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FIG 7.8. A decoupling system correctly matched to a coupled process can produce essentially independent control loops.
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PO0 1 Multiple-loop Systems
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Consider how a decoupling control system might be designed for the blending process of Example 7.4. The mathematical model is solved for ml and m2 in terms of F and x: ml = F x m2 = F - ml = F(l - x) (7.21)
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In the first-expression, if both F and x were set points, the manipulation of ml would be open-loop. Again, if extreme accuracy could be realized, this would be sufficient. But for the usual case these terms ought to be controller outputs. In this example, then, let m,, the output of the composition controller, and mF, the output of the flow controller, take the place of F and x in the model. T h e n
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ml = mFmz
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m2 is the product of the flow-controller output with the complement of the composition-controller output:
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m2 = mF(1 - m,)
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FiguT 7.9 illustrates both the coupling of the process and the decoupling of the control system. Notice the similarity to the system shown in Fig. 6.10. To appreciate how the system works, envision the process responding from a controlled condition. If any change occurs in mF, ml and m2 will both change proportionately and in the same ratio that is already maintaining composition control. Should m, be required to change, ml wiI1 move i one direction and m2 in the other, to avoid upsetting total flow. 1 Each loop now responds independently of the other. Notice that a change in set point or measurement of either controller will move both valves in the appropriate direction. If set points had been used as decoupling signals, only one valve would be moved when a load change occurred in its loop, which would upset the other loop. Also, decoupling would be defeated if either controller were placed in manual.
Process - -
FIG 7.9. The decoupling control system should be the image of the process.
Multivariable Process Control
Suppose the measurements of x and F were to be used to decouple the controller outputs, such that
ml = Pm, m2 = (1 - x)mi4
On startup, F would be zero, causing ml to be zero: thus only m2 would be manipulated by the flow controller. As x increased toward its set point, m2 would be decreased by the decoupling signal. This action is clearly in the wrong direction, since ml and m2 should move in opposition to maintain constant flow. The decoupling signal has formed a positive feedback loop.
In many applications, such as this blending system, complete decoupling is unnecessary. As pointed out in the discussion of dynamic effects, very fast loops are scarcely disturbed by their coupling to slow loops. Therefore that part of the decoupling system designed to liberate flow, pressure, and level loops can ordinarily be omitted. In Fig. 7.9, the multiplier whose output is m2 can be eliminated with little detriment. Every change in m, will now upset total flow, but its rate of change is severely restricted by the dynamics of the composition loop. In effect, the composition controller manipulates both valves already-one through the action of the flow. feedback loop. The flow controller manipulates m2 directly, and ml through the remaining multiplier.
Decoupling Half-coupled Loops
Half-coupled loops such as the one shown in Fig. 7.5 are simple to decouple, and litt le risk is involved. The decoupling is in one direction only, and there is no possibility of a positive feedback loop. Therefore a measurement of the independent controlled variable can be used to decouple the output of the dependent loop. From the model of the half-coupled blending system,
l - x
Since the output of the composition controller need not equal x, but only be a reasonably linear function thereof, it is sufficient that
ml = Ym, (7.25)
The resulting system is diagrammed in Fig. 7.10. In the foregoing examples, the manipulated variables were identified as valve position. But in each case, a linear characteristic was assumed, with a constant pressure drop for the blending system. If a control