Proportional Time Control in Visual Studio .NET

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Proportional Time Control
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In systems where on-off conk01 produces a limit cycle that is both t oo long and too high, certain modifications may be applied. Proportional time control is a technique by which the on-off output is modulated with a signal of fixed period but variable on time. The percentage of each period during which the controller output is maximum is proportional to deviation. Thus the average value of output is the same as it would be with a proportional controller. Figure 5.10 shows the relationship between deviation and controller output. Because this type of controller naturally oscillates, the loop is forced to limit-cycle at the period of modulation, 7m. This period should be selected so that, the gain of the process is low enough for its limit cycle to be of negligible amplitude. Having this assurance, the loop approaches proportional control with a linear final operator. The criteria for adjust-
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FIG 5.10. Percentage on time varies as the measurement passes through the proportional band.
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ing the proportional band are essentially the same as with a linear loop, and offset is encountered for the same reason.
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A Constant-speed Motor as the Final Operator
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Up to this point only two-state on-off control has been presented, where the final operator would be a solenoid valve or an electrical heating element. These are either on or off. But very often a constant-speed reversible motor is used to drive a valve or to position a lever. This type of operator has three states: drive upward (opening), stop, and drive downward (closing). So the controller must be similarly arranged. The simplest controller for this function consists of two on-off devices whose inflection points are separat#ed by a dead zone. Wit,hin this zone, the motor would be stopped. In practice, each on-off device also contains a small differential gap. The input-output relation is pictured in Fig. 5.11. A finite dead zone must exist, first to ensure that the on-off switches do not overlap, for this would energize both windings of the motor simultaneously and could result in damage. But beyond this, it provides a
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FIG 5.11. The three-state controller is comprised of two on-off devices separated bg a dead zone 2.
Deviation
Nonlinear Control Elements
state of rest for the loop that did not exist with previous on-off configurations. Thus the control system is not bound to limit-cycle. The dead zone Z is defined as the change in deviation between deenergizing one output and energizing the other. To avoid a limit cycle, the dead zone must be wide enough SO that energizing one output will not change the controlled variable enough to energize the other. The amplitude of whatever limit cycle might be generated would be 400G,/a, where G, is the process gain at the natural period including the phase lag of the gap and the motor. Therefore, if limit cycling is to be avoided, z > 400 G, a (5.5)
Along with the stability given by the dead zone comes offset. Instead of a limit cycle of amplitude 400G,/a, an equal offset is encountered. At least the limit cycle had an integrated error of zero; but offset integrates to infinity. The limit cycle with a constant-speed motor will always be centered around the set point, because the motor is an integrator. So the average value of the error will always approach zero-this was not so with twostate on-off control. But the presence of an integrating element (the motor) in the loop doubles its natural period.
Adding Other Control Modes
If limit cycling and offset are both unacceptable, a two-mode controller can be added to the loop. This controller would actuate the three-state device which in turn drives the valve, as shown in Fig. 5.12. Reset action will keep driving the output of the controller out of the dead zone until the error is reduced to zero. Only then will the loop reach a steady state. Proportional action is necessary for stability, for without it, the double integration of reset and motor would cause an undamped cycle. The availability of a proportional band adjustment eliminates the need for an adjustable dead zone, since the two effects are similar. Derivative may also be added to advantage. It contributes about as much to the loop as it does to a linear loop by reducing the period of oscillation and allowing a narrow proportional band. Derivative cancels, to some extent, the sluggishness of the constant-speed motor.
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