стр. 68 |

SP

SP

LC LT

AC

AT

Product

(c)

SP

PC

PT

Coolant

SP

SP

AC

LC

LT

AT

6 Reflux

5 Distillate

4

Feed

3

SP 2

SP

Steam

1

AC

LT

LC

AT

Bottoms

(d)

Figure 9-1.1 Continued.

183

PAIRING CONTROLLED AND MANIPULATED VARIABLES

m1 c1

G11

G21

G12

c2

m2

G22

Figure 9-1.2 Block diagram of a 2 ВҐ 2 multivariable process.

m1 m2

c1 K11 K12

c2 K21 K22

Figure 9-1.3 Steady-state gain matrix.

this notation the п¬Ѓrst n is the number of controlled variables and the second n is

the number of manipulated variables.)

If we donвЂ™t know how to decide but a decision has to be made, it makes sense to

control each controlled variable with the manipulated variable that has the great-

est inп¬‚uence on it. In this context, inп¬‚uence and process gain have the same

meaning; consequently, to make a decision we must п¬Ѓnd all process gains (four gains

for a 2 ВҐ 2 system) of the process. The following are the open-loop process gains of

interest:

Dc1 Dc1

K11 = K12 =

Dm1 Dm2

m2 m1

Dc 2 Dc 2

K 21 = =

K 22

Dm1 Dm2

m2 m1

where the notation Kij refers to the gain relating the ith controlled variable to the

jth manipulated variable.

The four gains can be arranged in the form of a matrix to give a more graphical/

mathematical description of their relationship to the controlled and manipulated

variables. This matrix is called the steady-state gain matrix (SSGM) and is shown in

Fig. 9-1.3.

From this SSGM the combination of the controlled and manipulated variables

that yields the largest absolute number in each row may appear to be the one that

should be chosen. For example, if |K12| is larger than |K11|, m2 is chosen to control

184 MULTIVARIABLE PROCESS CONTROL

m1 m2

c1 m11 m12

c2 m21 m22

Figure 9-1.4 Relative gain matrix.

c1. However, this method of choosing the pairing of controlled and manipulated

variables is not correct because it suffers three weaknesses: (1) the comparison of

the second row may yield the use of the same manipulated variable; (2) under

closed-loop operation the gains may vary; and more important, (3) the gains may

have different units. Thus it is not a fair comparison; the matrix, as it stands, is depen-

dent on units.

A technique developed by Bristol [1] has been proposed to normalize the terms

in the matrix, making them independent of the units, taking into consideration the

closed-loop gains, and ensuring that no manipulated variable is chosen more than

once. This technique, called the relative gain analysis or interaction measure, yields

the relative gain matrix (RGM), which is then used to reach a decision. The RGM

is shown in Fig. 9-1.4.

The relative gain terms in the RGM are deп¬Ѓned as follows:

Dc1 Dm2

в€‚ c1 в€‚ m2 K12

m1 m1

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