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Þ
1 p c ð z Þ q ð z Þ
q
ð
z
G c ð
z
Þ ¼
ð
19
Þ
where
p 1
c M
q
ð
z
Þ ¼
ð
z
Þ
f
ð
z
Þ
ð
20
Þ
and the
filter f(z) is given by:
Þ ¼ ð
1
z
f
ð
z
ð
21
Þ
z
a
for a given value of
ʱ
. Meanwhile, the anti windup compensator
filter R(z) is given
by:
1
R
ð
z
Þ ¼
ð
22
Þ
a 1 z
þ
a 0
where a 1 , a 0 > 0. The saturation function is obtained by the scalar sign function
(Zhang et al. 2011 ) taking into account the following sign function representation:
1
if Re(z)
0
[
sign
ð
z
Þ ¼
ð
23
Þ
1
if Re(z)
0
\
so for j = 1 the following representation of the saturation model is implemented:
saturation
ð
z
Þ ¼
U max sign 1 ð
z
Þ
ð
24
Þ
where U max is the saturation limit and
sign 1 ð
z
Þ ¼
z
ð
25
Þ
In this section in order to design the anti windup control system for discrete time
models
the following
first order plus
time delay discrete time model
is
implemented:
k
1 z N
G p ð
z
Þ ¼
ð
26
Þ
s
z
þ
Where k is the system gain,
is the time constant and N > 0 is an integer which
indicates the number of time delays. In order to obtain the internal model controller
it is necessary to get the equivalent transfer function p ʳ
˄
* (z) taking in count the
compensator and saturation in order to obtain this transfer function:
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