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In the case of a dead-time process with perfect complementary feedback, a step disturbance in load would produce an error corrected one dead time later. Figure 4.14 shows the results. The magnitude of the error is AC = Am K,, and its duration is rd. Therefore the integrated error per unit load change is (4.12) To properly evaluate this response, a two-mode controller will be applied to the same process, but it must be adjusted so that the error will not cross zero during recovery. Thus integrated error is actually IAE, permitting comparison of loops with different damping. A phase
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FIG 4.14. A load-induced error prevails for one dead time.
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AmKp -----------A
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Time
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1 Selecting the Feedback Controller
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lag of 22.5 is chosen for the controller, such that r0 = 2.3rd. The tangent of 22.5 is 0.414. Reset time is then 2.3rd R = 2n(0.414) = 0.887d Quarter-amplitude damping requires loop gain to be 0.5.
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GPRKp = 0.5 = 100Kp4 1 + (0.414)2 P P = 2OOK,4i3 = 216K,
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Integrated error is then E am = 100 = (2.16KP)(0.887d)
= 1.90KPTd
This indicates that complementary feedback significantly reduces IAE, as well as settling time. It is important to see over what range of processes complementary feedback has an advantage over two-mode control. A single-capacity plus dead-time process will respond to a step load change under complementary feedback as shown in Fig. 4.15. Without going into the derivation of the load-response curve, it turns out that the integrated area per unit load change is (4.13) A proportional-plus-reset controller applied to the same process, and adjusted to produce 22.5 phase lag, can serve as a reference for comparison. The values of reset time and proportional band required for >i-amplitude damping were calculated for selected ratios of T~/TI in the process. The integrated error per unit load change was then found as the PR product, to compare with that obtainable through complementary feedback. This information is plotted in Fig. 4.16, with coordinates
FIG 4.15. The peak of the loadresponse curve occur8 7d min from its ri*e.
Time
Linear Controllers
FZG 4.16. Complementary feedback is superior to two-mode control for processes more difficult than 7d/T1 = 1.3.
w $ aI 0 1
2 w=1 3 4
normalized as E/Am Kprd vs. rd/rl. plots as
The complementary feedback curve
E __- =1+71 Am Kprd Td
It was pointed out earlier that, as in regard to closed-loop gain, complementary feedback was superior for processes more difficult than Td/71 = 1.2. But the comparison was not exactly on the same basis, because the proportional band was selected for s/4-amplitude as opposed to critical damping. The comparison shown in Fig. 4.16 is limited in the same way, but the agreement of the two methods is evident. The intent. has been to prove in two ways that, complementary feedback should be reserved for only the most difficult applications.
Practical Considerations
Pure dead time cannot be generated by analog means, therefore a dead-time complementary analog controller will never be available. Dead time can be generated digitally, so the possibility exists for direct digital control systems. But in view of the problems that can be anticipated from a mismatch of process and controller dynamics, complementary feedback is of questionable value for any pure dead-time process. (A similar and more reliable method will be presented laterin this chapter.) In process-control work, capacity nearly always exceeds dead time, hence loop response generally has a moderate tolerance for controller maladjustment. This is fortunate, because to make the dynamics of the controller duplicate those of the process is not really possible. The foregoing discussion on complementary feedback was based primarily on critically damped response. With a pure dead-time process, this was the best obtainable. But with less difficult processes, lower damping will enhance recovery from load disturbances due to greater controller gain. In general, better performance will be obtained on more difficult applications by using delayed reset,4 as shown in Fig. 4.17. This is obviously a compromise between two-mode control and complementary feedback,
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