Design and Modeling of Anti Wind Up PID Controllers - Complex System Modelling and Control Through Intelligent Soft Computations - page 41

Appendix 2

In this appendix the internal model PID controller, explained in Sect. 4 the gain and
time constant are found and shown in the following equations. Consider the fol-
lowing representation in Taylor series of the digital PID controller ( 31 ) based on the
analog controller design shown in (Lee et al. 1998 )

f 00 ð

f ð z Þ 1 1 Þ

f 0 ð

2

G c ð s Þ ¼

1 ¼ 1 ð

f ð 1 Þþ 1 Þð z 1 Þþ ð z 1 Þ þÞ ð 89 Þ

z z 2

f ð z Þ
z

Due to G c ð s Þ ¼

1 the following equation can be considered:

Þ ¼ ð

aÞ

P c A ð

aÞ

z 1 z

ð ð Þ

D z 90

z 1

because of ( 30 ) can be represented by:

zP 1
c M
ð z aÞ P c A ð 1 aÞ z

ð aÞ

1

G c ð z Þ ¼ ð 91 Þ

The design procedure of the discrete time SISO controller is similar to the
continuous time SISO case, (Lee et al. 1998 ) where ( 90 ) can be represented by:

N ð z Þ

D ð z Þ ¼ ð 92 Þ

z 1

where

P c A ð N ð z Þ ¼ ð z aÞ 1 aÞ z ð 93 Þ

Then by the Taylor series expansion of D(z) the following equation is obtained:

N 00 ð N 000 ð

1 1 Þ 1 Þ

2 3

N 0 ð D ð z Þ ¼

1 ð

N ð 1 Þþ 1 Þð z 1 Þþ ð z 1 Þ þ ð z 1 Þ þÞ

z 2 6

ð 94 Þ

Considering that N(1) = 0, ( 94 ) becomes in:

N 00 ð N 000 ð

1 Þ 1 Þ

N 0 ð

2

ð Þ ¼ Þþ ð Þþ ð Þ þ ð Þ

D z 1 z 1 z 1 95

2 6

Expanding D(z) in Taylor series expansion as an only term, the following result
is obtained: