compensation of dead time in pid controllers

8/9/2019 Compensation of Dead Time in PID Controllers

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2006-12-06 Page 1 of 25

Advanced Application Note 003 Rev A FisherRosemount Systems, 2007. All Rights Reserved.

Compensation of Dead Time in PID

Controllers

AdvancedApplication Note


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Compensation of Dead Time in PID Controllers2006-12-06 Page 2 of 25


Table of Contents:

1 OVERVIEW............................................................................................................................................3

2 RECOMMENDATIONS .........................................................................................................................6

3 CONFIGURATION.................................................................................................................................7

4 TEST RESULTS.................................................................................................................................. 11

5 REFERENCES ....................................................................................................................................25


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1 Overview

PID controllers with dead time compensation are reported to eliminate dead time in terms of a controllerseeing the effect of changes in its controller output. For set point changes where all the controller needsto be concerned with is how its output responds to a new set point, the results are impressive for an exact

knowledge of the process dead time. However, for unmeasured load disturbances at the process input,the ultimate performance is set by the total dead time from the process equipment, piping, control valves,instrumentation, and digital devices. This application note shows that a dead time compensator can offersome improvement in load rejection by facilitating more aggressive tuning of the PID but with aconsiderable risk of oscillations from an inaccurate dead time.

The ultimate performance achievable in terms of load disturbance rejection depends upon the dead time.In the Theory section of Chapter 2 of Advanced Control Unleashedequations are developed that showthe minimum peak error is proportional to the dead time and the minimum integrated error is proportionalto the dead time squared for unmeasured load upsets. How close the actual performance of a controlloop comes to this ultimate performance depends upon PID structure, tuning, and enhancements. Thisblog focuses on the effect of variations in dead time on the performance and robustness of dead timecompensation as an enhancement and Lambda as a tuning rule for disturbance rejection. The two

predominant methods of dead time compensation studied here are the Smith Predictor PID and the PIDwith a delayed external reset.

The Smith Predictor was extensively documented in the 1970s. It provides a new controlled variable thatis the response of the process variable to its controller output without dead time. It requires entry of threeparameters commonly known as process gain, dead time, and time constant. The Smith Predictor usesthese parameters to create models of the process from the controller output. In its most documentedform, the Smith predictor subtracts a model of the process with dead time from a model of the processwithout dead time and adds the net result to the measured process variable to create a new controlledvariable. If the model is perfect, the new controlled variable has zero dead time in terms of the controllerseeing the effect of its own controller output. Since the maximum allowable controller gain is inverselyproportional to dead time, the controller gain can theoretically increased without limit for a perfect modelprovided you ignore extenuating circumstances, such as loop interaction, measurement noise, and final

element dead band and resolution. One of the practical issues with the Smith Predictor is that the newcontrolled variable of the PID is no longer the actual process variable. The original process variable mustbe restored for the operator interface to the PID. Also, performance monitoring or trending must look atthe original process variable rather than the new controlled variable used by the PID. Terry Blevinsproposed in the 1979 ISA paper Modifying the Smith Predictor for an Application Software Package amultiplicative and additive correction of the process variable to deal with changes in the slope (gain) andintercept (bias), respectively in the process model.

The PID with a delayed external reset was informally presented in the 1980s and published in the early1990s. It simply consists of putting a dead time (DT) block in the external reset. This method only requiresthat a single parameter commonly known as process dead time be entered as the dead time in the DTblock. Terry Blevins documented in the early 1990s how the Smith Predictor for a particular Lambdatuning reduces to this PID with a delayed external reset.

The delayed external reset method of dead time compensation has several advantages readily evident.The user is not required to identify or estimate the process gain or process time constant. Also, the actualprocess variable is left intact. Other possibilities also exist for a more informative external reset signalthan just the Analog Output (AO) block BKCAL_OUT, such as the use of read back actual valve positionfrom a Digital Valve Controller (DVC) or a secondary loops process variable for a cascade control loop.These alternative external reset (ER) signals may potentially be able to compensate for dynamics anddisturbances of control valves or secondary loops. These ER signals can additionally protect the primarycontroller from outrunning the slewing rate of the control valve or the response of the secondary loop andprevent the walk off of override controllers. However, if a feedforward multiplier or split range block is


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used, the signal used for external reset must be converted back to its original basis as the controlleroutput (e.g. divided by feedforward multiplier). For more details on the value and use of external reset,see the article The Power of External Reset Feedback in the May 2006 issue of Control magazine.

