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or

W1 x1 + W2 x 2

x=

W

and substituting Eq. (9-1.3) into the above,

W1 x1 + W2 x 2

x= (9-1.4)

W1 + W2

In this 2 ВҐ 2 system there are four gains of interest: KW1, KW2, Kx1, and Kx2. From Eq.

(9-1.3) the п¬Ѓrst two gains can be evaluated:

в€‚W в€‚W

K W1 = = 1 and K W2 = =1

в€‚ W1 в€‚ W2

W2 W1

From Eq. (9-1.4) the other two gains are evaluated:

W2 (x 2 - x1 ) W1 (x1 - x 2 )

в€‚x в€‚x

K x1 = = and K x 2 = =

2 2

в€‚ W1 в€‚ W2

(W1 + W2 ) (W1 + W2 )

W2 W1

The steady-state gain matrix is then written as

W1 W2

W 1 1

W1 (x1 - x 2 )

W2 (x 2 - x1 )

x 2

2

(W1 + W2 )

(W1 + W2 )

For this blending process, development of the set of describing equations and

evaluation of the gains were fairly simple. For some processes these are not easily

done; examples are the chemical reactor and the distillation column, shown in Fig.

9-1.1. Fortunately, however, the design of most processes is usually done with the

use of п¬‚owsheet simulators, such as ASPEN, HYSIM, CHEMCAD, and ProII. From

these simulators it is usually simple to evaluate the required gains. For a 2 ВҐ 2 system

three computer runs sufп¬Ѓce to obtain the four gains. In this case the following

approximation is used: Kij ВЄ Dci /Dmj|m.

Once the open-loop gains Kij have been obtained, evaluation of the closed-loop

/

gains Kij and the relative gain terms mij is fairly straightforward. For the closed-loop

gain there is no need to actually go to the process and evaluate it. We show next

how to obtain this closed-loop gain and the relative gain for a 2 ВҐ 2 process; the

method is then extended to any higher-order process.

Consider the block diagram for a 2 ВҐ 2 process shown in Fig. 9-1.2. The effect of

a change in both manipulated variables on c1 is expressed as follows:

188 MULTIVARIABLE PROCESS CONTROL

Dc1 = K11Dm1 + K12Dm2 (9-1.5)

Similarly, on c2 we have

Dc2 = K21Dm1 + K22Dm2 (9-1.6)

To obtain the gain в€‚c1/в€‚m1|c 2 ВЄ Dc1/Dm1|c 2, Dc2 in Eq. (9-1.6) is set to zero:

0 = K21Dm1 + K22Dm2

or

K 21

Dm2 = - Dm1

K 22

Substituting this expression for Dm2 in Eq. (9-1.5) yields

K12 K 21

Dc1 = K11Dm1 - Dm1

K 22

and п¬Ѓnally, we obtain

Dc1 K11 K 22 - K12 K 21

/

K11 = = (9-1.7)

Dm1 K 22

c2

Note that this closed-loop gain can be evaluated simply by a combination of open-

loop gains. There is no need to close any loop nor to have the process operating.

Then

K11 K11 K 22

m11 = = (9-1.8)

/

K11 K11 K 22 - K12 K 21

A similar procedure on each of the other combinations yields

K12 K12 K 21

m12 = = (9-1.9)

/

K12 K12 K 21 - K11 K 22

K 21 K12 K 21

m 21 = = (9-1.10)

/

K 21 K12 K 21 - K11 K 22

K 22 K11 K 22

m 22 = = (9-1.11)

/

K 22 K11 K 22 - K12 K 21

and the relative gain matrix is

189

PAIRING CONTROLLED AND MANIPULATED VARIABLES

m1 m2

K11 K 22 K12 K 21

c1

K11 K 22 - K12 K 21 K12 K 21 - K11 K 22

K12 K 21 K11 K 22

c1

K12 K 21 - K11 K 221 K11 K 22 - K12 K 21

From this matrix, using the pairing rule presented earlier, the correct combination

of controlled and manipulated variables is chosen. It is easily shown from the matrix

that the terms in each row and each column add up to 1. The dimensionality con-

sistency of each term is also easily shown.

Applying the relative gain matrix to the blending process yields

W1 W2

W1 W2

W

W W

W2 W1

x

W W

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