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of Fig. 17.14. Of course, the terminology can be applied only to responses that resemble damped sinusoids. Values of the various parameters determined for the responses of Fig 17.14 am summarized in Table 17.4. Offset, realized only with proportional control, is included for completeness. It can be seen from Fig. 17.14 and Table 17.4 that addition of integral action eliminates offset at the expense of a more oscillatory response. When derivative action is also included, the response is much faster (lower rise time) and much less oscillatory (lower response time). The large overshoots realized in all three cases are characteristic of systems with relatively large time delays. In this case the controller is receiving information about the concentration in the second reactor that was true 1 min ago. This is to be compared with the reactor time constants of 1 and 2 min. Hence, it is not surprising that the system overshoots before the controller can take sufficient action. Figure 17.15 is presented for two purposes: (1) to illustrate that the ZieglerNichols controller settings should be regarded as first guesses rather than fixed values and (2) to show the effects of changing the various controller settings. These figures, which were obtained on a computer, am transient responses to step changes in set point for the three-mode PID control. They show the effects of individually varying the three control parameters Kc, ~1, and ~0. As an example of the use of these figures, suppose that it is decided that the maximum overshoot that can be tolerated is 25 percent. Figure 17.15a shows that overshoot may be reduced by decreasing K, at the expense of a considerably mom sluggish response. From Fig. 17.15b, we see that overshoot may be reduced by increasing ~1 (decreasing integral action) at a lesser expense in speed of response. Thus, for ~1 = 5 min, the overshoot is reduced to 20 percent without a serious sacrifice in speed. The overshoot cannot be significantly reduced by changing Q , as can be seen from Fig. 17.15~. However, the speed of response may be significantly increased by increasing the derivative action, at the expense of mom oscillation before the response has settled (higher decay ratio, lower period). From this brief study of these figures, it may be concluded that, to decrease overshoot without seriously slowing the response, a combination of changes should be made. A possible combination, which should be tried, is to reduce K, slightly and to increase TI and TO moderately. These changes would probably be tried on the actual reactor system when it is put into operation. Such adjustments from the preliminary settings are usually made by experienced control engineers, using trial procedures that are more art than science. For this reason, we leave the problem of adjustment at this point.
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TABLE 17.4 R rlod of oscillation, InhI 0.26 0.29 0.05 1.3 1.5 0.9 10.4 11.8 4.9 5.0 5.5 5.0
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Effeas of varying controller settings on system response. (2.4 indicates response usmg Llr;l;=rNichols settings.)
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17.1. Calculate the value of gain K, needed to produce continuous oscillations in the control system shown in Fig. P17.1 when (a) n is 2. (b) n i s 3 . Do nor use a graph for this calculation.
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FIGURE P17-1
17.2. (a) Plot the asymptotic Bode diagram IBk 1versus o for the in Fig. Pl7.2.
COMOI SYS@II
shown
CONTROL SYSTEM DESIGN
BY FRPQUENCY
RKWGNSE
FIGURE P17-2 (b) The gain K, is increased until the system oscillates continuously at a frequency
of 3 tad/mm. From this information, calculate the transportation lag parameter
17.3. The frequency response for the block G,in Fig. P17.3 is given in the following table:
GdIl 0.06 0.08 0.10 0.15 0.20 0.30 0.40 0.60 0.80 1350 1.40 1.m 0.84 0.61 0.35 0.22 0.11 0.066 -68 -88 - 105 - 145 -177 -235
Gp contains a distance velocity lag eSTs with T = 1 (this transfer function is included in the data given in the table). (a) Find the value of Kc needed to produce a phase margin of 30 for the system
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