Dynamic Elements in the Control Loop in Visual Studio .NET

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Dynamic Elements in the Control Loop
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FIG 1.26. The proportional band required for >i-amplitude damping for any combination of dead time and capacity can be selected from this chart.
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shift takes place in the dead-time element. ro = 2.67rd Substituting, T$ = 2a 71 = 2.35~~ 2.67
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This point lies on the abscissa of Fig. 1.26 at, T~,!T~ = 2.35. It may be recalled that the gain of a first-order lag at 70 = 2~7~ is l/d2. If the loop is to be damped, G19 = 0.5 Therefore, P = J$ = 100 &2 = 141 It is interest,ing to note the comparison between the controllabilit y of this process and the two-capacity process. Taken on the basis of an equal ratio of secondary to primary element, the dead-time plus capacit y process is 400/al6 or 8 times as difficult to control. Recall that the pure dead-time process was 12.5 times as difficult to control as the most difficult two-capacity process.
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The Effect of Derivative
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Derivative is the inverse of integral action. In theory, it is characterized by a 90 phase lead, although because of physical limitations 45 is about all that cm be expected. If perfect derivative (90 lead) were available, it could halve the period of the dead-time plus capacity loop by allowing the dcnd time to contribute all 180 . Remember t hat perfect derivative applied to the tn-o-capacity process provided critical damping with zero proportional band. But lcig. 1.27 indicates that perfect derivative is limited to zero damping at a period of i)rd with zero proport ional band.
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Time
Derivative is lead action, which has been described as the inverse of lag. That is why it could nearly cancel the effect of the secondary lag in a two-capacity process. But derivative is not the inverse of dead time. Nothing is, since no one can make time. Derivative is a poor subst,itute and consequently is only partially effective in improving the performance of the loop. Limited to 45 of phase lead, furthermore, proportional-plus-derivative controllers can only reduce the period of a dead-t ime plus capacity loop to 2.67~d. As pointed out earlier, derivative contributes gain as well as phase lead:
Since the gain of the process capacity decreases at the same rate,
Reducing 7O produces no net change in loop gain. Consequently, adding derivative does not allow a reduction in proportional band, as it did with the two-capacity process. Thus derivative is scarcely effective at all in the presence of dead time. The derivative mode exhibits a phase lead of 45 at 7O = 2aD. To take advantage of this lead, the derivative time should be set to locate this phase lead at the period of the loop after derivative has been added (2.67ra) : 2rD = 2.67rd For s/4-amplitude P = 4ooz1 damping, D = 1.33 2
This derivative sett,ing is contrasted with that recommended for the twocapacity process, that is, D = TZ.
Dynamic Elements in the Control Loop
SUMMARY
A careful reading of this chapter should disclose the dependence of control performance on what have been termed the secondary dynamic elements in the loop. The largest time constant has been defined as the primary element, and all others as secondary. The term difficult has been used to describe control of certain processes. The proportiona band required for a particular damping serves as an index of difficulty. There is good reason for this, for the proportional band is a measure of how much influence a controller has over a process. The derivation of proportional offset bears out this relationship. If the proportiona band is 100 percent, the controller and the load have equal influence over the controlled variable. At 200 percent band, t he load has twice as much influence. Figure 1.7 is a good illustration. Control probIems of principal interest are those invoIving two dynamic elements. Loops comprised of only one element are nothing more than limits of two-element loops. The difficulty of each of these processes is found to be proportional to the ratio of the secondary to the primary element. In addition, the period of the closed loop is a function of the secondary element alone. A performance index can be envisioned which would combine the sensitivity of the loop to disturbances with the time required to recover from them. This index would vary as the square of the secondary element. The significance of secondary elements is paramount. Settings of reset and derivative time are also directly related to the value of the secondary element. This rule seems as illogical as that governing the period of the pendulum, which varies with length, not with mass. Visualize length as the secondary element and mass as the primary, as a memory aid. Hopefully, the reader has observed how the open-loop characteristics of a process determine its closed-loop response. And how little influence the controller has over this response. It is particularly true for processes of increasing difficulty, where problems begin to appear.
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