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FIG 4.10. The complementary controller has a model of the process in its positive feedback loop.
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FIG 4.11. With complementary feedback, m images r, producing response which appears to be open loop.
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The output change Am proceeds through the process to cause AC = Am K,g, = 100 Ar K, $f Since lOOK,/P = 1, the set-point response is
AC = Ar g,
Passing through the subtracting junction, e = Ar - AC = Ar -- Am K,g, The signal then appears at the summing point as
Kpgp = 100 $ - Am g, 100; = 100 $ - 100 Am p
Within the controller, complementary feedback is sending +Am g, to the same summing point, such that Am will retain its original value of 100 Ar/P as e returns to zero. The controlled variable responds as it, would if the loop were open, because the output of the controller is constant after the new set point has been inserted. Overshoot is impossible with this control arrangement, so critical damping is always attained. Figure 4.11 shows how a typical process might respond to a step set-point change. The principal advantage of this type of control scheme is that critical damping can be achieved with a proportional loop gain of unity; in most control loops, the proportional gain exceeds unity. In a single-capacity process, for example, the proportional band may be reduced to zero, placing the proportional loop gain at infinity. Yet there are some processes, notably those dominated by dead time, in which the loop gain must be much less than unit,y to obtain the desired damping. Figure 1.26 shows the required proportional band for s/4-amplitude damping for any combin:tt,ion of dead time and capacity. A band of 100 percent (proportional gain of 1.0) is seen to be required for a process whose T~/T~ = 1.2. But with complementary feedback, the same proportional gain could produce c~ritjcal damping. Complementary feedback is, by this token, of advant,agc in the most dificult processes.
1 Selecting the Feedback Controller
For Dead Time
In theory, complementary feedback is capable of critically damping a process consisting of pure dead time. Following the lines of the example given for proportional control of dead time in Chap. 1, the advantages of complementary feedback will be demonstrated. For simplicity, let K, = 1. The controller output is
wl* =
9 (1 - c ) + m,-1
where n = t/Td.
cm = Inn-l
The process responds:
Starting at conditions t . = co = )/lo = 0 percent, P = 100 percent, a set-point change of 50 percent initiates the following sequence: 1 0 = 0%
1 1 = 5 0 co = 07 Cl = 0
c2 f 5 0
1111 = 1.0(50 7122 = 1.0(50
= o( c
- 0) + 0 = 50 - 50) + 50 = 50
The process comes to rest in one dead time. The best value of lOOK,/P is 1.0. At 2.0, the loop becomes undamped, while at <l.O, damping is heavier than critical. Figure 4.12 illustrates the effect of changing gain. So some variation in gain can be tolerat ed. Unfortunately the same is not true for the complementary feedback term g,. If the positive feedback arrives at a different time than the negative feedback from the process, the loop mill break into oscillations of two periods, which are the sum and the difference of the two dead . times. Perhaps the most significant features of complementary feedback, as brought out in this example, are: 1. No offset 2. Fast response (TV = 27d) 3. Availability of critical damping
FIG 4.12. The amount of damping varies inversely with loop gain.
0 1 2 3 4 n=t/r* 5 6 7 0
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FIG 4.13. A proportional-plus-reset controller is the complement of a single-capacity process.
Integral control of dead time was able to eliminate offset, but at 7O = 4rd. Neither proportional nor integral control was capable of critically damping the loop.
For Single Capacity
Although there is no need to use a complementary controller on simple processes, it is nevertheless interesting to speculate on its configuration. If the process is a first-order lag, its complementary controller turns out to be proportional-plus-reset. In fact, pneumatic two-mode controllers are made this way, as shown in Fig. 4.13. A single-capacity process can tolerate zero proportional band and zero reset time. Compared to a dead-time process, it can be concluded that the easier the process is to control, the less critical are its mode adjustments.
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