Page 136 - Practical Control Engineering a Guide for Engineers, Managers, and Practitioners

A New Do11ain and More Process Models m

4·5·3 Proportional-Integral Control of the FOWDT Process
Adding integral control causes the open-loop transfer function to
become

_ -sD g ks +I
G G ( ) s -e ----
pc -rs+l s
Applying s = jro gives a relatively messy expression-but the
reader might try wading through the following algebra-it's worth it:

G G ('ro)=e-it»D_g_kjro+I
p c J t'jOJ+ 1 jOJ

~(kro)2 +I e-itoO/tan-•(~)
= g
1
~( -rro)2 + 1 OJ eitan- (r<»>ei~r
= g ~(kro)2 +I e-jcoD+jtan-•(k~}jtan-•(r<»Hi
2
~(-rro) + 1 OJ
= 1Giei9

2
G _ g ~(kro) +I (4-19)
I 1- ~( -rro)2 + 1 ro

k(J)) tr
8= -roD- tan- (-rro)+ tan- -I- -2
(
1
1

This is the most complicated transfer function yet. Note how each
term was treated as a compte~ number z with a magnitude lzl and an
angle cp according to the 1zleJ" format discussed in App. B. Then the
magnitudes multiplied or divided and the angles added or sub-
tracted. The idea therefore is to break a relatively complicated struc-
ture up into its factors in the numerator and denominator, convert
each factor into a complex quantity with a magnitude, and an angle
and then combine the quantities according to basic algebra.
Using the Bode plot to find the correct control gains k and I is not
particularly fruitful. Instead, I used time domain simulation to find