controls lab
DESCRIPTION
Process Controls Laboratory ReportTRANSCRIPT
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Process Controls Laboratory Report
Modern process control is based upon the principle of automatic feedback loops
which continually measure process variables and adjust the controller output to achieve
desired set point values. In the majority of process feedback controllers, proportional-
integral-derivative (PID) controllers are used. Mathematically, the dependent, ideal PID
controller equation can be expressed as:
(1)
where u(t) is the controller output signal (CO), ubias is the controller bias, e(t) is the
controller error, and y(t) is the measured process variable (PV) [1]. This equation
includes the three different types of controllers; P-only control which includes only the
proportional term, PI control which includes both the proportional and integral terms, and
PID control which includes both aforementioned terms plus the derivative term.
To optimize a PID controlled process, controller tuning may be required. PID
controller tuning refers to the adjustment of the controller gain, Kc, reset time, τI, and
derivative time, τD. A set of correlations used for tuning the P-only, PI and PID
controllers include the ITAE (Integral Time Averaged Error) and IMC (Internal Model
Control) correlations (Appendix C) [1]. Using these correlations, a range from
conservative to aggressive tuning parameters can be calculated and used in the controller
tuning equation. To assess the performance characteristics of each controller, such terms
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as rise time, offset, peak overshoot, decay ratio and settling time are considered. A brief
description of these terms is included in Appendix B.
One widely used method to deduce the initial values for tuning variables is called
the First Order Plus Dead Time (FOPDT) model. The equation for the time domain form
for the FOPDT model is:
τ P
dy (t)dt
+ y ( t )=K Pu (t−ϴP ) (2)
By fitting an FOPDT model to process data, values for the process gain, KP, process time
constant, τP, and process dead time, ϴP, can be determined. The process gain describes
how the process variable changes with controller output. The process time constant is a
measure of how fast the process variable responds and is the time it takes the process
variable response to reach 63.2% of its total change in response to a change in controller
output. The process dead time is the amount of time that passes from the start of a step
change in controller output to the time the process variable shows a clear response. The
equations to calculate these FOPDT parameters are included in Appendix C. These
FOPDT variables describe the dynamic behavior of the process and are important
because they are used in the ITAE and IMC correlations to compute the controller tuning
values [1].
In this experiment, the dynamic behavior of a liquid level control process was
analyzed and FOPDT based controller tuning parameters determined to assess the
performance of P, PI and PID controllers. The apparatus consisted of two draining tanks,
4 in. diameter and 24 in. length, in vertical series above a 12 gallon holding tank. Water
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was drawn from the holding tank and pumped into the top tank. A controller was
connected to the lower tank level transmitter and the pump. The controller could be
placed in both manual and automatic mode to control the process. The process gain, reset
rate (1/τI) and derivative time could also be input into the controller to have it function in
P, PI or PID mode. The diagram of the apparatus is shown in Figure 1. To collect and
analyze the process data, the software program Loop Pro was used.
The experimental procedure consisted of first determining the dynamic nature of
the process by performing step tests in manual mode. Following this, an FOPDT model
was fit to the process by performing a doublet bump test. From this model, initial tuning
parameters were determined for a moderately-aggressive controller. Set point tracking
performance of moderately-aggressive P, PI and PID controllers were then examined by
altering the appropriate tuning parameters. Finally, the disturbance rejection performance
of the PI controller was analyzed by introducing a 500mL aliquot of water to each of the
top and center tanks. For further details on the procedure see reference [2].
Figure 1: Experimental apparatus for liquid level control experiment.
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To convey the cause and effect relationship of the process, a control loop block
diagram as shown in Figure 2 can be used to describe the liquid level controlled, closed
loop operation.
As most real world processes are non-linear in nature, the draining tanks
experiment was examined to assess its dynamic behavior. Three successive step tests were
performed in 10% increments for a controller output from 20 to 50% (Figure 3) while in
manual (open loop) mode. For each step, the process gain, time constant and dead time
were calculated by Loop Pro utilizing the equations in Appendix C. The resulting values
are shown in Table 1. An example of how the KP, τP and ϴP can be calculated graphically
is included in Appendix C.
Figure 2: Control loop block diagram for liquid level controlled draining tanks in series.
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Controller Step Kp (in/%) τp (s) ϴp (s)20-30% 0.33 45.0 14.030-40% 0.42 60.9 16.640-50% 0.45 79.0 15.9
If this process were linear, then the process gain, time constant and dead time for
each successive step test would be equivalent. It is shown however, that this process is
non-linear as these values vary as the CO changes.
Due to the non-linearity of this process, the best approach to determine initial
tuning parameters would be to perform a doublet test which steps both above and below
the process design level of operation (DLO) to account for the dynamic behavior.
