Thus the units show that the term indicates how much the feedforward con-

troller output, mFF(t) in %CO, changes per unit change in transmitter™s output, D in

%TOD.

Note the minus sign in front of the gain term. This sign helps to decide the

“action” of the controller. In the process at hand, KD is positive, because as f2(t)

increases, the outlet concentration x6(t) also increases because stream 2 is more

concentrated than the outlet stream. KM is negative, because as the signal to the

151

BLOCK DIAGRAM DESIGN OF LINEAR FEEDFORWARD CONTROLLERS

valve increases, the valve opens, more water ¬‚ow enters, and the outlet concentra-

tion decreases. Finally, KTD is positive, because as f2(t) increases, the signal from the

transmitter also increases. Thus the sign of the gain term is negative:

+

KD

Æ =-

+-

KT K M

D

A negative sign means that as %TOD increases, indicating an increase in f2(t), the

feedforward controller output mFF(t) should decrease, closing the valve. This action

does not make sense. As f2(t) increases, tending to increase the concentration of the

output stream, the water ¬‚ow should also increase, to dilute the outlet concentra-

tion, thus negating the effect of f2(t). Therefore, the sign of the gain term should be

positive. Notice that when the negative sign in front of the gain term is multiplied

by the sign of this term, it results in the correct feedforward action. Thus the nega-

tive sign is an important part of the controller.

The second term of the feedforward controller includes only the time constants

of the GD and GM paths. This term, referred to as lead/lag, compensates for the dif-

ferences in time constants between the two paths. In Section 7-3 we discuss this term

in detail.

The last term of the feedforward controller contains only the dead-time terms of

the GD and GM paths. This term compensates for the differences in dead time

between the two paths and is referred to as a dead-time compensator. Sometimes

the term toD - toM may be negative, yielding a positive exponent. As we learned in

Chapter 2, the Laplace representation of dead time includes a negative sign in the

exponent. When the sign is positive, it is de¬nitely not a dead time and cannot be

implemented. A negative sign in the exponential is interpreted as “delaying” an

input; a positive sign may indicate a “forecasting.” That is, the controller requires

taking action before the disturbance happens. This is not possible. When this occurs,

quite often there is a physical explanation, as the present example shows.

Thus it can be said that the ¬rst term of the feedforward controller is a steady-

state compensator, while the last two terms are dynamic compensators. All these

terms are easily implemented using computer control software; Fig. 7-2.7 shows the

implementation of Eq. (7-2.5). Years ago, when analog instrumentation was solely

used, the dead-time compensator was either extremely dif¬cult or impossible to

implement. At that time, the state of the art was to implement only the steady-state

and lead/lag compensators. Figure 7-2.6 shows a component for each calculation

needed for the feedforward controller, that is, one component for the dead time,

one for the lead/lag, and one for the gain. Very often, however, lead/lags have

adjustable gains, and in this case we can combine the lead/lag and gain into only

one component.

Well, enough for this bit of theory, and let us see what results out of all of this.

Returning to the mixing system, under open-loop conditions, a step change of 5%

in the signal to the valve provides a process response form where the following ¬rst-

order-plus-dead-time approximation is obtained:

-1.095e -0.9 s %TO

GM = (7-2.6)

3.50 s + 1 %CO

152 FEEDFORWARD CONTROL

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 )

L/L Lead/lag

FT

f 2 (t )

DT Dead time

mFB ( t ) SP

mFF ( t ) AC

FC

% CO

Gain

K

% CO

f 1 (t )

Figure 7-2.7 Feedforward control.

Also under open-loop conditions, f2(t) was allowed to change by 10 gpm in a step

fashion, and from the process response the following approximation is obtained:

0.032e -0.75 s %TO

GD = (7-2.7)

2.75s + 1 gpm

Finally, assuming that the ¬‚ow transmitter in stream 2 is calibrated from 0 to 3000

gpm, its transfer function is given by

100%TO D %TO D

HD = = 0.033 (7-2.8)

3000 gpm gpm

Substituting the previous three transfer functions into Eq. (7-2.5) yields

3.50 s + 1ˆ -( 0.75-0.9 )s

FFC = 0.891Ê e

Ë 2.75s + 1¯

The dead time indicated, 0.75 to 0.9, is negative and therefore the dead-time

compensator cannot be implemented. Thus the implementable, or realizable,

feedforward controller is

153

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 )

L/L Lead/lag

FT

f 2 (t )