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R dc/dt e=p=Rdm e dt =
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Reset time R must be adjusted for damping as with any other integrating controller. But the fact, that this is an integrating controller places an important limitation on its service: it cannot be used on non-selfregulating processes. The divider in Fig. 6.24 must operate in all four quadrants, because either the denominator or the numerator or both may be negative. A s the error, which is the denominator, passes through zero, the gain of the divider changes from plus to minus infinity or vice versa. Obviously, then, the system is extremely sensitive to noise around the point of equilibrium, i.e., at zero error. This has some undesirable features, but unfortunately is necessary for the system to function. In the steady state, dc/dt = 0 as does dm/dt. Therefore if the original state of the system is at rest at the wrong value of gain, it will not change its state without a disturbance. The signs applied to the summing junction in Fig. 6.24 would be used on a process whose gain decreased as 111 increased. The process-characteristic curve (c vs. 712) could go through a maximum, and control could be effected at that point, in which case K, = 0. If the process gain were to increase with 111, the signs at the summing junction would have to be reversed. It should be pointed out that equipment limitations prevent the use of this system on very slow processes. As R increases, differentiation becomes less accurat,e. Differentiation is at best an approximation,
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1Multiple-loop Systems
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anyway, because of the filtering that must be used for noise reduction and stability. This filtering is normaIly a lag of value about O.lR. The presence of this additional lag causes the phase of the controller to go beyond -90 at the natural period of the loop, making it a rather poor controller, from the point of view of stability.
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To overcome t#he equipment-limitation problem on slow processes, the optimizing search may be carried out discretely, using sampled data.5*6 This amounts to supplanting the differentials in the previous example with differences:
AC AC/At ATYL = Am, At
(6.22)
The sampling interval At must be long enough to let the process return to equilibrium after each change in controller output. When At has expired, the most recent AC/AM is calculated and compared to K,: e, = $$ - K, n Next, the output of the controller is stepped proportional to the error signal by a gain K,: Am n+~ = Keen The effect#ive reset time is related inversely to gain and directly to sampling interval At: (6.23) The sampling optimizer is not affected by the same equipment limitations as its continuous counterpart, but its other characteristics are similar. The sampling is of some advantage on processes dominated by dead time, but introduces the same uncertainty factor as other sampling controllers encountered. I\ otice the similarity of the control programs to that of the incremental DDC algorithms. Naturally a DDC computer may be readily programmed with this optimizing function. Sampling, however, still does not permit its use on processes without self-regulation.
A Peak-seeking Controller7
The heart of a peak-seeking controller is a LLone-way storage device: it accepts only increasing inputs. This function can be performed by a capacitor charged through a diode, as shown in Fig. 6.25. In an optimizing control system, the difference between the input, and output of the
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peak storage circuit determines the direction of control action. The output acts as the set point, while the input is the controlled variable; their difference is the error. put q+p + AS long as the error is zero, the manipucl 0 lated variable is being driven in such a way FIG 6.25. The peak storage as to increase the controlled variable. But circuit must be capable of the appearance of an error indicates that the ~ ~~~ ~~~~nto accept new . controlled variable has started to fall. At this point, the direction of the manipulated variable must be reversed to relocate the peak. This reversal has to be maintained long enough for the process to return to its peak value, so a time-delay lock on the reversal switch must be enforced. The peak storage circuit also requires resetting upon each reversal, otherwise it would cease to function. The manipulat,ed variable is driven at a constant speed either up or down, as the reversal switch dictates; it never comes to rest. As a result, the process will limit-cycle about the peak. The period of the limit cycle is that of the time delay, if it is set long enough to allow the process to recover from a reversal. Otherwise the process will cycle at its natural period. As with other self-optimizers, this, too, is an integrating device. I t therefore cannot be used on non-self-regulating processes. This class of processes has no equilibrium, no steady state, so the characteristic curve never comes to rest-it is always floating. Unfortunately, floating control action cannot hold it.
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