Process Control

54 www.aiche.org/cep June 2008 CEP

Because most processes are self-regulating, it cansometimes be challenging to tune a controller for anintegrating process. The principal characteristic of a

self-regulating process is that it naturally seeks a steady-state operating level if the controller output and disturbancevariables are held constant for a sufficient period of time.

For example, a car’s cruise control is self-regulating.By holding the fuel flow to the engine constant (assumingthe car is traveling on flat ground on a windless day), thecar is maintained at a constant speed. If the fuel flowrate

is increased by a fixed amount, the car will accelerate andthen settle at a different constant speed.

The temperature of a process stream exiting a heatexchanger is also self-regulating. If the shellside coolingfluid flowrate is held constant and there are no significantexternal disruptions, the tubeside exit stream temperaturewill settle at a constant value. If the cooling flowrate isincreased, allowed to settle, and then returned it to itsoriginal value, the tubeside exit stream temperature willmove to a new operating level during the increasedflowrate and then return to its original steady-state.

Tanks that have a regulated exit flow stream do not nat-urally settle at a steady-state operating level. This is acommon example of what process control practitionersrefer to as a non-self-regulating (or integrating) process.

Integrating processes can be remarkably challenging tocontrol. This article explores their distinctive behaviors.

Armed with this knowledge, you may come to realizethat some of your facility’s more-difficult-to-control level,temperature, pressure and other loops have such character.

Integrating (non-self-regulating) behavior in manual mode

The top plot of Figure 1 shows the open-loop (manualmode) behavior of a self-regulating process. In this ideal-ized response, the controller output (CO) signal and meas-ured process variable (PV) are initially at steady state. TheCO is stepped up from this steady state and then backdown. As shown, the PV responds to the step, and ulti-mately returns to its original operating level.

The bottom plot of Figure 1 shows the open-loopresponse of an ideal integrating process. The distinctivebehavior occurs when the CO returns to its original value and the PV settles at a new operating level.

Use this tuning recipe for the classic integrating process control challenge.

Robert Rice

Douglas J. Cooper

Control Station, Inc.

Improve Controlof Liquid

Level Loops

■ Figure 1. Integrating processes are characterized by the processvariable moving to a new value when the controller output returns toits starting value. In an ideal self-regulating process, the process vari-able returns to its original value when the controller output is steppedback down.

Self-Regulating

IntegratingBehavior

Time

PV tracks up anddown with CO

PV at new valuewhen CO returns

CO

CO

PVPV

Reprinted with permission from CEP (Chemical Engineering Progress), June 2008.Copyright © 2008 American Institute of Chemical Engineers (AIChE).

CEP June 2008 www.aiche.org/cep 55

The integrating behavior plot is somewhat misleading, asit implies that for such processes, a steady controller outputwill produce a steady process variable. While this is possi-ble with idealized simulations like that used to generate theplot, such “balance point” behavior is rarely found in inte-grating processes in industrial operations.

More realistically, if left uncontrolled, the lack of a bal-ance point means that the process variable of an integratingprocess will naturally tend to drift up or down, possibly toextreme and even dangerous levels. Consequently, integrat-ing processes are rarely operated in manual mode for long.

P-only control behavior is differentTo appreciate the difference in controlled behavior for

integrating processes, first consider the proportional, or P-only, control of an ideal self-regulating simulation. As

shown in Figure 2 (p. 56), when the setpoint (SP) is ini-tially at the design level of operation (DLO) in the firstmoments of operation, then PV equals SP (the DLO iswhere the setpoint and process variable are expected to beduring normal operation when the major disturbances areat their normal or typical values).

The setpoint is then stepped up from the DLO on theleft side of the plot. The simple P-only controller is unableto track the changing SP, and a steady error, called offset,results. The offset grows as each step moves the SP fartheraway from the DLO.

Midway through the process, a disturbance occurs, asshown in the middle of the plot. (Its size was predeter-mined for this simulation to eliminate the offset.) Whenthe SP is then stepped back down (on the right) the offsetshifts, but again grows in a similar and predictable pattern.