Therefore the process was stepped ± 10% CO around the DLO of 37% (10 in.) and a
moderately- aggressive FOPDT model was fit to the data as shown in Figure 4. Utilizing
Loop Pro’s Design Tools, the P-only, PI and PID tuning parameters for the FOPDT
model were determined and are shown in Table 2. Loop Pro utilizes the ITAE and IMC
tuning correlations given in Appendix C to compute each tuning parameter.
Figure 3: Step tests to examine the dynamic behavior of the draining tanks process.
Table 1: FOPDT Model Parameters
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Controller KC τI (s) τD (s)P-only 1.97 - -
PI 1.57 45.74 -PID 2.34 58.06 9.71
The performance of each of the three types of controllers varies due to the differing
components of controller equation (1). In P-only control, the only adjustable tuning
parameter is KC as the proportional term is the only term in the corresponding controller
equation. The proportional term adds or subtracts from CObias based on how far the PV is
from the SP at any instant of time. The setpoint tracking of a P-only controller was
examined by changing KC to ½ its original value and also doubling it as shown in Figure 5.
Figure 4: FOPDT model fit to doublet test for moderately aggressive controller.
Table 2: Moderately aggressive tuning parameters from FOPDT model.
Figure 5: Setpoint tracking of P-only controller by changing tuning parameter KC: 0.98, 1.97, 3.93.
Initial Case
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The advantage of P-only control is that there is only one tuning parameter to adjust
and therefore the best tuning values are obtained rather quickly, however as Figure 5
demonstrates, the disadvantage to P-only control is that it permits offset. To minimize
offset, KC may be increased, however this results in greater oscillatory behavior. Offset for
KC values of 0.98, 1.97 and 3.93 were 1.7 in., 1.2 in., and 0.2 in. respectively.
In PI control, there is both the proportional and integral term of equation (1). The
integral term considers the error (e(t) = PV-SP) over time and based on the tuning
parameters for KC and τI, will continually add or subtract from CObias and cause the
controller output to change. The advantage to PI control is that it eliminates the offset
present in P-only control by minimizing the integrated area of error over time. To assess the
effect changing the two tuning parameters has on a PI controller performance, both KC and
τI were halved and doubled as shown in Figure 6.
Figure 6: PV SP tracking of PI controller by changing tuning parameters KC: 0.787, 1.574, 3.148 and τI: 22.87s, 45.74s, 91.48s. KC increases along the positive y-axis and τI increases along the positive x-axis.
Initial Case
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From Figure 6 it can be determined that the greatest oscillatory behavior occurs
when KC is at its largest value (3.148) and τI is at its smallest value (22.87s). In this process,
using these tuning parameters actually resulted in increased magnitude of oscillations over
time and thus an unstable system. Either lowering τI, or increasing KC from the initial value
resulted in a greater peak overshoot, larger settling time and larger decay ratio (Appendix
C). However it can be seen that halving τI has more of an effect than doubling KC as the
decay ratio is larger with 60% versus 43%, the settling time is longer with 675s versus
440s, and the peak overshoot is larger with 83% versus 70% (while holding the other
variable at the initial value).
In PID control all three terms in equation (1) are utilized. The function of the
derivative term is to determine the rate of change of the error (slope) and then add or
subtract to CObias and thus influence the controller output. A rapidly changing error will
have a larger derivative and therefore a larger effect on controller output. The derivative
term will therefore work to decrease the oscillatory behavior in the process variable. To
assess the effect of changing derivative time, a comparison of the tuning parameter τD was
made for the PID controller by halving and doubling the initial value of 9.71s (Figure 7).
Figure 7: PV SP tracking of PID controller changing tuning parameter τD: 0.081s, 0.162s, 0.324s.
Initial Case
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From Figure 7 it can be determined that increasing the derivative time results in
less oscillatory behavior of the process variable however there is also an increased noise
in the controller output. Increasing τD also increase rise time, settling time, and decreases
peak overshoot as shown in Table 4.
Performance Parameter
τD
(0.081s)τD
(0.162s)τD
(0.324s)Rise Time (s) 35 43 60
Settling Time (s) 105 128 170Peak Overshoot (%) 37.5 32.5 20
In normal operation, disturbances to a process may occur. To assess the
disturbance rejection performance of this experiment, a 500mL aliquot of water was
added to the lower tank while operating a PI controller in moderate, moderately
aggressive and aggressive modes (Figure 8). The tuning parameters used for the moderate
PI controller and aggressive PI controller can be found in Appendix C.
From the figure above it can be seen that as the controller becomes more aggressive
oscillatory behavior increases, settling time decreases, the decay ratio increases, while peak
overshoot remains generally the same in response to the disturbance. As the moderate PI
controller exhibits the best behavior with regards to setpoint tracking, its response to further
Table 4: Performance comparison of PID tuning parameter derivative time.
