pid deadtime control of distributive process

Control Engineering Practice 9 (2001) 1177–1183

PID-deadtime control of distributed processes

F.G. Shinskey*

Process Control Consultant, 260 Whiteface Rd., North Sandwich, NH 03259, USA

Received 6 April 2001; accepted 6 April 2001

Abstract

While model-based controllers have been used successfully to control paper machines and other processes dominated by timedelay, matching the model to the process gives poor load regulation over lag-dominant processes. An important class of lag-

dominant processes including heat exchangers and distillation columns consists of distributed lags. A PID controller having time-delay compensation, while functionally similar to a model-based controller, is a much better load regulator, and twice as effective asa conventional PID controller on these processes. It is applied in this paper to regulate steam superheat temperature. r 2001

Elsevier Science Ltd. All rights reserved.

Keywords: Adaptive control; Distillation columns; Distributed-parameter systems; Heat exchangers; Load regulation; Model-based control; PID

control; Steam plant; Temperature control; Time delay

1. The PID-deadtime controller

The insertion of a time delay into the integralfeedback circuit of a PID controller produces a transferfunction similar to a Smith predictor or an InternalModel Controller. Its internal configuration is shown inFig. 1. Integration is accomplished by positive feedbackof controller output m through time delay td andintegral lag I. The controlled variable c passes through afiltered derivative block of time constant D and filter aD;but set point r does not. An additional filter tf is locatedso as to be in both the process and integral feedbackloops. The transfer function of m responding to c is

mðsÞcðsÞ

¼Kcð1þDsÞð1þ IsÞ

ð1þ aDsÞ½ð1þ IsÞð1þ tf sÞ � e�td s�; ð1Þ

where s is the Laplace operator and Kc is theproportional gain of the controller.This PID form is known as the ‘‘interacting’’ or series

controller; time-delay compensation can also be addedto the ‘‘noninteracting’’ or parallel controller. Theconnection from the controller output to the time-delaycompensator is shown dashed because it can be broken,allowing another signal to be substituted for the outputsignal. In this way, the controller can be protected

against integral windup and prepared to resume controlwhen the feedback connection is restored.An internal model controller (IMC) is shown in Fig. 2

for comparison. Its transfer function is

mðsÞcðsÞ

¼1

Kmðgm * =gf � gmÞ; ð2Þ

where Km is the steady-state gain and gm is the dynamic-gain vector of both the process and its model; gm* is theinvertible part of gm, and gf is a filter. For the simplestcase where the process is a first-order lag with deadtime,the IMC function becomes

mðsÞcðsÞ

¼1þ t1s

Kmð1þ tf s� e�td sÞ; ð3Þ

where t1 is the process lag, tf the time constant of a first-order filter, and td the process deadtime. If the integraltime of the PID controller with deadtime were set tozero as well as its derivative filter, the transfer functionof Eq. (1) would reduce to that of Eq. (3). The PIDcontroller with deadtime is then seen to be similar to asecond-order IMC controller. However, this paper isintended to illustrate their differences more than theirsimilarities.

*Tel/fax: +1-603-284-6404.

E-mail address: [email protected] (F.G. Shinskey).

0967-0661/01/$ - see front matter r 2001 Elsevier Science Ltd. All rights reserved.

PII: S 0 9 6 7 - 0 6 6 1 ( 0 1 ) 0 0 0 6 3 - 6

2. Limitations of IMC

While model-based controllers are intended to matchthe process parameters as closely as possible, so as toproduce a particular set-point response, the author(Shinskey, 1996) has shown that this results in poor loadregulation for processes that are lag-dominant. Theproblem is illustrated in reference to the block diagramof IMC in Fig. 2. In parallel with the process is itsmodel, also identified as Kmgm, assuming a perfectmatch. This produces an estimated value of c designatedas ce, which is subtracted from its real value. TheirdifferenceFgiven a matched modelFis the process loadq after passing through its steady-state and dynamicgains Kqgq.The IMC is designed to produce a desired set-point

