FIG 1.24. Too much as well as too little derivative degrades the stability of the loop. in VS .NET

Painting QR Code ISO/IEC18004 in VS .NET FIG 1.24. Too much as well as too little derivative degrades the stability of the loop.

FIG 1.24. Too much as well as too little derivative degrades the stability of the loop.
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In mpst controllers, the derivative mode operates on the output rather than on the error. Ordinarily, this presents no problem. But upon startup, or following gross set-point changes, the measurement will be outside the proportional band, causing the output to saturate. If derivative operates on the output, which is steady, rather than on the changing input, it is disabled. Derivative will suddenly be activated again when the measurement reenters the band. So if overshoot is to be avoided upon startup, the band must be wide enough to activate the derivative before the primary variable crosses the set point. The band will have to be at least as wide as that shown in Fig. 1.22:
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P = 1002
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(1.28)
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In controllers where derivative happens to operate directly on the measurement or error, P should be >io what was required for proportional control alone, that is, 20~47~. The reduction in band allowed through the use of derivative can in some applications eliminate the need for reset. If a choice between derivative and reset should ever be presented, the former should be selected because it can enhance both speed and stability at the same time. COMBINATIONS OF DEAD TIME AND CAPACITY Occurrences of either pure dead-time or ideal single-capacity processes are rare. The reasons for this are twofold: 1. llIass has the capability of storing energy. 2. JIass cannot be transported anywhere in zero time. Between the most and least difficult elements lies a broad spectrum of moderately dificult processes. Although most of these processes are dynamically complex, their behavior can be modeled, to a large extent, by a combination of dead time plus single capacity. The proportional band required to critically damp a single-capacity process is zero. F o r a dead-time process, it is infinite. It would appear, then, that the proportional band requirement is related to the dead time in a process, divided by its time constant. Any proportional band, hence any process, would fit somewhere in this spectrum of processes. A discussion of multicapacity processes in Chap. 2 will reaffirm this point.
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E ortunately we already investigated this problem when we discussed integral control of dead time. Figure 1.25 indicates the similarity of the loops. If the process is non-self-regulating (integrating), the representa- tion is exact. Because the phase lag of the dead time is limited to 90 ,
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Ud erstanding Feedback Control n
the period of the proportional loop is 47d. In the former case, for s/4amplitude damping, 2rd/ ~R was set equal to 0.5. Since the time constant R is no longer adjustable, but is now TV, part of the process, proportional adjustment must set the loop gain for +a-amplitude damping. Therefore,
2Td 100 --= 0.5 7rTl P P = 4003&
(1.29)
Notice that as 71 approaches zero, P approaches infinity. This is much worse than having no capacity at alI, i.e., dead time alone. The reason is that this expression holds only for a non-self-regulating process whose gain varies inversely with the time constant without limit. Fortunately, non-self-regulating processes dominated by dead time are virtually nonexistent. For the self-regulating process, gain is limited to that of the steady state, nominally 1.0. (Actual contributions of steady-state gain will be evaluated at length in the next chapter.) If the maximum gain of the self-regulating process is 1.0, the proportional band required for >/4-amplitude damping with dead time in the loop will approach 200 percent as 71 approaches zero. The proportional band setting can then be approximated by the asymptotes: (1.30) In Fig. 1.26, the locus of gain, G,, of the capacity, and P for j/4-amplitude damping are plotted vs. T~/T~; the asymptotes are indicated. A point midway between the asymptotes is found where the phase contribution of 71 is 45 . This occurs where 70 = 2~7~. Here 135 of phase
FIG 1.25. Zntegral control of dead time (aboue) is the same as proportional control of a dead-time plus integrating process (below).
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