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In-Depth Information
T
and x
1
(k) is the angular velocity and x
2
(k) is the
armature current u(k) is the input voltage.
For the discrete time SOF the following values of F and the gain matrices k
p
, k
I
and k
D
are obtained by solving the LMI (
73
) with a
where x
ð
k
Þ
¼½
x
1
ð
k
Þ;
x
2
ð
k
Þ
ʓ
value of
0
:
002
0
C
¼
ð
78
Þ
0
0
:
002
3
:
1623
0
3
:
1623
0
3
:
1623
0
10
8
F
¼
0
3
:
1623
0
3
:
1623
0
3
:
1623
ð
79
Þ
and the following PID controller gain matrices are:
3
:
1623
0
10
8
k
p
¼
0
3
:
1623
10
10
8
k
D
¼
ð
80
Þ
0
1
50
0
10
8
k
I
¼
0
50
The same PID controller gains are implemented for the system when there is no
anti windup compensation. In the case of the PID anti windup controller by H
∞
synthesis the following matrix F is obtained by solving the LMI shown in (
75
)
3
:
1604
0
3
:
1604
0
3
:
1604
0
10
8
F
¼
0
3
:
1604
0
3
:
1604
0
3
:
1604
ð
81
Þ
3
:
1604
0
10
8
k
p
¼
0
3
:
1604
1
:
5802
0
10
8
ð
Þ
k
D
¼
82
0
1
:
5802
1
:
5802
0
10
8
k
I
¼
0
1
:
5802
With these results, a numerical simulation of the DC motor with anti windup and
no anti windup compensation was done with the PID anti windup controller gain
matrices achieving the following outcome.
In Fig.
23
the respective angular velocities when a anti windup and no anti
windup controllers are implemented in the feedback loop of the DC motor, the
system is stabilized at
the nominal speed 1,750 RPM (183.26 rad/s) when a
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