Integral (Reset) Control of Dead Time in Visual Studio .NET

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Integral (Reset) Control of Dead Time
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Proportional control is obviously rejected for most applications demanding a band wider than a few percent. So another control mode is needed. An integral controller is a device whose output is the time integral of the deviation:
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1 = e dt R /
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(1.6)
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where R is the time const.ant of the controller, known as integral or reset time. As long as a deviation exists, this controller will change
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FIG 1.7. The response to a load change illustrates how the proportional band affects both damping and offset.
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Dynamic Elements in the Control Loop
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FIG 1.8. The output of an integrator will change by an amount equal to its input in time R.
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its output, hence it is capable of driving the deviation to zero. of change of output is proportional to the deviation:
The rate
(1.7) Response to a step input is shown in Fig. 1.8. Before using an integral controller in a closed loop, its gain and phase characteristics must be defined. Again we are primarily interested in these properties at the natural period of the loop, TV. Introducing a sinusoidal input to the controller, e = A sin 2a t To The controller output mill be the time integral of the input: 1 112 = edt=i R / R A sin 2a 4 dt 70 >
Extraction of the appropriate item from a table of definite integrals enables us to solve the above equation: m = g+os2*;) +wlo
where 71~0 is the output at time zero. In order to evaluate phase and gain properties, the output must be reduced to the same form as the input, using the trigonometric identity -cosz=sin(-5+X) We can convert 112 into a sine function:
Understanding
Feedback
Control
The phase shift of the integrator is the angle of the output minus the angle of the input:
n--z
-90
(1.8)
An integrator exhibits a phase lag of 90 regardless of the period of the input. The gain of an integrator is the amplitude of the output over the amplitude of the input: G
= A70/2aR A =2:R
(1.9)
e FIG 1.9. Adjusting reset time affects the damping.
Dynamic Elements in the Control Loop
FIG 1.10. Increasing reset time
trades recolrery for damping, although 7O is unaffected.
Time
In closing the loop, the sum of the phase shift of the dead time and the integral controller must equal -7r at the natural period TV:
lr 2lrTd -lr= ---~ 2 70
-180 = -90 - 360 z
Solving for 70, 7 o -- 4Td (1.10)
Notice that the period is twice that for proportional control, because only 90 of phase shift was allowed to take place in the dead-time element. To sust ain oscillations, the loop gain must be 1.0. Since the dead-time gain is already 1.0, the integrator gain for this condition must also be 1.0. Solving for reset time, GR = pR = 1 . 0 lr R=+ T To summarize, a dead time of 1 min would cycle with a period of 4 min, sustained by a reset time of 2/r, or about 0.63 min. Quarter-nmplitude damping can be achieved by halving the gain, which means doubling the reset time. Figure 1.9 shows the entire situation. Again, the controller has but one adjustment, which only affects damping. The period of oscillation and the integral time for f/l-amplitude damping have been established by the process. Use of the integral controller has avoided the previously encountered proportional offset, but at the cost of reduction in speed of response. The response of a dead-time process under integral control to a gradual load change is pictured in Fig. 1.10. The rate of recovery is slow xhen the reset time is too long. With a proper amount of reset, the measurement will cross the set point during the first cycle, exhibiting $i-amplitude damping.
Proportional-plus-reset Control
(1.11)
This controller combines the best features of the proportional and integral modes in that proportional offset is eliminated with little loss
-2-w +PR
Reset
---Gspasc \ +p=o
GR~100~,/2,,RP +R =-90"
E - - - - - - -
G,,,~w J~+OZ
.#&on-%,,/2nR
Proportionaltreset
FIG 1.11. The resultant gain is the square root of the sum of the squares of the components.
of response speed.
The controller is represented as follows:
m=T(e+$/edt) Having already found the performance characteristics of each of the modes individually on a dead-time process, intuition dictates that the performance of the combination will be somewhere in between, e.g.,
depending on the particular combination of sett ings of proportional and reset. An infinite combination of settings can be found t,o provide constant damping. We have already seen that for s/4-amplitude damping, 100=05 P * or &= 0.5
depending on the control mode used. For the two-mode controller, then, the sum of the gains must equal 0.5. The proportional and integral components of gain are out of phase with each other, however. So their resultant gain must be the vector sum of the two components. Figure 1.11 shows the relationship between the vectors.
200 P B 0 _ 100 P /_ _N / / / / 0 0.5 1.0 ro/2rR /
A/ -//Reset Proportional _
FZG 1.12. A plot of gain vs. 7O for the proportional-plus-reset controller shows the contributions of the components.
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