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Then the reset time of the integrator is
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To be sure, the valve position changes in discrete steps, as does the output of the up-down counter. The steps are about 1 percent, and the valve will attempt to follow each pulse registered by the counter. But because most valves are incapable of following the pulse rate of the counter, which may be as high as 100 pulses/set, delivery is reasonably smooth. Although integral control is capable of reducing a flow-rate error to zero, integral offset can exist. Examine the flow delivery following startup as shown in Fig. 6.12. The set flow rate is already at its proper value, but it takes time to bring the measured flow rate to the same value. Eventually there will be no rate error, but in the meantime a significant volume error appears as the area between the curves. The integrated error E required to generate a given controller output is
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E = 1 e dt = Rim
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FIG 6.12. An integral control sgstern is subject to volume offset.
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1 6 6 1 Multiple-loop Systems
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The actual volumetric offset is the percent integrated error E times the maximum flow rate: EF = mV (6.8)
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Each valve position, hence each flow rate, has a related volume offset. On the face of each digital control station are two counters. One registers the total demand, while the other registers the total delivered flow. Volume offset appears as a difference of perhaps 5 or 10 between the two counters. In the steady state, the two count rates are identical, so the volume offset is constant. In a continuous process, volume offset is of little significance, representing only a few counts out of increasing thousands. But in small batches of only a few hundred counts, it can be cause for concern. The obvious solution is to add a second integral mode. But double integral by itself is unstable, in that it produces 180 phase lag at all periods. But if the first integral, i.e., the volume error, is acted upon by a proportional-plus-reset controller, the system can be stable. S u c h an arrangement is functionally described in Fig. 6.13. Mathematically, the up-down counter totals up the difference between the demand and the measured rates, appropriately scaled: EF = /- F(r - c) dt The D-A converter fixes the range V of volume error E over which the control valve is driven. Substituting Eq. (6.7) and adding proportionalplus-reset (&) action:
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(6.9)
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The expanded control equation is (r - c) dt + & /-I (r - c) dt ] Fig. 6.14 shows how the second integral works to eliminate volume off set. Flow loops with conventional controllers, i.e., proportional-plus-reset, have their natural period in the region of 1 to 10 sec. The phase contribution of a properly adjusted controller at the natural period is normally 30 or less. But t,he blending controller contributes about 100 lag,
FIG 6.13. A second integration is required to eliminate the volume error.
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FIG 6.14. The second integral caneels volume deficiency with an equal excess, but takes longer to reach a steady state.
E 2 LL
Time
generally doubling the period of a loop. Thus blending loops will oscillate from about a 5- to a 20-set period. Presence of any valve hysteresis, in combination with the two integrations, will cause a limit cycle, as explained in Chap. 5. Although a valve positioner is not ordinarily recommended for flow, the period of a blending loop is usually far enough removed from that of the position loop for one to be safely used.
SELECTIVE CONTROL LOOPS
Frequently a situation is encountered where two or more variables must not be allowed to pass specified 1imit.s for reasons of economy, efficiency, or safety. If the number of controlled variables exceeds the number of manipulated variables, whichever ones are in most> need must logically be selected for control. (This is the case of the squeaky wheel getting the grease.) Automatic selector units are available for this type of service. They are employed in four basic areas of application: 1. Protection of equipment 2. Auctioneering 3. Redundant instrumentation 4. Nonlinear control functions As an example of how equipment might be protected by a selective control system, consider a compressor whose discharge is ordinarily on
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