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FIG 1.2. The response of the weigh cell to a change in solids flow is delayed by the travel of the belt.
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Dynamic Elements in the Control Loop
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FIG 1.3. Pure dead time transmits the input delayed by T+
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tute the delay. Dead time is also called (pure delay, transport lag, or distance-velocity lag. As with other fundamental elements, it rarely occurs alone in a real process. But there are few processes where it is not present in some form. For this reason, any useful technique of control system design must be capable of dealing with dead time. An example of a process consisting of dead time alone is a weightcontrol system operating on a solids conveyor. The dead time between the action of the valve and the resulting change in weight is the distance between the valve and the cell (feet), divided by the velocity of the belt (ft/min). Dead time is invariably a problem of transportation. A feedback controller applies corrective action to the input of a process based on a present observation of its output. In this way the corrective action is moderated by its observable effect on the process. A process containing dead time produces no immediately observable effect-hence the control situation is complicated. For this reason, dead time is recognized as the most difficult dynamic element naturally occurring in physical systems. So that the reader may begin without illusions about the limitations of aut,omatic controls in their influence over real processes, the difficult clement of dead time is presented first. The response of a dead-time element to any signal whatever will be the signal delayed by that amount of time. Dead time is measured as shown in Fig. 1.3. Notice the response of the element to the sine wave in Fig. 1.3. T h e delay effectively produces a phase shift between input and output. Since one characteristic of feedback loops is the tendency toward oscillation, the property of phase shift becomes an essential consideration.
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The Phase ShiFt of Dead Time
We are primarily interested in phase characteristics of elements at the natural period of the loop. Assume, to begin, that a closed loop containing dead time is already oscillating uniformly. The input to the process is the sine wave
n 1 Ud erstanding Feedback Control
FIG 1.4. The manipulated variable is cycling with an amplitude of A at the natural period.
where m = manipulated variable whose average component is m. A = amplitude t = time 7O = period Phase angles will be expressed both in degrees and in radians for reasons that will become clear later.
2&/r,
t/r*
0 s/4 34 1
sin 27d/ro Degrees 0 90 180 270 360 Radians 0 0
31r;z 2T
0 -1 0
This wave, passing through a dead time, will be delayed by an amount Ed, but will be undiminished, so that the output will be c= Asin27r~+m0 TO The input angIe subtracted from the output angle yields the phase shift &:
= -2=7d = -360 7--d 70 70
(1.1)
The negative sign indicates a lag in phase. Because dead time does not alter the shape or amplitude of a signal, its gain Gd is unity to all periodic waves:
Gd = 1 . 0
(1.2)
Dynamic Elements in the Control Loop
Proportional Control of Dead Time
Having defined the process, the next step is the selection of a suitable controller. A proportional controller will be chosen first, because of its simplicity. It contains no dynamic elements. Output and input are related by the expression
Wl+ +b
(1.3)
where P = proportional band, Y0 e = error or deviation of the measurement from set point b = output bias As P approaches zero, the gain of the proportional controller approaches infinity. At 100 percent band, the gain is 1.0. The output of the controller equals the bias when there is no error. Because there are no dynamic elements in the proportional controller, the entire 180 phase shift will take place in the dead-time element. This determines the natural period: C$d = -180 = -- T
Substituting for the previously determined &, -zn7d =
70 -T
-360':
-180'
Solving for 70,
70 = %Td (1.4)
The relationship is as plain as it appears. A I-min dead-time process will cycle with a 2-min period under proportional control. This is not an approximation-it is exact. Next it is important to estimate the proportional band necessary to sustain oscillation. Dead time offers no gain contribution, so if the loopgain product is to be 1.0, the controller proportional band must bc 100 percent. To dampen the oscillations, the band must be increased, thus att,enuating the input cycle. Figure 1.5 illustrates how a proportional band of 200 percent reduces the amplitude of each successive half-cycle by one-half, resulting in >i-amplitude~damping of each successive cycle. This degree of damping is generally accepted as nearly optimum throughout the industry. Notice that ,there is only one adjustment available, and it affect s the damping. Given a process consisting of a I-min dead time to be COW
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