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4.2 Example 2
In this subsection the stabilization of a discrete time SISO system is done when
saturation is found in the model. The system to be stabilized is the following:
z
2
10z
þ
1
G
p
ð
Þ
¼
ð
Þ
z
33
with the following saturation,
filter and anti windup compensator parameters
(Table
3
).
Using the formulae shown in Appendix 2and a sampling period of T = 1 s, the
controller gain and parameters are found as shown in Table
4
.
With this control systems parameters, the system output with AWC compen-
sation and with no AWC compensation are shown in the
figure below.
The system response shown in Fig.
11
corroborates that a small overshoot and
small settling time is obtained when an internal model anti windup controller is
implemented, in contrast when there is not anti windup compensation. These results
are expected due to the anti windup compensator reduces the integral action when
the system is saturated, so a smaller overshoot and smaller settling time is obtained
when the internal model controller is implemented.
In Fig.
12
the non saturated input of the system with anti windup compensation
is shown where this signal reaches the necessary output value to obtain the required
value.
In Fig.
13
the non saturated input, when there is no anti windup compensation, is
shown. As can be noticed, the required input signal is applied to the system until the
required output value is obtained.
In Fig.
14
the saturated input value is depicted, where the limiter imposed by the
saturation makes the system to reach the desired value and as it is compared with
Fig.
15
the saturated signal is better when an anti windup controller is implemented.
Table 3 Filter, compensator
and saturation parameters
Parameter
Value
ʱ
0.2
10
−
7
a
0
4
×
10
−
7
a
1
5
×
U
max
10.9
Table 4 Parameters with
AW and no AW
compensation
Parameter
Value with AW
compensation
Value with no AW
compensation
3.4362
×
10
7
3.1062
×
10
8
K
c
˄
1.6869
2
i
10
−
6
˄
0.4265
1
×
d
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