Dynamic Elements in the Control Loop in VS .NET

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Dynamic Elements in the Control Loop
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FIG 1.20. Because the displacement chamber cannot fill instantaneously, it introduces a second capacity.
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So in this example, the metering pump is used as a load, and the time constant for the vessel is 71 = V/F. Let us study the effect of zero proportional band on this process. The set-point response is given in Fig. 1.21. When the measurement is below the set point, the fill valve will be wide open, delivering flow F. If the load (outflow) is 50 percent of F, the rate of rise of level will be dh - = ; (100 - 50) dt 50% =71
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But the measurement c lags behind the level by 72: dc c+.,,,=h It can be shown that if dc/dt is constant, it is equal to dh/dt. dh h-c = 2~ = 5 0 % ; Then
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FIG 1.21. Zero proportional band will cause a two-capacity process to overrun the set point.
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This is the difference in value between the intermediate variable h and the measurement. Their difference in time is simply the amplitude difference divided by t,he rate of rise:
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The controller will not close the valve until the measurement reaches the set point. Notice that bhe intermediate variable has exceeded the set point by 50~~/7~ at this time. When the valve is shut, outflow will exceed inflow by 50 percent and the level will descend at the same rate. As long as the level is higher than the measurement, the measurement will continue to rise. The measurement will stop rising when it equals the level. The time elapsed between actuation of the controller and the peak of the measurement represents >i-cycle. From inspection of the figure, this time is somewhere between 0.572 and ~2 min. It has been calculated at 0.7~~. This would make the period of the first cycle about 2.5~ because the later portions of the cycle are shorter. Notice that the period is proportional to TV, and the amplitude proportional to ~2/7~. These relationships will appear repeatedly in subsequent examples. We know from phase and gain characteristics of the process that it cannot sustain oscillations. This means that each cycle must be successively smaller. But because the inflow is either on or off, the rate of change of level is constant for each cycle. Hence, the period must also decrease. Finally the loop oscillates at zero amplitude and zero period as was anticipated. This unusual property is found only in two-capacity processes.
Proportional bond 1 0 0 rZ/Tl
Time
FIG 1.22. A proportional band of ~OOT~/T, not wide enough to prevent overshoot.
Dynamic Elements in the Control Loop
Proportional
Control
If overshoot is undesirable, the proportional band must be widened. So that there will be no offset at the normal load, the controller must be biased accordingly. In this example the bias would be 50 percent. When the error is zero, t herefore, the inflow will be 50 percent. With the lower edge of the proportional band 50rJr1 percent away from the set point, the tank level will just reach the set point as the valve begins to throt tle. This clearly will not prevent overshoot, for the valve will deliver more than 50 percent flow as long as the measurement is below the set point, raising the level farther. In order to bring the level back down to the set, point, the measurement must overshoot, so as to reduce the inflow below 50 percent. Consequently a proportional band of 1007.J~~ (5072/~~ on either side of 30 percent flow) is not wide enough. In Fig. 1.23 the example is repeated with the proportional band at ZOOTJT~. Throttling begins when the intermediate variable is 30r2!r1 below the set point, where the rate of rise starts to decrease. This allows the measurement to overtake the tank level, and both will come to rest at the set point. This no overshoot characteristic is called critical damping. In these examples the load was 50 percent. If the load were instead 80 percent, the rate of rise of level would be only 20C;,/r1. But the controller would be biased by 80 percent, so that only 20 percent of the proportional band would be below the set point. With a band setting of 20072/7,, this would leave 4072, ~~ belolv the set point. This throttling
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