/

Dc1 Dm2

‚ c1 ‚ m2 K12

c2 c2

or, in general,

‚ c i ‚ m j m K ij

m ij = =/ (9.1-2)

‚ci ‚m j c K ij

Let us make sure that we understand the meaning and signi¬cance of all the terms

in Eq. (9-1.2). The numerator

Dc i

‚ci

ª

Dm j

‚m j m m

is the open-loop steady-state gain Kij, de¬ned previously. That is, this is the gain of

mj on ci when all other manipulated variables are kept constant. The denominator

Dc i

‚ci

ª

Dm j

‚m j c c

/

is the closed-loop gain, Kij. That is, this is the gain of mj on ci when all other loops

are closed and all controllers have integral action, thus returning the controlled vari-

ables to their corresponding set points. Therefore, we can write

185

PAIRING CONTROLLED AND MANIPULATED VARIABLES

gain when all other loops are open K ij

m ij = =/

gain when all other loops are closed K ij

As seen from the de¬nition of mij, this term takes into consideration the gain

under closed-loop conditions and is a dimensionless number. In addition, a prop-

erty of the RGM is that the summation of all the terms in each row and in each

column must equal 1. (This means that for a 2 ¥ 2 system only one term has to be

evaluated, and the others can be obtained using this property. For a 3 ¥ 3 system,

which has nine terms in the matrix, only four independent terms have to be evalu-

ated, and the other ¬ve can be obtained using this property.) Thus the RGM resolves

all three weaknesses of the SSGM, and therefore it can be used to decide how to

pair controlled and manipulated variables.

A complete understanding of the meaning/signi¬cance of mij is important before

proceeding. From the de¬nition of mij, notice that it is essentially a measure of the

effect of closing all other loops on the process gain for a given controlled and manip-

4

ulated variable pair. That is, if m12 = “ = 0.8, it means that when the other loops are

5

closed, the “effect” of a change in m2 on c1 is larger than when the other loops are

open. Speci¬cally, the value says that the gain when the other loops are open is only

80% of the gain when the other loops are closed. Thus the numerical value of mij is

a measure of the interaction between the control loops.

If mij = 1, the process gain is the same with all other loops open or closed; of

course, this is good! This indicates either no interaction between the particular loop

and all other loops, or possible offsetting interactions. The greater the deviation

from 1, the greater the loop interaction.

If mij ª 0, it may be due to one of two possibilities. One is that the open-loop gain,

‚ci/‚mj|m, is either zero or very small. In this case mj does not affect ci, or hardly any,

when all other loops are open. Alternatively, the closed loop is so large that mij ª 0.

This means that to keep the other controlled variables constant, the other loops

interact signi¬cantly with the loop in question. In either case, this is no good! Either

of the two possibilities indicates that ci should not be controlled manipulating mj.

If mij ª • (very large), it may be due to one of two possibilities. One, the closed-

loop gain, ‚ci/‚mj|c, is either zero or very small. This means that when the other loops

are in automatic mode, the loop in question cannot be controlled because mj does

not affect ci, or hardly any. Alternatively, the open-loop gain is very large. This means

that when the other loops are in manual mode, the effect of mj on ci is very large,

whereas it is not the case when the others are in automatic. Obviously, this condi-

tion is also no good!

The preceding discussion has illustrated the signi¬cance of mij. In general, values

of mij close to 1 represent controllable combinations of controlled and manipulated

variables. Values of mij approaching values of zero or in¬nity represent uncontrol-

lable combinations.

With this background we can understand the pairing rule ¬rst presented by

Bristol [1]: To minimize the interaction between loops, always pair on RGM elements

that are closest to 1.0. Avoid negative pairings. The proposed pairing rule is easy and

convenient to use. Realize that only steady-state information is needed. This is cer-

tainly an advantage since this information can even be found during the process

design stage. Thus, it does not require the process to be in operation. In the next

section we discuss in more detail the calculation of these gains.

186 MULTIVARIABLE PROCESS CONTROL

To close this presentation let us look at two possible RGMs to further under-

stand what the mij terms are telling us about the control system. Consider the fol-

lowing RGM:

m1 m2

c1 0.2 0.8

c2 0.8 0.2

1

The terms m11 = m22 = 0.2 = “ indicate that for this pairing the gain of each loop

5

increases by a factor of 5 when the other loop is closed. The terms m12 = m21 = 0.8 =

4

“ indicate that for this pairing the gain increases only by a factor of 1.25. This

5

explains why the c1 - m2 and c2 - m1 pairing is the correct one.

Consider another RGM:

m1 m2

-1

c1 2

-1

c2 2

The terms m11 = m22 = 2 = 1/0.5 indicate that the gain of each loop is cut in half when

the other loop is closed. The terms m12 = m21 = -1.0 indicate that the gain of each

loop changes sign when the other loop is closed. Certainly, this is undesirable

because it means that the action of the controller depends on whether the other

loop is closed or open. This explains why the correct pairing is c1 - m1 and c2 - m2.

9-1.1 Obtaining Process Gains and Relative Gains

The ¬rst information needed to obtain the relative gain terms is the steady-state

open-loop gains, Kij. There are three different ways to calculate these gains:

1. Using the step test method learned in Chapter 2. This is the method used to

obtain the information to tune feedback and cascade controllers and to design

feedforward controllers. Thus we are quite familiar with it.

2. Starting analytically from the equations that describe the process.

3. By the use of a ¬‚owsheet simulator.

To obtain these steady-state open-loop gains analytically, the equations that

describe the process are written ¬rst. From these equations the gains are then eval-

uated. Using the blending process of Fig. 9-1.1a as an example, the outlet ¬‚ow W

and the outlet mass fraction of salt x are to be controlled. Because there are two

components, salt and water, two independent mass balances can be written. A

steady-state total mass balance provides the ¬rst equation,

W1 + W2 = W (9-1.3)

A mass balance on salt provides the other equation,

187

PAIRING CONTROLLED AND MANIPULATED VARIABLES