response:

cðsÞrðsÞ

¼ gf ðsÞgmðsÞgm * ðsÞ

ð4Þ

which includes only the process deadtime and the filter.The load response, however, includes the load dynamicsas a dominant term:

cðsÞqðsÞ

¼ KqgqðsÞgf ðsÞgmðsÞgm * ðsÞ

� 1

� �: ð5Þ

Following a step change in load, after the deadtime inthe load path elapses the controlled variable will beginto deviate from set point. At that time, controller outputm will step an amount equal to the step in q multipliedby Kq/Km. Fig. 3 illustrates the case where Kq=Km.

Given only first-order dynamics, c will follow anexponential trajectory until the deadtime in gm elapses,then returning to set point on another exponentialtrajectory, both governed by gq. The time lag in the loadpath for the curve in Fig. 3 is only twice the deadtime.Also shown is a recovery trajectory which returns to setpoint at the end of the deadtime, a response identified asthe ‘‘best’’ achievable. It is obtained by causing m toovershoot the step in q, with the overshoot sustained forone deadtime.The ratio of the integrated error IE for the two

responses is

IEbest

IEIMC¼1� e�td=tq

1þ tf =td; ð6Þ

where tf is the time constant of a first-order filter in theIMC controller. Even without a filter, the curves inFig. 3 show an integrated error for IMC that is 2.5 timesthe ‘‘best’’. For most fluid processes where temperatureand composition are controlled, the ratio of timeconstant in the path of the load input to the deadtimeis 5–10, giving integrated-error ratios of 5.5–10.5.The advantage of the PIDtd controller lies in that it is

tunable, much like a conventional PID controller. Andwhen tuned to minimize integrated absolute error (IAE)or some similar objective function, it outperforms bothPID and model-based controllers, approaching the‘‘best’’ load response without compromising set-pointresponse. In this paper, the PID controller with time-delay compensation is evaluated as a regulator fordistributed-lag processes.

3. Distributed-lag processes

Multiple-stage chemical processes consist of a seriesof connected first-order lags of similar time constant. If

Fig. 1. Time-delay compensation is added to the integral feedback

loop of the PID controller.

Fig. 2. Internal model control configuration.

Fig. 3. For lag-dominant processes, IMC gives unacceptable load

response.

F.G. Shinskey / Control Engineering Practice 9 (2001) 1177–11831178

information can flow in both directions through such aprocess, these lags are considered to be ‘‘interacting’’. Ina distillation column, for example, a step change inreflux flow entering the top will affects the compositionof the overhead vapor almost immediately. But it alsochanges the flow and composition of the liquid leavingthe top tray, which subsequently affects the compositionof the vapor leaving the next tray. This new vaporcomposition then shifts the equilibrium on the top tray,causing a secondary change to the composition of theoverhead vapor. Each change in either the flow orcomposition of either the vapor or liquid thus propa-gates both upward and downward, ultimately affectingthe composition of both overhead and bottom productsin a series of steps.If the stages are few in number, the individual steps

can sometimes be seen in the response of productcompositions to step disturbances. But most columnscontain 20 or more trays, whose capacity smoothens thecomposition response curves into long exponential lags.An electrical analog of such a multistage process is aseries of identical resistors with identical capacitorsconnected across the line at each junction, forming a‘‘ladder network’’. This network has also been used tosimulate the response of electrical and pneumatictransmission lines.