Level control in a surge tank for a single-valve kegging (SVK) system

Surge tanks are designed to counteract fluctua-tions in flow characteristics that would other-

wise disrupt upstream or downstream systems.Surge tanks are often installed between twoprocess systems with incompatible flow patternsto provide flow smoothing. The “wild stream” hasflow control requirements that are difficult to influ-ence, and the controller then adjusts the controlledstream to maintain the liquid level in the tank.

The primary objective of a surge tank is toabsorb the fluctuations of the wild stream withoutsignificantly impacting the controlled stream. Tobest achieve this result, the level in a surge tankshould be allowed to swing between an upperand a lower level limit. The more the tank isallowed to swing, the larger the surge capacity ofthe tank. Often, however, these swinging tanksare viewed as poor performers and are thentuned for tight performance, counteracting theintended design objective.

A major beer brewer uses an SVK system tofill several lanes of kegs (top). Because the keg-filling lanes are operated in an on/off fashion, thewild stream flowrates requested by the SVK sys-tem can quickly vary from 0 to 180 gal/mindepending on the number of kegs being filled atany point in time.

As shown in the figure on the bottom, adjust-ing the flow of beer pumped from the large stor-age tanks controls the level in the surge tank.Due to the sensitive nature of beer and of theanalytical instrumentation involved, a surge tankis installed to dampen the large demand fluctua-tions required by the keg-filling system. By allow-ing the surge tank to swing more freely betweenits constraints, the control changes sent to thelarge storage tank are reduced.

Surge Tank Performance

Upper Constraint

Lower Constraint

Leve

l PV

/ SP,

%W

ild-S

tream

Flow

, gal

/min

CO

, %

Time, h

Aggressively Tuned PI Controller Conservatively Tuned PI Controller

0 5 10 15 20 25 30

70

60

50

40

40

20

140120100806040200

■ Tuning a control system for a beer-keg filling line (top) to allow the surge tank to fluctuate morebetween its constraints (bottom) reduces the control changes sent to the storage tank.

DraftBeer

StorageBeer Pump

CO2

FLOW

TEMPPRES BALL

Beer Valve

PSI O2

BeerSurgeTank

Racker Pump

TEMP pH

FLOW

O2

PRES

LIC

BALL

FT01

TT01PT01

CT01

AT01 CT02PT02

TT02 AT02 PT03

CT03

FT02

CT04

FC

Single Valve Kegging System

LIC

56 www.aiche.org/cep June 2008 CEP

With this as background, consider an ideal integratingprocess simulation under P-only control. Even under simpleP-only control, as shown on the left of Figure 3, the processvariable is able to track the setpoint steps with no offset. Thisbehavior can be quite confusing, as it does not fit the expect-ed behavior of the more-common self-regulating process.

This happens because integrating processes have a natu-ral accumulating character (and is, in fact, why “integratingprocess” is used as a descriptor for non-self-regulatingprocesses). Since the process integrates, it appears that thecontroller does not need to.

Yet the setpoint steps in the right of Figure 3 show this isnot completely correct. Once a disturbance shifts the baselineor balance-point operation of the process (shown roughly atthe midpoint in the plot), an offset develops and remains con-stant even as SP returns to its original design value.

Controller output behavior is tellingThe CO plots in Figures 2 and 3 demonstrate an interest-

ing feature that distinguishes self-regulating from integratingprocess behavior. In the self-regulating process plot, theaverage CO value tracks up and then down as the SP stepsup and then down. In the integrating process plot, the COspikes with each SP step, but then in a most unintuitive fash-ion, returns to the same steady value. It is only the change inthe disturbance flow that causes the average CO to shift mid-way through the plot, where it then remains centered aroundthe new value for the remainder of the SP steps.

PI control behavior is differentThe dependent, ideal form of a proportional-integral (PI)

controller (1) is one of numerous algorithms that are widelyemployed in industrial practice:

CO CO K e tKT

e t dtbias cc

i= + + ∫( ) ( ) (1)

Process Control

■ Figure 3. Unlike an ideal self-regulating process, P-only control foran ideal integrating process shifts the baseline operation of theprocess, producing a sustained offset even as the setpoint returns toits original value.