The results presented here show that for a perfect model and the same controller tuning the PID with adelayed external reset performed better for processes with a small dead time to time constant ratio (timeconstant dominant), whereas the Smith Predictor performed better for processes with a large dead time totime constant ratio (dead time dominant). The Smith Predictor did not do as well for small dead time totime constant ratios because the control error seen in the controlled variable by the PID is much smallerthan the actual control error in the process variable. In both cases, the improvement was not asimpressive as the improvement gained from setting Lambda equal to the dead time rather than the timeconstant. Surprisingly the improvement in load disturbance rejection from dead time compensation wasgreater for processes with small dead time to time constant ratios. This goes against the conventionalwisdom that the best opportunity for dead time compensation is for dead time dominant loops. The resultscan be explained in terms of the ultimate limit for performance of dead time dominant loops being lower.The reduction in the peak excursion from more aggressive tuning settings is negligible for dead timedominant processes because the peak error is essentially the open loop error.

Another startling result was how quickly a Smith Predictor erupted into rapidly growing oscillations in thecontroller output when the model dead time was more than twice the actual process dead time. The fast

full scale oscillations in the controller output resembled on-off control. While it is relatively well known thatdead time compensators are sensitive to model mismatch, the effect was expected to be gradual andthought to be more in terms of a model dead time being too small. The concern for rapid deterioration fora model dead time being too large was raised in Good Tuning a Pocket Guideand was documented formodel predictive control in Models Unleashed. While a PID with delayed external reset is also adverselyaffected by a dead time mismatch in both directions, this PID develops a small amplitude high frequencydither rather than a full scale oscillation in controller output for an excessively high model dead time. Theconsequence is less severe and may be adequately handled by the addition of a small dither filterinserted in the PID controller output, but this was not tested.

Another major point here is that PID controller tuning for self-regulating processes without extenuatingcircumstances can only initiate oscillations for an identified (modeled) process dead time or gain that istoo small or an identified time constant that is too large. For PID controllers with dead time compensation

or model predictive controllers, high and low identified (model) values can cause oscillations.

In order to get the performance benefit from dead time compensation, the PID must be tuned moreaggressively. In other words, a PID with dead time compensation will perform the same as a PID withoutdead time compensation if they are tuned the same. While the improvement in integrated absolute error(IAE) for load upsets from more aggressive tuning (higher controller gain and lower reset time) can beaccurately estimated for a regular PID, the equation does not work well for a dead time compensator.Furthermore, a dead time compensator soon reaches a point of diminishing returns. For example, theimprovement in load rejection of a Smith Predictor from a controller gain that is quadrupled may not benoticeable whereas for a regular PID, it normally results in a four fold reduction in IAE. It is important toremember there is a tradeoff between performance and robustness for any feedback controller in that asyou make controller tuning more aggressive to improve load rejection you make the controller moresensitive to changes in the process gain, dead time, or time constant.

A nonlinear gain from the installed characteristic of a control valve has been widely discussed. However,the nonlinearity of the process gain of the temperature or composition response is the inverse andconsequently the combined effect is less than documented when these loops directly manipulate a controlvalve. The variability of dead time is often larger than the variability of the process gain or time constantbecause the dead time is inversely proportional to a rate (e.g. flow rate or pumping rate or rate of changeof a signal) and has many different sources (e.g. valve deadband or resolution, piping transportationdelay, mixing delay, process lags in series, sensor lags, signal filters, and discrete communication or scanintervals). Thus, it is problematic to compute the dead time accurately enough to get the benefit of a deadtime compensator.


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The calculation of process dead times and the role of PID structure will be addressed will be detailed inAdvanced Application Notes 004 and 005, respectively.

Only a small portion of the test results are included here as figures. A more complete compilation of testresults is posted on the web site under the category of Continuous Control. Information on disturbances,dead time, and controller tuning is also posted on the this website under the categories of Plant Designand Tuning and Control System Performance.

http://ModelingandControl.com


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2 Recommendations

To improve the performance of a PID controller for load disturbances at the process input:

(1) First improve the PID controller tuning before even considering dead time compensation. SettingLambda equal to the maximum dead time (Lambda factor equal to the maximum dead time to timeconstant ratio) is effective for load disturbances at the process input if there are no extenuatingcircumstances.

(2) Add feedforward control whenever it is possible to measure or infer load disturbances at the processinput.

