Design and Modeling of Anti Wind Up PID Controllers - Complex System Modelling and Control Through Intelligent Soft Computations - page 16

U max a 1 z 2 N U max a 0 z 1 N k ð þ Þ

p c ð z Þ ¼

ð 27 Þ

ðð a 1 U max Þ z þ a 0 Þðs z þ 1 Þ

With a sample time T. Where this transfer function is divided as explained in
( 17 ) and ( 18 ) as:

U max a 1 z 2 Þ

ðð a 1 U max Þ z þ a 0 Þðs z þ

k ð þ U max a 0 z

p c M ð

Þ ¼ ð Þ

z 28

1 Þ

p c A ð z N

z Þ ¼ ð 29 Þ

Then using ( 19 ) the following internal model controller is obtained:

ð 1 aÞ z ðð a 1 U max Þ z þ a 0 Þðs z þ 1 Þ

G c ð z Þ ¼ ð 30 Þ

ð U max a 1 z 2 þ U max a 0 z Þð z aÞð U max a 1 z 3 N þ U max a 0 z 2 N Þ

In order to obtain the internal model anti windup controller, it is necessary to

de ne the following standard PID controller:

1

G c ð z Þ ¼ K c ð 1 þ

Þ þ s d ð

z 1 ÞÞ ð 31 Þ

s i ð z 1

Due to the integral term of G c (z) the controller gain and parameters using a
similar procedure like the continuous time counterpart. Implementing the Taylor
series expansion, similar as the previous section the following constant and time
constants of the PID controller are found:

f 0 ð K c ¼ 1 Þ

f 0 ð

Þ

1

s i ¼

ð 32 Þ

f ð 1 Þ

f 00 ð 1 Þ

s d ¼

2f 0 ð 1 Þ

where f and its derivatives are de ned in Appendix 2. This equations are valid for

any sampling period T and the resulting equations are shown in Appendix 2. The
proposed control strategy explained in this section meets the robustness and internal
stability properties that make them suitable for the anti windup control of discrete
time SISO systems. In the next subsection, an illustrative example is shown, to test
the system performance by a numerical example.