VARIATIONS OF THE ON-OFF CONTROLLER in Visual Studio .NET

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VARIATIONS OF THE ON-OFF CONTROLLER
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A pure on-off control function produces an output of either 0 percent (off) or 100 percent, (on), according to the sign of the error. Because a sinusoidal error of any amplitude will produce a square wave of unit amplitude, the device is said to have a variable gain. In order to establish a figure of gain for any device, its out put should be expressed in the same terms as its input. So instead of a 100 percent square wave, the output will be thought of as the fundamental sinusoidal component of the square wave. This component has an amplitude of 4, ~ times that of the square wave. Then the gain of the on-off cont roller, G,, becomes the output amplitude 400/a over t,he input amplitude A, expressed in percent:
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An on-off controller employed on any process capable of shifting phase beyond 180 will cause a limit cycle. The loop gain G,G:, under these conditions is 1.0, GP representing the process gain at its natural period. Therefore the peak-to-peal; amplitude of the limit cycle, expressed in percent, is
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Figure 5.7 shows the traject,ory of a limit cycle under on-off control. Because the load is low, the limit cycle is not centered about the set point.
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1 Selecting the Feedback Controller
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m Differential
100%
FIG 5.7. An input-output graph for on-off control.
There is no such thing as a perfect on-off device, and if there were, it would be useless. If it could change stat& instantaneously with an infinitesimal input, the presence of any noise whatever would cause constant chattering. Therefore real on-off controllers are all limited in their action by one of the following mechanisms: 1. Gain sufficiently low to require a significant input to cause saturation, as in a controller with a very narrow (but not zero) proportional band. 2. Positive feedback, causing the device to lock in either output condition. This action is called differential gap, and is characteristic of switching devices. Differential gap is hysteresis in an on-off controller. A signal must change by an amount equal to the gap H in order to change the state of the output. The input-output characteristic of an on-off controller with differential gap appears in Fig. 5.8. The phase characteristics of differential gap are the same as those of hysteresis. But because of the on-off action, gain differs. An input signal of amplitude lower than the widt#h of the gap produces no output; otherwise, gain is that of an ideal on-off controller. Any process whose dynamic elements are capable of shifting phase in excess of - 180 will limit-cycle under on-off control. Presence of differ100
s i 2
FIG 5.8. The state of the output will not be changed until the deviation exceeds the width of the gap.
Deviation
- o +
Nonlinear Control Elements
Time
FIG 5.9. In the presence of differential gap, even a single-capacity process will limit-cycle.
ential gap will increase the amplitude of the cycle by approximately the width of the gap and will lengt,hen its period proportionately because of the additional phase lag. For an exact prediction, the phase and gain formulas given in Eqs. (5.1) and (5.3) should be applied. But because an on-off controller forces a process in a full-scale square wave, its resulting error signal in many casts will be far from sinusoidal. Under these conditions the equations given for sinusoidal response are inexact. A case in point is the control of a single-capacity integrating process by a device with differential gap. The resulting limit cycle is triangular, as shown in Fig. 5.9. With only 90 phase lag in the process, the full 90 lag in the controller is consumed in a limit cycle of A = H. Because the gain of this function to signals smaller than a gap width is zero, it also acts as a nonlinear filter, rejecting noise of low amplitude.
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