(3) If there is economic justification for further improvement and the dead time can be updated within 25%accuracy for varying operating conditions, trial test and closely monitor a PID with delayed external resetfor low dead time to time constant ratios.

(4) For loops with high dead time to time constant ratios, multiple manipulated variables, interactions, orconstraints, consider model predictive control.


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3 Configuration

A DeltaV library module template titled PID_DEADTIME is available for implementing the Smith Predictorin DeltaV. It provides additional features such as a bias or process gain correction to the process modelfor major load disturbances. This study used a simple Smith Predictor of the form commonly documented

in the literature.

Figure 3-1a shows the module used for testing the AC2 PID with a Smith Predictor. It includes a simpleprocess simulation that consists of an actuator and a first order plus dead time process. The threeprocess parameters are PROCESS_GAIN, PROCESS_DELAY, and PROCESS_LAG. The actuator cansimulate stroking lags, delay, deadband, and stick-slip but these were all set to zero or negligible valuesfor this study. The module also includes noise added to the AT2 block output and a periodic load upsetadded to the process input. The load upset goes through a filter block whose filter time is the upset lag(UPSET_LAG). If the upset lag is zero, the upset is a step change at the process input.

The Smith Predictor is a composite block shown in Figure 3-1a. The output of the AV2 block is the inputsignal to the Smith Predictor. The three parameters provided to the Smith Predictor are MODEL_GAIN,MODEL_DELAY, and MODEL_LAG. For a perfect model, these parameter values are equal to their

corresponding process parameter values. A DITHER_FILTER block was inserted between the AC2 OUTand the AV2 block CAS_IN but it did nothing in these tests (filter time constant was kept at zero).

Note that in control literature, different nomenclature is often used for the three dynamic parameters of afirst order plus dead process or model. For each parameter below, the first name is the most common andthe last name is most specific.

process gain = plant gain = open loop gain

process dead time = plant delay = total loop dead time

process time constant = plant lag = open loop time constant

Figure 3-1b shows the drill down into the composite block. The Smith Predictor input signal is multipliedby the model gain in the MLTY1 block and then goes through a filtered with a process lag in the FLTR1block. The output of FILTR1 block is delayed via the DT1 block and subtracted in the SUB1 block from itsundelayed output. The Smith Predictor output is added in the main module via the SUM2 block to theoutput of the PV2 block, which has added noise to the AT2 block output. If the model is perfect and thereare no disturbances or noise, the model with the delay cancels out the measured process variable fromthe AI block. What is left is the process model without any dead time (delay). Any loop without dead timeand extenuating circumstances can have its controller gain increased without limit and still be stable.

A common mistake is to forget that the process variable used by PID as the controlled variable is nolonger the actual process variable but a model of the process response without dead time based solely onthe controller output. Trends of the AC2 PID PV will show a much smaller deviation from the set point anda false sense of actual control loop performance.


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Smith PredictorOutput

Figure 3-1a Test Module with Embedded Smith Predictor Composite Block

Smith PredictorEmbedded Composite

Figure 3-1b Drill Down of Embedded Smith Predictor Composite Block


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Figure 3-2a shows the test module for AC2 PID with delayed external reset. A dead time blockDELAY_COMP was inserted between the BKCAL_OUT of the AV2 block and the BKCAL_IN of the AC2block. This method of dead time compensation requires just one parameter, which is the model dead time(MODEL_DELAY). A DITHER_FILTER block was inserted between the AC2 OUT and the AV2 blockCAS_IN but it did nothing in these tests (filter time constant was kept at zero). This test module employsthe same type of process simulation used in the test module for the Smith Predictor.

Figure 3-2b shows that in order for external reset signal to be used, the Dynamic Reset Limit box must bechecked in FRSPID_OPTS for the AC2 PID.

This method dead time compensation has several advantages readily evident. The user is not required toidentify or estimate the process gain or process time constant. Also, the actual process variable is leftintact. Other possibilities also exist for a more informative external reset signal than just the AnalogOutput (AO) block BKCAL_OUT, such as the use of read back actual valve position from a Digital ValveController (DVC) or a secondary loops process variable for a cascade control loop. These alternativeexternal reset (ER) signals may potentially be able to compensate for dynamics and disturbances ofcontrol valves or secondary loops. These ER signals can additionally protect the primary controller fromoutrunning the slewing rate of the control valve or the response of the secondary loop and prevent thewalk off of override controllers. However, if a feedforward multiplier or split range block is used, the signalused for external reset must be converted back to the original basis of the controller output (e.g. divided

by feedforward multiplier).