3.1. Open-loop response

Fig. 4 shows a step response of a network of 20 equalinteracting lags. Note that it is dominated by theexponential response of an equivalent first-order lag,but also with some initial time delay. The time scale isnormalized so that 63.2% response occurs at a normal-ized value of 1.00. The time at this point is

St ¼ tiðn2 þ nÞ=2; ð7Þ

where ti is the value of the individual time constant andn is their number. The factor (n2+n)/2 is equal to the

sum of the numbers from 1 to n. This squaredrelationship can give a 100-tray distillation column afour-times slower response than a 50-tray column.Interestingly, there is almost no difference in the shapeof the response curve when n=100Fit does not changesignificantly for values of n>20. As a consequence, the20-lag model is satisfactory for simulating much higher-order systems.The highest-order interacting system is the distributed

lag, consisting of an infinite number of infinitesimal lags.Distillation columns containing packing rather thandiscrete trays behave as distributed lags. Perhaps themost common distributed process is the heat exchanger:heat is transferred across a broad area having adistributed temperature gradient and distributed heatcapacity as well.

3.2. Variable parameters

Distillation columns tend to operate with internalcirculation rates of liquid and vapor which are muchhigher than their throughput, especially when there aremany trays. As a result, their dynamics tend to beunaffected by production rate. Another example of adistributed lag with constant dynamics is a stirred tank,providing that its internal circulation rate is reasonablyconstant and greater than throughput.By contrast, heat exchangers generally operate on a

once-through basis, neither fluid being recirculated.Given that an exchanger has a fixed heat capacity, thevariable flow rate of the fluids causes residence time Stto vary inversely with flow. The shape of the responsecurve does not change, but both its equivalent timedelay and first-order lag change with flow, while keepingthe same ratio between them.The steady-state response of extrinsic variables such

as temperature and composition also depends on flowrates. Consider the following steady-state model of heattransfer between steam flowing at rate W and having alatent heat of H, and a liquid flowing at rate F andhaving a specific heat of C:

Q ¼WH ¼ FCðT2 � T1Þ; ð8Þ

where Q is the rate of heat flow and T1 and T2 are therespective inlet and outlet temperatures of the liquid.The steady-state gain of exit temperature in response tosteam flow is seen to vary inversely with liquid flow:

Kp ¼dT2

dW¼H

FC: ð9Þ

As liquid flow increases, the exit temperatureresponds to changes in steam flow faster and with lessgain, making it easier to control. However, control-loopstability must also be provided at low flow rates, wherethe response is slower and the gain is higher.

Fig. 4. The step response of 20 interacting lags; the shape does not

change significantly above 20 lags.

F.G. Shinskey / Control Engineering Practice 9 (2001) 1177–1183 1179

3.3. Closed-loop response

Modeling a distributed-lag process as first-order plustime delay is not very satisfactory in predicting itsclosed-loop response. The distributed lag is much moreresponsive to derivative action in the controller. Forexample, the step load response of a first-order processwith deadtime can be reduced in integrated error by afactor of 2.25 by changing from PI to interacting PIDcontrol. But for a distributed process, the reductionachieved by the same change in controllers is 2.89. Theoptimum controller tuning is somewhat different as well.Proceeding from a PID controller to one with time-delaycompensation is even more striking: the integrated erroris reduced by a factor of 1.5 for the first-order-plus-delayprocess and 2.5 for the distributed process. As aconsequence, the PIDtd controller seems particularlywell-suited to regulating distributed processesFoverseven times as effective as PI control on the basis ofintegrated error.Fig. 5 compares the step load response of a distrib-

uted process (simulated by 20 interacting lags) usingthree different controllers: PI, interacting PID, andinteracting PIDtd : All controllers were tuned to mini-mize integrated absolute error (IAE), Table 1 lists theirmode settings in terms of the process parameters Kp andSt: Proportional band P is expressed in percent. (Theproportional gain Kc=100/P.) The last column in thetable is the integrated error (IE) per unit change incontroller output m required to meet the load change,calculated as

IE

Dm¼

P

100ðI þ td Þ: ð10Þ

Integrated error can be related to operating cost inmost product-quality loops, where it represents valuableingredients given away or excessive energy consumptionused to assure that specifications are met. In someapplications, however, the peak deviation may be asimportant. Examples are the control of boiler pressure,

to avoid lifting safety valves, and steam superheat,where extreme temperatures shorten the life of metalheat-transfer tubes. The size of the peak deviationfollowing a load change is directly proportional to theproportional-band setting. So the progression of pro-portional-band settings in the table relates to the peakdeviations shown in Fig. 5.

