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FIG 4.19. Flow blending processes are often dominated by a discontinuous analyzer.
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Although feedback control can be exercised over a process containing a sampling element, the effect is somewhat different than what is normally experienced. Actually, the control loop is open except for an instant at the start of each sampling interval. The response of this sort of loop is not difficult to visualize, because it is nothing more than a series of open-loop responses. Under proportional control, a loop containing a dominant sampling element behaves just as if the sampling interval were pure dead time. For zero damping, P = 100
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70 = 2 At
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(4.14)
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Proportional control is obviously insuficient, as it was with pure dead time, so reset action is necessary. To facilitate identification of the influence of a sampling element, a process consisting of pure dead time and a gain of unity will be selected. The controlled variable will then follow the manipulated variable one dead time later. Figure 4.20 shows the first case, where 7d = 0. The measurement CT is seen by the controller for the entire sampling The interval between n = 1 and n = 2; t hen c* is changed to c,*. sampled variable c* normally appears in histogram form, as shown in Fig. 4.18, but to simplify this and following figures its value at the beginning of each sample interval will be indicated by a circle. The results shown in Fig. 4.20 are identical to those obtained with complementary feedback on a dead-time process: a loop gain of 2.0 produces uniform oscillation at a period of twice the delay element, and a loop gain of 1.0 gives critical damping. But the case of zero dead time is purely hypothetical. To be of value, any method for estimating control-loop performance cannot be so limited. Figure 4.21 gives the conditions for zero damping if the dead time of the process is one-half or one sampling interval
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2 s :: ct 0 2 3 n = t/At (a) 4 5 0 2 3 n = t/At (b) 4 5
FIG 4.20. When dead time is zero, (a) a reset time of At/2 produces uniform oscillations, (b) reset of At gives critical damping.
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2 3 n= t/nt (a)
n= t / A t (b)
FIG 4.21. Zero damping results when (a) R = T<, = At/2 or (b) R = Ed = At.
The period of oscillation shown in Fig. 4.21~ is 4 At, while that in Fig. 4.21b is 6 At. Add to these the period of 2 At when dead time is zero, and the formula for 70 with integral control is readily derived: 70 = 2 At + 4~~ (4.15) The contribution of the sampling element is evident, in that the natural period of a dead-time process with integral control is 4rd without sampling. Sampling adds 2 At to the period. The phase shift +A introduced by the sampling element can then be related to the period of the loop:
4A = -,A! = -1*(yE
(4.16)
The existence of any dead time whatever in the loop precludes critical damping with integral control. The value of reset time necessary for zero damping was At/2 for both Figs. 4.20~ and 4.21~2, although their periods of oscillation differed. But as oscillation becomes more sinusoidal, i.e., as more sampling intervals make up a period, for zero damping R approaches r,/2a, just as in a continuous loop. The sampled wave in Fig. 4.20~ is square, while that of 4.21~ contains three steps, such that R departs somewhat from 7,/2a. For >i-amplitude damping, R is to be doubled. The above estimates of reset time are based on unit process gain. They must be multiplied by the process steady-state gain K, in order to arrive at the required loop gain.
Two-mode Control
Two-mode control combines the speed of response of proportional action with the elimination of offset brought about by automatic reset. The proportional mode is just as valuable in a sampled dead-time loop as it was in one without sampling. In fact, proportional action enables any loop whose dead time is less than the sampling interval to he critically damped. Figure 4.22 shows how this is done.
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