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DelayedExternal

Reset

Figure 3-2a Test Model for PID with Delayed External Reset

Figure 3-2b Enabling of Dynamic Reset Limit in FRSPID_OPTS for PID with Delayed External Reset


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4 Test Results

In all of the test results the AC1 used for comparison purposes is always an uncompensated PID withLambda equal to the process time constant (lag), which is equivalent to a Lambda factor of one. All testshave an unmeasured load disturbance at the process input. The first upset in each figure is for a step

disturbance. The second upset in each figure is for a disturbance that has been slowed down by a timeconstant (upset lag) that is twice the original process dead time.

The first set of test results in Figure 4-1a through 4-1f illustrates the effect of different tuning for differentmodel accuracies for a low and high dead time to time constant ratio. Here AC2 is an uncompensatedPID with Lambda equal plant delay (process dead time) which is equivalent to a Lambda factor set equalto the delay/lag ratio (dead time to time constant ratio).

Figures 4-1a through Figure 4-1c is for a control loop with a 0.2 delay/lag (dead time to time constant)ratio. This ratio of 0.2 is seen in concentration and temperature control of columns and vessels wherethere is a low degree of back mixing. Single large gas pressure volumes and highly agitated vessels witha low turnover time can have a much lower delay/lag ratio if there is no significant additional dead timeassociated with an analyzer, sensor, or wireless communication time. The improvement in the integrated

absolute error (IAE) is about 66% from setting the Lambda equal to the process dead time instead of theprocess time constant and matches well the improvement predicted by Equations 2-2a and 2-2b in thebook New Directions in Bioprocess Modeling and Control. This IAE improvement holds up even for the50% increase in plant delay in Figure 4-1b. For the 50% decrease in plant delay in Figure 4-1c, the IAE isslightly less for both AC1 and AC2 but the per cent improvement is basically the same.

Figure 4-1d through Figure 4-1f is for a control loop with a 4.0 delay/lag (dead time to time constant) ratio.This ratio of 4.0 is seen in concentration and temperature control of plug flow pipelines and exchangervolumes where there is essentially no back mixing. This high ratio also occurs when chromatographs witha cycle time and sample transportation delay much larger than the residence time of an agitated volume.Even higher ratios occur for sheet thickness control. The improvement in the IAE is negligible but the AC2controller is less oscillatory for the original plant delay and a 50% increase in plant delay (process deadtime). An upset lag makes both loops less oscillatory. For a decrease in plant delay both controllers havea very smooth responses but the return to set point is noticeably faster for AC1.


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AC1

Original Delay/Lag Ratio = 0.2

Load Upset IAE

Improvement (%)

Step Upset

AC2

Slow UpsetUpset Lag = 2 Delay

AC1 = standard PI(Lambda = Lag and Reset = Lag)AC2 = standard PI(Lambda = Delay and Reset = Lag)

AC2AC1

Original Plant Delay

Figure 4-1a Uncompensated PID for 0.2 Delay/Lag Ratio and Original Plant Delay

AC1


Load Upset IAEImprovement (%)

Step Upset

AC2

Slow Upset

Upset Lag= 2

Delay


AC2AC1

50% Increase in Plant Delay

Figure 4-1b Uncompensated PID for 0.2 Delay/Lag Ratio and 50% Increase in Plant Delay


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AC1


Load Upset IAE

Improvement (%)

Step Upset

AC2


AC1 = standard PI(Lambda = Lag and Reset = Lag)

AC2 = standard PI(Lambda = Delay and Reset = Lag)

AC2AC1

50% Decrease in Plant Delay

Figure 4-1c Uncompensated PID for 0.2 Delay/Lag Ratio and 50% Decrease in Plant Delay


Step Upset Slow Upset

Upset Lag = 2 Delay



AC1

AC2

AC2

AC1

Figure 4-1d Uncompensated PID for 4.0 Delay/Lag Ratio and Original Plant Delay


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Step Upset Slow UpsetUpset Lag = 2 Delay


AC2 = standard PI(Lambda = Delay and Reset = Lag)


AC1

AC2

AC2

AC1

Figure 4-1e Uncompensated PID for 4.0 Delay/Lag Ratio and 50% Increase in Plant Delay



Upset Lag= 2

Delay



AC1

AC2

AC2

AC1

Figure 4-1f Uncompensated PID for 4.0 Delay/Lag Ratio and 50% Decrease in Plant Delay


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The second set of test results shows how well a Smith Predictor can do. Here AC2 is a Smith PredictorPID with the gain doubled and the reset time halved after Lambda has again been set equal to the plantdelay (process dead time). In other words, this AC2 has twice the proportional and integral action of theuncompensated AC2 in the first set of test results.

