Figure 72.4 Condensed block diagram.
A more condensed block diagram, shown in Fig. 72.4, can now be drawn. The
signi¬cance of each transfer function is as follows:
GM = transfer function that describes how the manipulated variable, m(t ), affects
the controlled variable, c (t ) (in this case, GM = GV GT1 H )
GD = transfer function that describes how the disturbance, f2 (t ), affects the
controlled variable (in this case, GD = GT2 H )
To review, the objective of feedforward control is to measure the inputs, and if a
change is detected, adjust the manipulated variable to maintain the controlled vari
able at set point. This control operation is shown in Fig. 72.5 and the block diagram
in Fig. 72.6. The signi¬cance of each new transfer function is as follows:
HD = transfer function that describes the sensor and transmitter that measures
the disturbance
FFC = transfer function of feedforward controller
Note that in Figs. 72.5 and 72.6 the feedback controller has been “disconnected.”
This controller will be “connected” again later.
Figure 72.6 shows that the way a change in disturbance, Df2, affects the controlled
variable is given by
Dc = GD D f2 + H D ( FFC)GM D f2
The objective is to design FFC such that a change in f2(t) does not affect c(t), that
is, such that Dc = 0. Thus
0 = GD D f2 + H D ( FFC)GM D f2
149
BLOCK DIAGRAM DESIGN OF LINEAR FEEDFORWARD CONTROLLERS
f 6 (t ) x6 ( t )
AT
x3 ( t ) x4 ( t ) c( t ),% TO
f 3 (t ) f 4 (t )
x5 ( t ) x7 (t )
x2 ( t ) f 7 (t )
f 5 (t )
FT D, % TOD
f 2 (t )
FFC
SP
AC
FC
m FF, % CO
f 1 (t )
Water
Figure 72.5 Feedforward control system.
f2
D
FFC HD gpm
%TOD
m FF
GD
%CO
m FB c
c set e GM
GC
% TO + %CO %TO
% TO

Figure 72.6 Block diagram of feedforward control system.
150 FEEDFORWARD CONTROL
Dividing both sides by Df2 and solving for FFC yields
GD
(72.1)
FFC = 
H DGM
Equation (72.1) is the design formula for the feedforward controller. We understand
that at this moment, this design formula does not say much; furthermore, you
wonder what is it all about. Don™t despair, let us give it a try.
As learned in earlier chapters, ¬rstorderplusdeadtime transfer functions are
commonly used as an approximation to describe processes; Chapter 2 showed how
to evaluate this transfer function from step inputs. Using this type of approxima
tion for this process,
 to D s
Ke %TO
GD = D (72.2)
t Ds + 1 gpm
 to M s
Ke %TO
GM = M (72.3)
t Ms + 1 %CO
and assuming that the ¬‚ow transmitter is very fast, HD is only a gain:
%TOD
H D = KTD (72.4)
gpm
Substituting Eqs. (72.2), (72.3), and (72.4) into (72.1) yields
K D t M s + 1 ( t )
to M s
FFC =  oD
e (72.5)
KT K M t D s + 1
D
We next explain in detail each term of this feedforward controller.
The ¬rst element of the feedforward controller, KD/KTDKM, contains only gain
terms. This term is the part of the feedforward controller that compensates for the
steadystate differences between the GD and GM paths. The units of this term help
in understanding its signi¬cance:
KD %TO gpm %CO
[= ] =
(%TO D gpm) (%TO %CO) %TO D
KT K M
D