3.4. Tuning vs. matching

Model-based control was developed primarily forprocesses having a pronounced time delay, the intentbeing to match the process delay with one in the controlsystem. Improved set-point response was promised,along with the elimination of tuning, except for that ofthe filter. In the distributed process, there is no true timedelay, yet a controller having a time delay is moreeffective regulating it than those that do not.Furthermore, if the PIDtd controller were replaced

with a model-based controller matching the processfaithfully, the load response would be much poorer, dueto the dominant lag in the load path. So the concept ofmatching the controller to the process must be rejectedon two counts: a higher performance can be achievednot only by tuning the controller instead of matching itsparameters to the process, but by using a controllerwhose structure does not match the process, either.In other words, the high performance of the PIDtd

controller on the distributed process refutes the notionthat model-based control is superior to other methods.What is proved instead is that the PIDtd controller has ahigh performance capable of maximizing load responseon a variety of processesFits structure does not dependon the nature of the process being controlled, althoughits tuning does. While elimination of tuning may bedesirable, this exercise shows that it is still required ifload rejection is to be optimized, especially for a lag-dominant process. (Although not a part of thisdiscussion, the set-point response of this controller isalso excellent with the tuning that is optimal for loadresponse.)

4. Control-loop robustness

Robustness has been defined as the minimum changein given process parameters which brings the loop to the

Fig. 5. Step load responses for a distributed lag; all three controllers

were tuned to minimize IAE.

Table 1

Controller settings and integrated error for optimum load regulation

of distributed processes

Controller P/Kp I=St D=St td=St IE/Dm

PI 20 0.54 F F 0.11 (KpSt)PID 15 0.25 0.10 F 0.038 (KpSt)PIDtd 6.6 0.067 0.16 0.16 0.015 (KpSt)


limit of stability, as identified by an undamped oscilla-tion. As a general rule, the higher the controllerperformance, achieved either through tighter tuning orthe addition of more modes, the lower the robustness ofthe loop. (The converse is not necessarily true, in that itis possible for a controller to exhibit low performanceand low robustness at the same time, which has been theexperience of some engineers attempting to control pHor steam temperature using a Smith predictor.)Adjustments to proportional, integral, and derivative

mode settings provide tighter control when moved inone direction (higher gain, faster integration, longerderivative time), and more robustness when moved inthe other (lower gain, etc.). Time-delay compensation,however, can cause instability when pushed too far ineither direction. Therefore, the increase in performancewhich it brings, comes with the requirement of veryprecise tuning.

4.1. Robustness considerations with heat exchangers

There are only two parameters required to specify thedistributed lagFKp and StFand as indicated in Table1, to tune the controller. However, in the case of a heatexchanger, they do not vary independently, buttogether, inversely with process flow. As a consequence,robustness properties for the three loops described inTable 1 and Fig. 3 need to be evaluated as these twoprocess parameters change together with flow.In simulations of the PI loop, the limit of stability was

reached when the flow fell to 63% of the value where thecontroller was optimally tuned. For the PID loop,instability was reached at 69% of the original flow. Withthe PIDtd controller in the loop, instability was reachedwhen the flow fell to 79% of the original, or rose to112% of the originalFthe last being the narrowest ofthe margins observed. For this application, then, therobustness of the PIDtd controller is only 12%,compared to 31% for the PID controller and 37% forthe PI controller.