1) No offset …

2) ...then disturbanceload changes …

Time

CO

, %D

, %PV

and

SP,

%

40 80 120 160

65605550

60

50

40

545250

3) … producingsustained offset

■ Figure 4. Increasing controller gain (Kc) for the PI control of an idealself-regulating process causes the process variable response to movefrom a sluggish to an oscillatory response behavior.

Kc = 0.3No oscillation

Time

CO

, %PV

and

SP,

%

150 300 450 600 750

5654525048

60

55

50

45

Kc = 0.3Modest oscillation

Kc = 1.2PV oscillates

■ Figure 5. For PI control of an integrating process, oscillatoryresponse behavior can occur both when the controller gain (Kc) of aPI controller is too small and when it is too large.

Kc = 1PV oscillates

Time

CO

, %PV

and

SP,

%

75 150 225 300

57

54

51

48

45

10080604020

Kc = 4Overshoot butno oscillation

Kc = 8PV oscillates

■ Figure 2. P-only control of an ideal self-regulating process shiftsthe offset caused by disturbances.

1) Offset grows …

2) … then disturbanceload changes …

Time

3) … shifting the offset

CO

, %D

, %PV

and

SP,

%

100 200 300 400 500 600

65605550

60555045

65

50

CEP June 2008 www.aiche.org/cep 57

Figure 4 shows an ideal self-regulating process simula-tion that is controlled using this PI algorithm. Reset time,Ti, is held constant throughout the simulation while con-troller gain, Kc, is doubled and then doubled again. As Kcincreases, the controller becomes more active, and, asexpected, this increases the tendency of the PV to displayoscillating (underdamped) behavior.

For comparison, consider PI control of an ideal inte-grating process simulation as shown in Figure 5. Ti, isagain held constant while Kc, is increased. A counter-intu-itive result is that as Kc becomes small and as it becomeslarge, the PV begins displaying an underdamped (oscillat-ing) response behavior. While the frequency of the oscilla-tions is clearly different between a small and large Kc,when seen together in a single plot, it is not always obvi-ous in what direction the controller gain needs to beadjusted to settle the process, in particular, when seeingsuch unacceptable performance on a control room display.

A tuning recipe provides benefit One of the biggest challenges for practitioners is recogniz-

ing that a particular process shows integrating behavior priorto starting a controller design and tuning project. This, likemost things, comes with training, experience and practice.

Once in automatic mode, closed-loop behavior of anintegrating process can be unintuitive, and even confound-ing. Trial-and-error tuning methods can lead one in circlestrying to understand what is causing the unacceptable control performance.

A formal controller design and tuning procedure for inte-grating processes helps overcome these issues in an orderlyand reliable fashion. Best practice is to follow a formal recipewhen designing and tuning any PID controller. A recipe-based approach causes less disruption to the productionschedule, wastes less raw material and utilities, requires lesspersonnel time, and generates less off-specification product.

The controller design and tuning recipe for integratingprocesses contains four steps, as follows (2):

1. Establish the design level of operation (the normal orexpected values for the setpoint and major disturbances).

2. Bump the process, and collect dynamic process data ofthe process variable response to changes in controller output.

3. Approximate the process data behavior with a first-order-plus-dead-time integrating (FOPDT integrating)dynamic model.

4. Use the model parameters generated in step 3 and thecorrelations in Table 1 to complete the controller tuning.

It is important to recognize that real processes are more complex than the simple FOPDT integrating model.In spite of this, the model does provide an approximationof process behavior that is sufficiently rich in dynamic

information to yield reliable and predictable control performance when used with the rules and correlations in Step 4 of the recipe.

The FOPDT integrating modelThe FOPDT dynamic model commonly used to approxi-

mate self-regulating dynamic process behavior has the form:

where Kp is the steady-state process gain, Tp is the overallprocess time constant, and θp is the process dead time. Yetthis model cannot describe the kind of integrating processbehaviors explored above. These dynamic behaviors arebetter described with the FOPDT integrating model form:

It is interesting to note when comparing these two mod-els that individual values for the familiar process gain, Kp,and process time constant, Tp, are not separately identifiedfor the FOPDT integrating model. Instead, an integratorgain, Kp*, is defined that has units of the ratio of the processgain to the process time constant, or:

