of the angle is
[
−
4
,
4
]
rad
seg
. Both membership
functions for the angle x and its derivative ˙xareshowninFigs.
1.3
and
1.4
respec-
tively.
If we apply the TS method directly to this example, then the condition number
of the matrix X is 3.4148e
rad
.
and the one of the angular velocity is
[−
5
,
5
]
015, which shows clearly a non reliable result. Using
the parameters' weighting method with weighting factor
+
γ
=
0
.
01, the inverted
pendulum fuzzy model can be represented as follows:
If
x
1
is
M
1
and
x
2
is
M
2
then
S
11
:
x
2
=−
8
.
1994
+
3
.
4151
x
1
−
0
.
2006
x
2
−
1
.
0443
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
12
:
x
2
=−
˙
8
.
3766
+
3
.
0426
x
1
+
0
.
0000
x
2
−
1
.
0443
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
13
:
x
2
=−
˙
8
.
1994
+
3
.
4151
x
1
+
0
.
2006
x
2
−
1
.
0443
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
21
:
x
2
=−
˙
0
.
0251
+
5
.
5416
x
1
−
0
.
0085
x
2
−
1
.
5332
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
22
:
x
2
=
˙
0
.
0225
+
6
.
1796
x
1
−
0
.
0000
x
2
−
1
.
5332
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
23
:
x
2
=−
˙
0
.
0251
+
5
.
5416
x
1
+
0
.
0085
x
2
−
1
.
5332
u
If
x
1
is
M
1
and
x
2
is
M
2
then
S
31
:
x
2
=
˙
8
.
1236
+
3
.
8634
x
1
+
0
.
2363
x
2
−
1
.
0903
u
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