5. Process identification

If the PID controller with time-delay compensation isto be used to control a heat exchanger, the dynamics ofthe process need careful identification across the entireoperating range. The earliest and easiest identificationmethod to administer is the step test used by Ziegler andNichols (1942). A step change is manually introducedinto the controller output from a steady state, andthe resulting response in the controlled variablerecordedFthe ‘‘reaction curve’’. The step-responsecurve for the distributed lag from Fig. 4 is analyzed inFig. 6.

5.1. Evaluating the reaction curve

Ziegler and Nichols apparently did not want to waitto achieve complete response of lag-dominant processes,and so terminated their test after observing the slope ofthe reaction curve pass its steepest point. This has merit,in that the dominant lag of a distillation column couldbe more than an hour, and five time constants arerequired for 99% complete response. In the meantime,other disturbances could arise.Their method, therefore, used only two features of the

curve upon which to base their controller settings: theestimated time delay and the steepest slope of the curve.A self-regulating process really requires three para-meters for identification: time delay, time constant, andsteady-state gain. However, an estimate of the gainrequires a return to the steady state, which may requiretoo much time, and may not be reached at all.Reasoning that the steepest slope was a function ofboth the gain and the time constant, they replaced thesetwo parameters with one.Fig. 6 shows that the estimated time delay tde lies

between the initiation of the step at time zero, and theintersection of the steepest slope with the baseline. In analternative to calculating the slope, Fig. 6 shows how toestimate the dominant time constant t1e; as if the steady-state gain were unity. The size of the step disturbanceDm is marked against the steepest slope, projected ifnecessary, and the time along that slope is the estimatedtime constant.

5.2. Application to the distributed lag

One of the concerns when using the Ziegler andNichols tuning rules is their limited scope. They workbest for step load changes applied to lag-dominantprocesses. In that the distributed lags are in fact lagdominant, having a consistently low ratio of time delay

Fig. 6. Estimating the time delay and time constant using an open-

loop step response.


to time constant, their method is reasonably effectivehere.To make use of the tuning rules given in Table 1

requires conversion from the parameters of tde and t1e toKp and St: Given that all distributed processes have thesame shape for their reaction curve, a direct correlationis possible:

Kp ¼ 7:5tde=tle St ¼ 7:0tde: ð11Þ

The distributed-lag process has, as seen in the secondcorrelation above, an equivalent first-order time con-stant of 6 times its delay. In this context, St representsthe sum of the equivalent first-order lag and delay. Thefirst correlation above has no other function than toidentify the steady-state gain.Earlier, it was mentioned that a first-order lag with

deadtime is not a particularly useful simulation of adistributed process. To demonstrate, if the deadtime andtime constant parameters estimated from the step-response curve are used to calculate the ‘‘best’’integrated error for a unit load change, as shown for afirst-order process in Fig. 3, its value comes out to be0.0219 KpSt: Then estimating the integrated errorproduced under PIDtd control by inserting its settingsfrom Table 1 into Eq. (10) gives a result of 0.0150 KpSt:This would indicate a controller performance of 0.0219/0.0150 or 146%, a value that is clearly unrealistic. The‘‘best’’ integrated error estimate really only applies toprocesses having deadtime, which the distributed lagdoes not. Controller performance on this process canonly be estimated relative to other controllers. There-fore, the use of the parameters estimated from thereaction curve is limited to determining optimumcontroller settings.

6. Controlling a superheater

One of the most difficult loops in a modern powerplant is that of controlling superheated steam tempera-ture. The efficiency of the generating station dependsstrongly on maximizing this temperature, but withinvery narrow constraints. Allowable temperature islimited by the ability of the steel-alloy heat-transfertubing to retain its strength. Excessive temperatures, andespecially temperature variations, cause stress anddistortion which can significantly shorten the life of asuperheater.The steam temperature is measured after leaving the

last of a series of heat-transfer sections, but is controlledby spraying attemperating water between upstreamsections. While the process is truly a distributed lag,and its parameters generally vary inversely with flow,the relationships in the context of the boiler aresomewhat more complex. As steam flow is increasedto satisfy increasing demand for electrical power, other

factors are also adjusted, including the tilt of the burnersand the recirculation of flue gas. Therefore, the processparameters cannot be expected to vary with steam flowfollowing a simple formula such as Eq. (9), but must bedetermined on-line by testing at several load levels.

