Design and Modeling of Anti Wind Up PID Controllers - Complex System Modelling and Control Through Intelligent Soft Computations - page 32

xk ð þ 1 Þ ¼ Ax ð k Þ B rð u ð k ÞÞ

ð 62 Þ

y ð k Þ ¼ Cx ð k Þ

Consider the following PID controller with anti windup compensation (Lim and
Lee 2008 ):

X

k 1

X

k 1

y ð k þ 1 Þ ¼ y ð k Þþ y ð k ÞCðrð u ð k ÞÞ u ð k ÞÞ

i¼0 i¼0

ð 63 Þ

k I X

k 1

u ð k Þ ¼ k p y ð k Þ y ð i Þ k D D y ð k Þ

i¼0

Due to P k 1
i¼

Þ P k 1
i¼ 0 y

ð k þ 1

0 y

ð k Þ ¼ y ð k Þ with y(0) = 0 subtracting the cor-

rection signal as done in the continuous time case explained in

the previous section (Cao et al. 2002 ). Where k p , k I , k D are diagonal matrices for the
proportional, integral and derivative parts of the PID controller,

Cðrð u ð k ÞÞ u ð k ÞÞ

ʓ is a positive

de . In order to design the PID controller

the following augmented variables are introduced to represent ( 62 ) and ( 63 )

nite matrix and D y ð k Þ ¼ y ð k Þ y ð k 1 Þ

2
4

3
5

x ð k Þ

P

k 1

x a ð

Þ ¼

k y ð k Þ

i ¼ 0

y ð k 1 Þ

2
4

3
5

ð 64 Þ

y ð k Þ

P

k 1

y a ð k Þ ¼

y ð k Þ

i¼0

D y ð k Þ

Then the augmented system is (Lim and Lee 2008 ):

2 3 2 3

2 3 2 3 2 3

x ð k þ 1 Þ x ð k Þ

A 00
CI0
C 00

B
C
0

0
0

4 5 ¼ 4 5 þ

P P

k 1 k 1

4 5 4 5 rð 4 5 u

y ð k þ 1 Þ y ð k Þ u ð k ÞÞ þ ð k Þ

i¼0 i¼0

y ð k Þ y ð k 1 Þ

ð 65 Þ

2 3

C 00
0

4 5 x a ð

y a ð k Þ ¼ I 0 k Þ ð 66 Þ

C 0 I