Design and Modeling of Anti Wind Up PID Controllers - Complex System Modelling and Control Through Intelligent Soft Computations - page 42

D 00 ð

1 Þ

D 0 ð

2

D ð z Þ ¼ D ð 1 Þþ 1 Þð z 1 Þþ ð z 1 Þ þ ð 96 Þ

2

Then associating the similar terms of ( 95 ) and ( 96 ) the following values for D(1)
and its derivatives are obtained:

N 0 ð

D ð 1 Þ ¼ 1 Þ

D 0 ð N 00 ð

1 Þ ¼ 1 Þ= 2 ð 97 Þ

D 00 ð N 000 ð

Þ ¼ Þ=

1 1 3

the values of D(1) and its derivatives can be found by:

D ð 1 Þ ¼ 1 þð N 1 Þð 1 aÞ ð 98 Þ

D 0 ð 1 Þ ¼ ð N ð N 1 Þð 1 aÞÞ= 2 ð 99 Þ

D 00 ð

Þ ¼ ðð þ Þ ð Þð aÞÞ= ð Þ

1 N 1 N N 1 1 3 100

Then the values for f(1) and its derivatives are found by (Lee et al. 1998 ):

1

f ð 1 Þ ¼ ð 101 Þ

p c M ð

ð 1 Þ=ð 1 aÞÞ D ð 1 Þ

Þ ¼ ð

p 0 c M ð

Þ=ð aÞÞ ð Þþð

p c M ð

Þ=ð aÞÞ

D 0 ð

Þ

1 1 D 1 1 1 1

f 0 ð

ð Þ

1 102

2

p c M ð

ðð 1 Þ=ð 1 aÞÞ D ð 1 ÞÞ

!

p 00
c

p 0 c M ð D 0 ð p c M ð D 00 ð ð ð 1 Þ=ð 1 aÞÞ D ð 1 Þþ 2 ð 1 Þ=ð 1 aÞÞ 1 Þþð 1 Þ=ð 1 aÞÞ 1 Þ

M

f 00 ð f 0 ð 2f 0 2 1 Þ ¼ 1 Þ þ ð 1 Þ= f ð 1 Þ

ð p 0 c M ð 1 Þ=ð 1 aÞÞ D ð 1 Þþð p c M ð 1 Þ=ð 1 aÞÞ D 0 ð 1 Þ

ð 103 Þ

With f(1) and its respective derivatives, the parameters of the digital PID con-
trollers can be found using ( 32 ).

References

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June 21 - 23, 1989.
Bateman, A., & Zongli, L. (2002). An analysis and design method for discrete-time linear systems
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1310.
Bohn, C., & Atherton, D. P. (1995). An analysis package comparing PID anti-windup strategies.
Control Systems Magazine, IEEE, 15(2), 34

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40.
Cao, Y.-Y., Lin, Z., & Ward, D. (2002). An antiwindup approach to enlarging domain of attraction
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145.

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