Figure 8: Moderate, moderately aggressive and aggressive response to unmeasured disturbance.
Moderately-AggressiveModerate Aggressive
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disturbance rejection was tested by adding a 500mL aliquot to the upper tank. This
response is shown in Figure 9 below.
Based upon the moderate PI controller response to the second disturbance, the
magnitude of the oscillations decreases, the settling time improves and the decay ratio
decreases. This would be expected to occur as the level transmitter is located on the lower
tank and therefore a change in the lower tank level would be immediately seen by the
controller whereas adding water to the upper tank does not significantly immediately
increase the flowrate into the lower tank.
In this study of P-only, PI and PID controllers on the draining tank process, there
were found to be certain advantages and disadvantages to each controller. The P-only
control exhibited offset in the process variable upon reaching steady state. In PI control,
integral action eliminated the offset but resulted in more oscillatory behavior in the
process variable. In PID control derivative action worked to eliminate the oscillations
however any noise present in the PV signal was reflected and amplified in the controller
output signal which would result in mechanical wear of final control elements.
Figure 9: Moderate PI controller response to upper tank disturbance.
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In choosing the ‘best’ performing controller it must be noted that best
performance is subjective, meaning that some processes may desire a PV response with
no overshoot, others may be able to tolerate overshoot and prefer faster rise times. For a
process that desires fast rise time with the minimal amount of oscillatory behavior and
overshoot it would be suggested to use a moderate to moderately aggressive PI controller.
This would results in good setpoint tracking, no CO noise, minimal oscillatory behavior,
fast rise and settling times along with good disturbance rejection. For this experiment this
would be a PI controller with KC, τI values of 3.148 and 91.48s or 1.575 and 91.48s or
1.574 and 45.74s or similar values within these ranges.
Appendix A: References
Hidden
Appendix B: Nomenclature
CObias: The value of the controller output signal when in manual mode, causes
the PV to settle at the DLO such that error = 0 when the disturbances
are at their normal or expected values.
Decay ratio: The size of the second peak above the new steady state divided by the
size of the first peak above the same steady state level.
Offset: Most notably associated with P-only controllers, is the difference from
the SP to where the PV settles out at a steady state value.
Peak overshoot: The size of the first peak above the new SP divided by the size of the
SP step.
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Rise time: The time from when the CO begins to step to when the PV to reaches
the setpoint (SP).
Settling time: The time at which the PV reaches ± 5% of the total change in the
process variable (ΔPV).
Appendix C: Data and Sample Calculations
ITAE (P-only) and IMC Tuning Correlations
Graphical determination of FOPDT parameters. Step test from 20-30% CO output.
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K P=ΔPVΔCO
=3.15∈.20 %
=0.1575∈./%
PV63.2 = PVinitial + 0.632 (ΔPV) = 3.5+0.632(3.15) = 5.49 in.
τ P=t 63.2−tPV start=91.5 s−46 s=45.5 s
ϴP=tPV start−tCO step=46 s−32=14 s
Performance Characteristics of Changing Tuning Parameters of PI Controller
DR: decay ratio, ST: settling time, PO: peak overshoot, RT: rise time
KC: 3.148 τI: 22.87s
UNSTABLESYSTEM
KC: 3.148 τI: 45.74s
DR: 43%ST: 440sPO: 70%RT: 30s
KC: 3.148 τI: 91.48s
DR: 27%ST: 232sPO: 44%RT: 38s
KC: 1.574 τI: 22.87s
DR: 60%ST: 675sPO: 83%RT: 34s
KC: 1.574 τI: 45.74s
DR: 21%ST: 235sPO: 48%RT: 45s
KC: 1.574 τI: 91.48s
DR: 11%ST: 174sPO: 23%RT: 50s
KC: 0.787 τI: 22.87s
DR: 10%ST: 323sPO: 49%RT: 57s
KC: 0.787 τI: 45.74s
DR: 0%ST: 300sPO: 23%RT: 75s
KC: 0.787 τI: 91.48s
DR: 0%ST: 110sPO: 0%
RT: 115s
Note: above values are estimated determined from graphical analysis
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Summary of Tuning Parameters
Tuning Controller KC τI (s) τD (s)Moderate
(Conservative) Controller
P-only 1.24 - -PI 0.59 45.74 -
PID 0.80 58.06 9.71Moderately- Aggressive Controller
P-only 1.97 - -PI 1.57 45.74 -
PID 2.34 58.06 9.71
Aggressive Controller
P-only 3.12 - -PI 3.11 45.74 -
PID 5.57 58.06 9.71
Appendix D: Lab Notebook