6.1. Plant identification

The PIDtd controller was applied to the superheatersof a 500MW power boiler in Ontario, Canada. Steamtemperature was to be controlled at 538751C. Becauseof the robustness limitations of the controller, theprocess parameters had to be determined preciselyacross the full load range, and the mode settings wouldneed to be gain-scheduled as a function of steam flow.This means adapting all the mode settingsFnot only thegainFbased on the observed response of the process, asit changes with load. Open-loop step tests were, there-fore, conducted at load levels from 100 through 530MW, with the time delay and time constant estimated asin Fig. 6. The results of those tests appear in Table 2.Over the range tested, generated load varied by 5.3 : 1.

However steam flow is the variable used to index thecontroller settings, and that changed over a 4.5 : 1 range.Based on Eq. (9), Kp would be expected to vary as much,but its actual variation was only 2.3 : 1, half as much.There is even a slight reversal in the estimate of Kp,probably caused by a change in another operatingvariable such as flue-gas recirculation. The dynamicparameter St followed steam flow more closely, show-ing a variation of 3.3 : 1. Yet even these moderatedvariations are huge compared to the narrow robustnesslimits demonstrated by the PIDtd controller in simu-lated closed-loop testing, where a 12% flow increasemoved the loop from optimum performance to un-damped cycling. The controller would have to remaintuned very precisely to deliver the expected performanceimprovement over PID control.

6.2. Implementation

The temperature controller was implemented byinserting a time-delay compensator in the integralfeedback loop of an interacting PID controller, just asshown in Fig. 1. After obtaining the estimated process

Table 2

Parameters estimated from step tests

Load (MW) Steam flow (%) tde (min) t1e (min) Kp St (min)

100 20 3.5 5.0 5.25 24.5

200 35 2.5 4.8 3.90 17.5

300 50 2.3 4.3 3.98 16.1

400 66 1.35 3.5 2.89 9.5

530 89 1.05 3.5 2.25 7.4


parameters in Table 2, the controller was then tunedfollowing the rules given in Table 1. After achievingstable, responsive control at one load level with thesesettings, the controller was retuned at each of the otherlevels, with similar results. Remarkably, no fine tuningwas required at any loadFthe rules which had beendeveloped by simulation of a process consisting of 20interacting lags proved to be entirely satisfactory. This isan important consideration, in that the PIDtd controlleris not easy to tune on-line. It is common experience tofind even the derivative setting of PID controllers left atzero by operators and technicians because it complicatesthe tuning procedure. Adding a fourth parameterFdeadtimeFcomplicates the procedure much more. Inaddition, cycling can develop if the deadtime setting iseither too long or too short, and it is not always clear inwhich direction to adjust it when attempting to stabilizea cycling loop.Finally, the optimum settings found for each of the

load levels were programmed into a gain-scheduler,which interpolated between the values obtained by

testing. In this way, the controller would remainoptimally tuned at all loads. Control of steam tempera-ture proved to be tighter than had ever been achieved forthat loop previously. At the time of writing this paper,the controller has been providing satisfactory perfor-mance for over a year.

Acknowledgements

The author gratefully acknowledges the work ofSigifredo Ni *nno of Foxboro Canada, who conductedthe plant tests and implemented the control system sosuccessfully.

References

Shinskey, F. G. (1996). Process control systems (4th ed.) (pp. 130–132).

New York: McGraw-Hill.

Ziegler, J. G., & Nichols, N. B. (1942). Optimum settings for automatic

controllers. Transactions of the ASME, 759–768.


pid deadtime control of distributive process

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