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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