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p = 1 + D/R
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R'=R+D
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D' = l/D + I/R
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Several important observations can be made from the above relationships: 1. Where D > R, derivative time is affected more by the reset setting, and vice versa. 2. It, is impossible to make the effective derivative time equal to or greater than the effective reset time. 3. As D approaches R, further adjustment will produce very little change in D', so there is little purpose in trying to fine tune an interacting controller. To illustrate the above points, Table 4.1 lists several combinations of settings together with their effects. This example was chosen to show how three markedly different combinations of adjustments can result in substantially the same effective values. Some engineers have prepared
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FIG 4.8. Three-mode controllers: (a) interacting, (b) noninteracting.
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TABLE 4.1 Effective Values of Modes at Various Settings
Controllers
P 20 15 13.3
P 10 10 10
R 1.0 1.5 2.0
R 2.0 2.25 2.67
D 1.0 0.75 0.67
D 0.5 0.5 0.5
tuning maps, in which the response of a given process to various combinations of proportional, reset, and derivative settings are compared. The results would bear some merit if a noninteracting controller were used, but interaction limits the range of effective settings too stringently. I t is doubtful whether any difference in response obtained with the three combinations in Table 4.1 would be noticeable. The effective values of proportional and derivative are the same in each case, while effective reset time only changes from 2.0 to 2.67 min-hardly noticeable. The integrated error is, of course, a function of the effective values of proportional and reset:
E P'R' -=Am
Substitution for P' and R' reveals the relationship of integrated error to the settings of the two modes: E -= A112 lOO(1 +
P(R+D) P R D/R) = ii%
The integrated area is unaffected by this interaction. A more severe form of interaction exists in pneumatic controllers whose reset and derivative circuits are in parallel feedback about the amplifier. Pneumatic controllers equipped with an antiwindup switch are connected in this way. The only difference between these and the conventional interacting controller is in the effective proportional band:
P'= P(gg)
(4.10)
When D = R, the effective proportional band is zero. And if D > R, the effective band actually becomes negative; negative proportional band in a negative feedback controller means positive feedback. Extreme care must therefore be exercised when adjusting one of these controllers, or the results could be disastrous.
Adjusting Twoand Three-mode Controllers
In order to formulate an effective procedure for adjusting controllers, it is first necessary to determine where the optimum values lie. To per-
1 Selecting the Feedback Controller
mit extension to a broad range of difficult processes, a dead-time plus integrating process will be used, with controller settings left in terms of 74 a n d 71. This is necessary because the natural period of a loop does vary with the controller settings. Table 4.2 has been prepared by equating the gain product of process and controller to O.Tj, with gain and phase for the controllers accurately calculated from the vector diagrams of Figs. 1.11 and 4.7. Several significant conclusions may be drawn from TabIe 4.2. Derivative is very effective in improving the performance of the loop, even though the dominant secondary element is dead time. But the reason that it is so effective is because it offsets the phase lag of reset, preventing the period and process gain from increasing in its presence. A striking characteristic of the optimum controller setting for both of the threemode controllers is that the phase contribution of the controller is zero at the natural period. Furthermore, derivative and reset times should be equal. Thus the optimum settings can be predicted with accuracy knowing either the dead time of the process or the period of oscillation under proportional control. From these conclusions, it is possible to formulate a rigorous adjustment procedure for three-mode controllers: 1. With maximum reset time and minimum derivative, excite the closed loop into oscillation by reducing the proportional band.
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