works both ways, the system is said to be fully coupled or interactive; other-

wise, the system is partially coupled. Interaction is not a problem in partially

coupled 2 ¥ 2 systems.

3. The interaction effect on one loop can be eliminated by interrupting the other

loop; this is easily done by switching the controller to manual. Suppose that

controller 2 is switched to manual; this has the effect of setting GC2 = 0, leaving

the characteristic equation as

1 + GC1GV1G11 = 0

which is the same as if only one loop existed. This may be one reason why

many controllers in practice are in manual. Manual changes in the output of

controller 2 simply become disturbances to loop 1. Usually, however, it is not

necessary to be this drastic to yield a stable system. By simply lowering the

gain and/or increasing the reset time of the controller, that is, detuning the

controller somewhat, the same effect can be accomplished while retaining

both controllers in automatic. The effect of doing this is to move all the roots

of Eq. (9-2.3) to the negative side of the real axis.

The preceding paragraphs have described how the interaction on a 2 ¥ 2 system

affects the stability of the system, which is probably the most common multivari-

able control system. For higher-order systems the same procedure must be followed.

However, the conclusions are not as simple to generalize.

9-3 TUNING FEEDBACK CONTROLLERS FOR INTERACTING SYSTEMS

The third question asked at the beginning of the chapter refers to tuning of the feed-

back controllers in a multivariable environment. The interaction among loops

makes the tuning of feedback controllers more dif¬cult. The following paragraphs

present some procedures for this tuning; for a more complete discussion, see

Shinskey [3] and Smith and Corripio [2]. We ¬rst discuss tuning for a 2 ¥ 2 system

and then discuss n ¥ n systems.

The ¬rst step, after proper pairing, is to determine the relative speed of the loops.

Then:

1. If one loop is much faster than the other one (say, the dominant time constant,

or the time constant of the ¬rst-order-plus-dead time approximation, is ¬ve

times smaller), the fast loop is tuned ¬rst, with the other loop in manual. Then

the slow loop is tuned with the faster loop in automatic. The tuning procedure

and formulas are the same as the procedure and formulas described in Chap-

ters 2 and 3.

2. If both loops are about the same speed of response, and one variable is more

important to control than the other one, detune the less important loop by

193

TUNING FEEDBACK CONTROLLERS FOR INTERACTING SYSTEMS

setting a small gain and a long reset time. This will reduce the effect of the

less important loop on the response of the most important loop because the

detuned loop will appear to be open.

3. If both loops are about the same speed of response and both variables are of

the same importance, each controller should be tuned with the other loop in

manual. Then the effect of interaction should be used to adjust the tuning.

(a) If the interaction is positive, the following is proposed:

/

KCi = KCimii (9-3.1)

(b) If the interaction is negative, the adjustment must be done by trial and

error after both loops are closed.

There is still another procedure, developed by Medina [4], that has proven to

work quite well and it is easy to apply. The procedure requires that we know the

¬rst-order-plus-dead time approximation to each of the four transfer functions that

Tuning a 2 ¥ 2 Multivariable Decentralized Feedback Controller

TABLE 9-3.1

Loop 1 is the loop with the smallest (to /t)ii ratio. Loop 2 is the one with the largest (to /t)ii

ratio. The formulas presented here are to tune loop 2. Loop 1 is tuned by the user by

whatever method he or she desires.

PI“PI Combination

t 22

KC 2 = t I 2 = t 22

A;

K 22

1 1 K 12K 21

A= g=

;

1 - g l + t o 22 K 11K 22

l Ê t 12t 21 ˆ

Ê to to ˆ Ê to ˆ Ê to ˆ

= 1.104g + 1.124g 2 + 0.066 Á 12 21 ˜ + 0.368 Á 11 ˜ - 0.237 Á 21 ˜ - 0.12

ln

Ë t 11t 22 ¯

Ë t o11 t o 22 ¯ Ë t o 22 ¯ Ë t o 22 ¯

t o 22

Formulas are good for g £ 0.8.

PI“PID Combination

Loop 1 is PI and loop 2 is PID.

t o12 + t o 21 - t o11

t 22

KC 2 = t I 2 = t 22 ; t D2 =

A;

K 22 2

1 1 K 12K 21

A= g=

;

1 - g l + t o 22 K 11K 22

l Ê t 11 ˆ Ê t 21 ˆ

Ê to to ˆ Ê to ˆ

= 1.283g + 1.014g 2 + 0.0675 Á 12 21 ˜ + 0.463 Á 11 ˜ - 0.319 - 0.771

ln

Ë t 22 ¯ Ë t 22 ¯

Ë t o11 t o 22 ¯ Ë t o 22 ¯

t o 22

Formulas are good for g £ 0.8.

194 MULTIVARIABLE PROCESS CONTROL

is, K11, t11, to11, K12, t12, to12, K21, t21, to21, and K22, t22, to22. Remember that all the gains

must be in %TO/%CO, as used in Chapter 3 to tune controllers. Table 9-3.1 shows

the formulas to use.

9-4 DECOUPLING

Finally, there is still one more question to answer: Can something be done with the

control scheme to break, or minimize, the interaction between loops? That is, can a

control system be designed to decouple the interacting, or coupled, loops? Decou-

pling can be a pro¬table, realistic possibility when applied carefully. The relative

gain matrix provides an indication of when decoupling could be bene¬cial. If for

the best pairing option, one or more of the relative gains is far from unity, decou-

pling may help. For existing systems, operating experience is usually enough to

decide. There are two ways to design decouplers: from block diagrams or from basic

principles.

9-4.1 Decoupler Design from Block Diagrams

Consider the block diagram shown in Fig. 9-4.1. The ¬gure shows graphically the

interaction between the two loops. To circumvent this interaction, a decoupler may

be designed and installed as shown in Fig. 9-4.2. The decoupler, terms D21 and D12,

should be designed to cancel the effects of the cross blocks, G21 and G12, so that each

controlled variable is not affected by the manipulated variable of the other loop. In

other words, decoupler D21 cancels the effect of manipulated variable m1 on con-

trolled variable c2, and D12 cancels the effect of m2 on c1. In mathematical terms, we

design D21 so that

Dc 2