Gain
K
m FF ( t ), %CO SP
m ( t ), % CO AC
FC
SUM
mFB ( t ), %CO
f 1 (t )
Figure 72.8 Implementation of feedforward/feedback controller.
3.50 s + 1ˆ
FFC = 0.891Ê (72.9)
Ë 2.75s + 1¯
Figure 72.8 shows the implementation of this controller. The ¬gure shows that the
feedback compensation has also been implemented. This implementation has been
accomplished by adding the output of both feedforward and feedback controllers
using a summer. Section 74 discusses how to implement this addition. Figure 72.9
shows the block diagram for this combined control scheme.
Figure 72.10 shows the response of the composition when f2(t) doubles from
1000 gpm to 2000 gpm. The ¬gure compares the control provided by feedback
control (FBC), steadystate feedforward control (FFCSS), and dynamic feedforward
control (FFCDYN). In steadystate feedforward control, no dynamic compensation
is implemented; that is, in this case the feedforward controller is FFC = 0.891.
Dynamic feedforward control includes the complete controller, Eq. (72.9). Under
steadystate feedforward the mass fraction increased up to 0.477 mf, a 1.05% change
from the set point. Under dynamic feedforward the mass fraction increased up to
0.473 mf, or 0.21%. The improvement provided by feedforward control is quite
impressive. Figure 72.10 also shows that the process response tends to decrease ¬rst
and then increase; we discuss this response later.
The previous paragraphs and ¬gures have shown the development of a linear
feedforward controller and the responses obtained. At this stage, since we have not
yet offered an explanation of the lead/lag unit, the reader may be wondering about
it. Let us explain this term before further discussing feedforward control.
154 FEEDFORWARD CONTROL
f2
D
FFC HD gpm
%TOD
m FF
GD
%CO
m FB m, % CO c
c set e GM
GC
%
% TO + %CO %TO

Figure 72.9 Block diagram of feedforward/feedback controller.
Figure 72.10 Feedforward and feedback responses when f2(t) changes from 1000 gpm to
2000 gpm.
155
LEAD/LAG TERM
73 LEAD/LAG TERM
As indicated in Eqs. (72.5) and (72.9), the lead/lag term is composed of a ratio of
two ts + 1 terms; or more speci¬cally, its transfer function is given by
O( s) t ld s + 1
= (73.1)
I( s) t lg s + 1
where O(s) is the Laplace transform of output variable, I(s) the Laplace transform
of input variable, tld the lead time constant, and tlg the lag time constant.
To explain the workings of the lead/lag term let us suppose that the input
changes, in a step fashion, with A units of amplitude. The equation that describes
how the output responds to this input is
Ê t ld  t lg  t t ˆ
O(t ) = A Á1 + e˜ lg
(73.2)
t lg
Ë ¯
Figure 73.1 shows the response for different values of the ratio tld/tlg while
keeping tlg = 1; the input is a step change of 5 units of magnitude. The ¬gure shows
that as the ratio increases, the initial response also increases; as time increases, the
response approaches asymptotically its ¬nal value of 5 units. For values of tld/tlg >
1 the initial response (equal to the input change times the ratio) at t = 0 is greater
than its ¬nal value, while for values of tld/tlg < 1 the initial response (also equal to
the input change times the ratio) is less than its ¬nal value. Therefore, the initial
response depends on the ratio of the lead time constant to the lag time constant,
tld/tlg. The time approach to the ¬nal value depends only on the lag time constant,
Response of lead/lag to an input change of 5 units, different ratios of tld/tlg.
Figure 73.1
156 FEEDFORWARD CONTROL
Response of lead/lag to an input change of 5 units, different ratios of tld/tlg.
Figure 73.2
tlg. Thus, in tuning a lead/lag, both tld and tlg must be provided. The reader should
use Eq. (73.2) to convince himself or herself of what was just explained.
Figure 73.2 is shown to further help in understanding lead/lags. The ¬gure shows
two response curves with identical values of the ratio tld/tlg but different individual
values of tld andtlg. The ¬gure shows that the magnitude of the initial output
response is the same, because the ratio is the same, but the response with the larger
tlg takes longer to reach the ¬nal value.
Equation (72.5) indicates the use of a lead/lag term in the feedforward con
troller. The equation indicates that tld should be set equal to tM and that tlg should
be set equal to tD.
74 EXTENSION OF LINEAR FEEDFORWARD CONTROLLER DESIGN
With an understanding of the lead/lag term, we can now return to the example of
Section 72: speci¬cally, to a discussion of the dynamic compensation of the feed
forward controller. Comparing the transfer functions given by Eqs. (72.6) and
(72.7), it is easy to realize that the controlled variable c(t) responds slower to
a change in the manipulated variable m(t) than to a change in the disturbance f2(t).
Recall that a design consideration for a feedforward controller is to compensate for
the dynamic differences between the manipulated and the disturbance paths, the
GD and GM paths. The feedforward controller for this process should be designed
to speed up the response of the controlled variable on a change in the manipulated
variable. That is, the feedforward controller should speed up the GM path; the result
ing controller, Eq. (72.9), does exactly this. First, note that the resulting lead/lag
term has a tld/tlg ratio greater than 1, tld/tlg = 3.50/2.75 = 1.27. This means that at the
157
EXTENSION OF LINEAR FEEDFORWARD CONTROLLER DESIGN
moment the signal from the ¬‚ow transmitter changes by 1%, indicating a certain
change in f2(t), the lead/lag output changes by 1.27%, resulting in an initial output
change from the feedforward controller of (0.891)(1.27) = 1.13%. Eventually, the