Figures 4-2a through Figure 4-2d is for a control loop with a 0.2 delay/lag ratio. The improvement in the

integrated absolute error (IAE) in Figure 4-2a is about 76% over the uncompensated conservatively tunedAC1, which is about 10% better than an uncompensated more aggressively tuned AC2 in the first set oftest results. This improvement holds up for 50% changes in the plant delay as illustrated by Figure 4-2band Figure 4-2c. However, if the model delay is more than 100% higher than the plant delay, the SmithPredictor breaks out into growing oscillations in the controller output as shown in Figure 4-2d. The risk ofessentially on-off control from a high model dead time may be an unacceptable risk.

Figures 4-2e through Figure 4-2h is for a control loop with a 4.0 delay/lag ratio. The Smith Predictor isless oscillatory than the uncompensated conservatively tuned AC1 for the original plant delay and aincrease in plant delay shown in Figures 4-2e and Figure 4-2f. However, for an decrease in plant delayshown in Figure 4-2g, which is normally thought of as a more stable condition, high frequency oscillationsstart to appear. If the model delay is more than 100% higher than the plant delay, the Smith Predictorbreaks out into growing oscillations in the controller output as shown in Figure 4-2h.


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AC2

AC1


Step Upset

AC2


AC1


AC1 = standard PI(Lambda = Lag and Reset = Lag)AC2 (AT2/PV)= Smith Predictor PI(Lambda = Delay and Reset = Lagbut with controller Gain increased

and Reset decreased by factor of 2)


Figure 4-2a Smith Predictor PID for 0.2 Delay/Lag Ratio and Original Plant Delay

AC2

AC1

Load Upset IAE

Improvement (%)

AC2AC1


Upset Lag= 2

DelayAC1 = standard PI(Lambda = Lag and Reset = Lag)AC2 (AT2/PV)= Smith Predictor PI

(Lambda = Delay and Reset = Lagbut with controller Gain increasedand Reset decreased by factor of 2)



Figure 4-2b Smith Predictor PID for 0.2 Delay/Lag Ratio and 50% Increase in Plant Delay


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Figure 4-2c Smith Predictor PID for 0.2 Delay/Lag Ratio and 50% Decrease in Plant Delay

Step Upset

AC2


AC1

AC1 = standard PI(Lambda = Lag and Reset = Lag)AC2 (AT2/PV)= Smith Predictor PI(Lambda = Delay and Reset = Lagbut with controller Gain increasedand Reset decreased by factor of 2)

110% Increase in Model Delay


Figure 4-2d Smith Predictor PID for 0.2 Delay/Lag Ratio and 110% Increase in Model Delay


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AC2

AC2

AC1


Upset Lag = 2 Delay




AC1

Figure 4-2e Smith Predictor PID for 4.0 Delay/Lag Ratio and Original Plant Delay

AC2

AC2

AC1


Upset Lag = 2 DelayAC1 = standard PI(Lambda = Lag and Reset = Lag)AC2 (AT2/PV)= Smith Predictor PI(Lambda = Delay and Reset = Lag

but with controller Gain increasedand Reset decreased by factor of 2)



AC1

Figure 4-2f Smith Predictor PID for 4.0 Delay/Lag Ratio and 50% Increase in Plant Delay


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AC2

AC2

AC1


Upset Lag = 2 Delay




AC1

Figure 4-2g Smith Predictor PID for 4.0 Delay/Lag Ratio and 50% Decrease in Plant Delay

Step Upset

AC2


AC1




Figure 4-2h Smith Predictor PID for 4.0 Delay/Lag Ratio and 110% Increase in Model Delay


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The third set of test results shows how well a simpler type of dead time compensated PID reset can do.Here AC2 is a PID with a delayed external reset. Like the Smith Predictor the gain is doubled and thereset time halved after Lambda has again been set equal to the plant delay (process dead time). In otherwords, this AC2 like the Smith predictor has twice the proportional and integral action of theuncompensated AC2 in the first set of test results.

