Design and Modeling of Anti Wind Up PID Controllers - Complex System Modelling and Control Through Intelligent Soft Computations - page 8

1

f ¼ ð 8 Þ

r

ðk s þ 1 Þ

for some positive constant r. The IMC PID anti windup controller G c is is given by
the following formulae

q 1

G c ð s Þ ¼ ð 9 Þ

1 G p1 q 1

where this controller is transformed into a PID form as shown in the rest of this
section. The transfer function G p1 is divided in the following parts as explained in
( 6 )

e h s p 1A ð s Þ ¼

ð 10 Þ

k ða þ bD / Þð a 1 s þ a 0 Þ

p 1m ð s Þ ¼

ð a 1 s þ a 0 ða þ bD / ÞÞðs s þ 1 Þ

Based on these equations q 1 is given by:

Þ ¼ ð

a 1 s þ a 0 ða þ bD / ÞÞðs s þ 1 Þ

q 1 ð s ð 11 Þ

r

ða þ bD / Þð þ

a 0 Þðk

þ Þ

k a 1 s s 1

Using these equations the controller G c is is given by:

1

G c ð s Þ ¼ ð 12 Þ

r

p 1m ððk

þ Þ

p 1a Þ

s 1

Substituting the functions p 1m and p 1A the following IMC anti windup controller
is found:

ð a 1 s þ a 0 ða þ bD / ÞÞðs s þ 1 Þ

G c ð s Þ ¼ ð 13 Þ

r

k ða þ bD / Þð a 1 s þ a 0 Þððk s þ 1 Þ e h s Þ

For the PID anti windup controller synthesis it is necessary to consider a PID
controller for G c (s) and then by Mclaurin series expansion the IMC anti windup
controller parameters are found (Shamsuzzoha and Lee 2007 ). For this purposes,
consider the following PID controller

1
s i s þ s d s

G c ð s Þ ¼ K c ð 1 þ Þ ð 14 Þ

where K c is the controller gain, ˄ i and ˄ d are the integral and derivative time constant
that must be obtained in order to get the IMC anti windup controller time constants.
The time constants of the IMC anti windup controller are found by the Mclaurin