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amlx
2
(
)
=
g
sin
(
x
1
(
t
))
−
t
)
sin
(
2
x
1
(
t
))/
2
−
a
cos
(
x
1
(
t
))
u
(
t
)
x
2
(
˙
t
aml
cos
2
4
l
/
3
−
(
x
1
(
t
))
where
x
1
(
is the
gravity,
m
is the mass of the pendulum,
M
is the mass of the cart,
l
is the length of
the pendulum,
u
t
)
is the angle of the pendulum,
x
2
(
t
)
is the angular velocity;
g
.
A fuzzy model which approximates the dynamics of the nonlinear plant for the
range
x
1
∈
(
−
ˀ/
(
t
)
is the force applied to the cart and
a
=
1
/(
m
+
M
)
2
, ˀ/
2
)
can be realized by the following two rules,
R
1
:
IF
x
1
is
about
0THEN
x
˙
(
t
)
=
A
1
x
(
t
)
+
B
1
u
(
t
)
R
2
:
IF
|
x
1
|
is
about
ˀ/
2THEN
x
˙
(
t
)
=
A
2
x
(
t
)
+
B
2
u
(
t
)
where,
0
1
0
A
1
=
B
1
=
g
g
0
−
4
l
/
3
−
aml
4
l
/
3
−
aml
0
1
0
A
2
=
B
2
=
g
ˀ(
4
l
/
3
−
aml
ʲ
2
a
ʲ
0
−
2
4
l
/
3
−
aml
)
88
ⓦ
)
and
ʲ
=
cos
(
. The membership functions are
R2
R1
x1
-90
90
0
Through the PDC control, the stability of the system is guaranteed for example
with
F
1
=
[
−
120
.
67
−
22
.
67]
,
F
2
=
[
−
2551
.
6
−
764]
where it is obtained
3
.
62 0
.
62
P
=
.
0
.
62 0
.
28
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