Figures 4-3a through Figure 4-3d is for a control loop with a 0.2 delay/lag ratio. The improvement in theintegrated absolute error (IAE) in Figure 4-3a is about 88% over the uncompensated conservatively tunedAC1, which is about 22% better than an uncompensated more aggressively tuned AC2 in the first set oftest results. It is anticipated that the improvement increases as the delay/lag ratio decreases. Theimprovement is large enough for this dead time compensator to be considered for low delay/lagapplications where the plant delay is accurately known. The improvement deteriorates and the responsebecomes oscillatory for a 50% increase in the plant delay as illustrated by Figure 4-3b. A decrease inplant delay has little as shown in Figure 4-3c. However, if the model delay is more than 100% higher thanthe plant delay, the PID with a delayed external reset exhibits some decaying oscillations in the controlleroutput as shown in Figure 4-3d.

Figures 4-3e through Figure 4-3h is for a control loop with a 4.0 delay/lag ratio. The PID with a delayedexternal reset does not have the slow oscillations seen in the uncompensated conservatively tuned AC1for the original plant delay and 50% changes plant delay shown in Figures 4-3f through Figure 4-3g.However, there is a persistent high frequency dither in the controller output. The a time constant couldhave been set in the DITHER_FILTER block to help smooth this out. Control valve dead band andresolution limits in the control valve may help prevent these oscillations from affecting the process. If themodel delay is more than 100% higher than the actual plant delay, the dither amplitude gets larger andslower as shown in Figure 4-3h.


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AC2

AC1



AC2


AC1



AC1 = standard PI(Lambda = Lag and Reset = Lag)AC2 = dead time compensated PI(Lambda = Delay and Reset = Lagbut with controller Gain increasedand Reset decreased by factor of 2)

Figure 4-3a PID with Delayed External Reset for 0.2 Delay/Lag Ratio and Original Plant Delay

AC2

AC1

Load Upset IAE

Improvement (%)

Step Upset

AC2


AC1

Slow Upset

Upset Lag = 2 DelayAC1 = standard PI(Lambda = Lag and Reset = Lag)AC2 = dead time compensated PI



Figure 4-3b PID with Delayed External Reset for 0.2 Delay/Lag Ratio and 50% Increase in Plant Delay


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AC2

AC1



AC2


AC1


AC2 = dead time compensated PI(Lambda = Delay and Reset = Lagbut with controller Gain increasedand Reset decreased by factor of 2)


Figure 4-3c PID with Delayed External Reset for 0.2 Delay/Lag Ratio and 50% Decrease in Plant Delay

Step Upset

AC2


AC1




Figure 4-3d PID with Delayed External Reset for 0.2 Delay/Lag Ratio and 110% Increase in Model Delay


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AC1

AC2


AC2


AC1

AC1 = standard PI(Lambda = Lag and Reset = Lag)AC2 = dead time compensated PI



Figure 4-3e PID with Delayed External Reset for 4.0 Delay/Lag Ratio and Original Model Delay

AC1

AC2


Upset Lag= 2

Delay

AC2


AC1



Figure 4-3f PID with Delayed External Reset for 4.0 Delay/Lag Ratio and 50% Increase in Plant Delay


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AC1

AC2


AC2


AC1

AC1 = standard PI(Lambda = Lag and Reset = Lag)AC2 = dead time compensated PI



Figure 4-3g PID with Delayed External Reset for 4.0 Delay/Lag Ratio and 50% Decrease in Plant Delay

Step Upset

AC2


AC1




Figure 4-3h PID with Delayed External Reset for 4.0 Delay/Lag Ratio and 110% Increase in Model Delay


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5 References

1-1 Boudreau, Michael, A. and McMillan, Gregory K.,New Directions in Bioprocess Modeling and Control Maximizing Process Analytical technology Benefits, Instrumentation, Automations, and Systems (ISA), 2006.

1-2 McMillan, Gregory, Good Tuning a Pocket Guide, 2nd edition,Instrumentation, Automations, and Systems

(ISA), 2005.1-3 McMillan, Gregory and Cameron, Robert,Models Unleashed Virtual Plant and Model Predictive ControlApplications, Instrumentation, Automations, and Systems (ISA), 2004.

1-4 Blevins, Terrence L., McMillan, Gregory K., Wojsznis, Willy K., and Brown, Michael W.,Advanced ControlUnleashed Plant Performance Management for Optimum Benefits,Instrumentation, Automations, and

Systems (ISA), 2003.

compensation of dead time in pid controllers

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