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0
0.5
1
1.5
2
2.5
3
10
10
5
5
0
0
x
2
-5
x
1
-5
-10
-10
Fig. 4.7
Temporal phase portrait
Fig. 4.8
Pendulum
M
θ
τ
The matrices
P
q
,
q
=
1
,...,
4 obtained are:
2
2
.
186 0
.
398
.
176 1
.
885
P
1
=
P
2
=
0
.
398 0
.
126
1
.
885 2
.
581
6
P
4
=
39
.
834 9
.
743
.
31
−
16
.
585
P
3
=
9
.
743 30
.
865
−
16
.
585
11
.
082
Figures
4.6
and
4.7
show, respectively, the outputs obtained for different initial
conditions and temporal phase portrait of the closed loop system formed for the real
plant with the fuzzy controller. Note how all paths converge to the origin of the state
space.
4.4.2 Pendulum
The dynamics of the pendulum in Fig.
4.8
is determined by the Eq. (
4.55
), where
M
is the mass of the pendulum,
L
the length of the bar,
g
the acceleration of gravity,
b
the viscous friction coefficient,
θ
the angle formed by the pendulum relative to
the vertical, and
τ
torque applied on it, and the state variables are
x
1
(
t
)
=
θ(
t
)
)
=
θ(
and
x
2
(
t
t
)
. The following values are considered:
M
=
1 kg,
L
=
0
.
5m,
8m/s
2
g
=
9
.
and
b
=
0
.
5 N/m/s.
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