Design and Modeling of Anti Wind Up PID Controllers - Complex System Modelling and Control Through Intelligent Soft Computations - page 40

synthesis ensuring the system stability and robustness properties of the closed loop
system. The PID controller gain matrices, in both cases, are found by solving the
respective linear matrix inequalities LMI

s in order to obtain the required gains to
stabilize the system. In both cases it is proved that the system performance is not
deteriorated by the windup phenomena when the input of the system is constrained
or saturated, then in comparison when only a PID controller is implemented, the
settling time and overshoot are smaller due to the anti windup characteristics of the
PID controller.

'

Appendix 1

In this appendix the internal model PID controller, explained in Sect. 3 the gain and
time constants are found with the following equations.
De

ne:

r

D ð s Þ ¼ ððk s þ 1 Þ P 1A ð s ÞÞ= s ð 83 Þ

and

K p ¼ p 1m ð 0 Þ ð 84 Þ

Then the following gain and time constants are obtained using ( 15 ) with the
following equations of the function f(s) and its derivatives (Lee et al. 1998 ):

1
K p D ð

ð Þ ¼ ð Þ

f 0 85

0 Þ

where D(0) is

k P 1A ð

D ð 0 Þ ¼ r 0 Þ ð 86 Þ

D

and D the derivatives of D(0), ð 0 Þ ð 0 Þ are shown in (Lee et al. 1998 ). Then the

derivative of f(0) is given by:

! r

a þ bD / a 1
a 0 a bD /

a þ bD / a 0 a 1
a 0 a bD /

a þ bD / a 0 s
a 0 a bD /

K K K

f

Þ 1 K 2
p

ð Þ ¼

2

ð k þ h

0

2 2

1 = 2rr ð 1 Þk 1 = 2 h

þ ð 87 Þ

2

K p r k þ h ð Þ

! !

Þ D K p D

€ p 1m ð 0 Þ D ð 0 Þþ 2 p 1m ð _ 0 ð 0 Þþ ð 0 Þ

f

Þ ¼f 2f

ð 0 ð 0 Þ þ ð 0 Þ= f ð 0 Þ ð 88 Þ

K p D p 1m ð _ 0 Þ D ð 0 Þþ ð 0 Þ