стр. 61 |

relates the manipulated variable f1(t) in terms of the disturbances f2(t) and x2(t). In

this equation we consider all other inlet п¬‚ows and compositions at their steady-state

values. That is,

f1 (t ) = f [ f5 , x5 , f2 (t ), x 2 (t ), f7 , x 7 , x 6 (t )]

set

where the overbar indicates the steady-state values of the variables.

Because we are dealing with compositions and п¬‚ows, mass balances are the

appropriate п¬Ѓrst principles to use. Since there are two components, A and water, we

can write two independent mass balances. We start with a total mass balance around

the three tanks:

r f5 + r f1(t ) + r f2 (t ) + r f7 - r f6 (t ) = 0 one equation, two unknowns [ f1(t ), f6 (t )]

(7-5.1)

Note that f2(t) is not considered an unknown because it will be measured and thus

its value will be known. A mass balance on component A provides the other

equation:

r f5 x5 + r f2 (t )x 2 (t ) + r f7 x 7 - r f6 (t )x 6 (t ) = 0

set

two equations, two unknowns

(7-5.2)

Because x2(t) will also be measured, it is not considered an unknown. Solving for

f6(t) from Eq. (7-5.1), substituting into Eq. (7-5.2), and rearranging yields

163

DESIGN OF NONLINEAR FEEDFORWARD CONTROLLERS FROM BASIC PROCESS PRINCIPLES

1 1

[ x 2 (t ) - x 6 (t )] f2 (t )

f1 (t ) = ( f5 x5 + f7 x 7 ) - f5 - f7 + set

(7-5.3)

x (t ) x (6)

set set

6 6

Substituting the steady-state values into Eq. (7-5.3) yields

1

[ 850 + f2 (t )x 2 (t )] - f2 (t ) - 1000

f1 (1) = (7-5.4)

x (t )

set

6

Equation (7-5.4) is the desired steady-state feedforward controller.

The implementation of Eq. (7-5.4) depends on how the feedback correction, the

output of the feedback controller, is implemented. This implementation depends

on the physical signiп¬Ѓcance given to the feedback signal; there are several ways

to do so.

One way is to decide that the signiп¬Ѓcance of the feedback signal is Df1(t) and use

set

a summer similar to that in Fig. 7-4.2. In this case we п¬Ѓrst substitute x 6 (t) = 0.472

into Eq. (7-5.4) and obtain

Г€ x 2 (t ) Л˜

f1 (t ) = 800.85 + f2 (t ) ГЌ (7-5.5)

- 1Л™

ГЋ 0.472 Лљ

This equation is written in engineering units. Depending on the control system being

used, the equation may have to be scaled before implemented. Assuming that this

is done, if needed, Fig. 7-5.1 shows the implementation of this controller; a multi-

plier is needed only with no dynamic compensation. Please note that because Eq.

(7-5.5) provides f1(t), a п¬‚ow loop has been added to stream 1. If it is decided not to

use the п¬‚ow loop, a conversion between f1(t) and the valve position should be

inserted in Eq. (7-5.5).

Another way to implement the feedback compensation is by deciding that the

set

signiп¬Ѓcance of the feedback signal is 1/x 6 . This signal is then input into Eq. (7-5.4)

to calculate f1(t). Thus, in this case the feedback signal is used directly in the feed-

forward calculation and not to bias it; Fig. 7-5.2 shows the implementation of this

controller. The п¬Ѓgure shows only one block referred to as CALC. The actual number

of computing blocks, or software, required to implement Eq. (7-5.4) depends on the

control system used.

Figure 7-5.3 shows the response of the process under feedback controller and the

two nonlinear steady-state feedforward controllers to disturbances of a 500-gpm

decrease in f2(t) and a 0.2-mf decrease in x2(t). The response FFCNL1 is obtained

when Eq. (7-5.5) is used (Fig. 7-5.1). The response FFCNL2 is obtained when

Eq. (7-5.4) is used (Fig. 7-5.2). The improvement in control performance obtained

with the nonlinear controllers is obvious. The improved performance obtained

with the second nonlinear controller is quite impressive. This controller describes

more accurately the nonlinear characteristics of the process and can provide better

control.

set

Instead of calling the output of the feedback controller 1/x 6 , we could have alter-

set

natively called it x 6 . The control performance would be the same, but what about

the action of the feedback controller in both cases? Think about it.

Previous paragraphs have shown two different ways to implement the nonlinear

164 FEEDFORWARD CONTROL

f 6 (t ) x6 ( t )

AT

x3 ( t ) x4 ( t ) c ( t ),% TO

f3 (t ) f 4 (t )

x5 ( t ) x7 ( t )

AT

x2 ( t ) f7 (t )

f 5 (t )

MUL

FT

f2 (t )

SP

f 1 (t ) AC

FC

FC SUM

mFB ( t ), %

FT

Figure 7-5.1 Nonlinear 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 )

AT

x2 (t ) f 7 (t )

f 5 (t )

стр. 61 |