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Fig. 3 Anti windup IMC PID
architecture
G p
1 s
)
r +
-
G sat
( s
)
G c
( s
)
G p
( s
)
+
~ 1 s
G p
(
)
+
Then the equivalent internal model anti windup control system is shown in
Fig. 3 . Where G p1 is is the equivalent plant given by G p1 (s)=G sat (s)G p (s)
G p is is represented by a
first order plus time delay function given by:
ke h s
s
G p ð
s
Þ ¼
ð
4
Þ
s
þ
1
where k is the gain of the transfer function,
θ
is the time delay and
˄
is the time
constant of the transfer function.
After
finishing the explanation of the anti windup controller by implementing a
model of the saturation nonlinearity, the IMC PID anti windup controller design can
be derived using the equivalent transfer functions of the original system, consid-
ering the saturation effects on the model. To start this process it is necessary to
obtain the equivalent transfer function of the anti windup controller, basically after
obtaining this transfer function G p1 , the design of the IMC PID controller is
straightforward because the equivalent transfer function is completely linear due to
the implementation of an equivalent model of the saturation nonlinearity. Consid-
ering the equivalent transfer function G p1
e h s
k
ða þ bD / Þð
a 1 s
þ
a 0 Þ
G p1 ð
s
Þ ¼
ð
5
Þ
ð
a 1 s
þ
a 0 ða þ bD / ÞÞðs
s
þ
1
Þ
riou 1989 ; Shamsuzzoha
and Lee 2007 ) dividing first the transfer function G p1 into two parts as the process
for designing a IMC controller with anti windup properties
Then an IMC controller is obtained (Morari and Za
G p1 ð
s
Þ ¼
p 1m p 1A
ð
6
Þ
where p 1a contains all the RHP poles and zeros with time delay and the portion p 1m
includes the rest of the transfer function. Now, de
ne the IMC controller q 1 as
shown in the following equation, considering a unit step input as the reference:
p 1
ð
Þ
q 1 ¼
1m f
7
where f is a
filter selected by the designer in the following form:
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