Tuning correlations for integrating processesThe FOPDT integrating model parameters Kp* and θp can

be computed using a graphical analysis of plot data, or in anindustrial setting by automated analysis using a commercialsoftware package. Once the model parameters are known, thetuning values for the dependent, ideal PI form, Eq. 1, as wellas the popular PID algorithm form, can be calculated:

For integrating processes there is no identifiableprocess time constant in the FOPDT integrating model.Thus, dead time, θp, is used as the baseline marker of time

CO CO K e tKT

e t dt K TdPVdtbias c

c

ic d= + + −∫( ) ( ) (5)

KKT

K PVCO timep

p

pp

* *=[ ] =[ ]×

or (4)

dPV tdt

K CO tp p( ) ( )*= − θ (3)

TdPV t

dtPV t K CO tp p p

( ) ( ) ( )+ = − θ (2)

Table 1. Use tuning correlations for PI and PID controllers for integrating processes.

K T T

K

T

TT

c i d

p

c p

c pc pPI 1 2

22* ( )+

++

θ

θθ

PID 1 20 5

20 25

22

2

K

T

TT

T

p

c p

c pc p

p c p* ( . )

.+

++

+θ

θθ

θ θ

TTc p+ θ

58 www.aiche.org/cep June 2008 CEP

for tuning. Specifically, θp is used as the basis for comput-ing the closed-loop time constant, Tc.

Building on the popular internal model control (IMC)approach to controller tuning, the closed-loop time con-stant is computed as Tc = 3θp (3).

The controller tuning correlations for integratingprocesses use this Tc, as well as the Kp* and θp from theFOPDT integrating model fit, in the correlations of Table 1.

A simulated example — the pumped-tank A pumped-tank simulation illustrates the design and tun-

ing of a controller for an integrating process. As shown inFigure 6, the process has two liquid streams feeding the topof the tank and a single exit stream pumped out of the bot-tom. The measured process variable (PV) is the liquid levelin the tank. To maintain the liquid level, the controller out-put (CO) signal adjusts a throttling valve at the discharge ofa constant-pressure pump to manipulate the flowrate out ofthe bottom of the tank. This approximates the behavior of acentrifugal pump operating at relatively low throughput.

Note that a pump strictly regulates the dischargeflowrate out of the tank. As a consequence, the physics donot naturally work to balance the system when any of thestream flowrates change. This lack of a natural balancingbehavior is why the pumped tank is classified as an inte-grating process. If the total flow into the tank is more thanthe flow pumped out, the liquid level will rise and contin-ue to rise until the tank fills or a stream flow changes. If

the total flow into the tank is less than the flow pumpedout, the liquid level will fall and continue to fall.

Figure 7 is a plot of the pumped-tank behavior with thecontroller in manual mode (open-loop). The CO signal isstepped up, increasing the discharge flowrate out of thebottom of the tank. The flow out becomes larger than thetotal feed into the top of the tank and, as shown, the liquidlevel begins to fall. As the situation persists, the liquidlevel continues to fall until the tank is drained. The saw-toothed pattern occurs when the tank is empty because thepump briefly surges every time enough liquid accumulatesfor it to regain suction.

Figure 7 does not show that if the controller output wereto be decreased enough to cause the flowrate out to be lessthan the flowrate in, the liquid level would rise until thetank was full. If this were a real process, the tank wouldoverflow and spill, creating safety and profitability issues.

Graphical modeling of integrating process dataThe graphical method of fitting an FOPDT integrating

model to process data requires a data set that includes atleast two constant values of controller output, CO1 andCO2. As shown in Figure 8 for the pumped tank, both mustbe held constant long enough that a slope trend in the PVresponse (tank liquid level) can be visually identified.

An important difference between the traditional processreaction curve graphical technique for self-regulatingprocesses and integrating processes is that integratingprocesses need not start from a steady-state value before abump is made to the CO. The graphical technique discussedhere is only concerned with the slopes (or rates of change)in PV and the constant controller output signal that causedeach PV slope.

The FOPDT integrating model describes the PV behaviorat each value of constant controller output, CO1 and CO2, as:

Process Control

■ Figure 7. With the simulated level control in manual mode, the liquid level falls as the controller increases the flowrate out of thebottom of the tank.

Time, min

CO

, %Le

vel,

m

20 25 30 35 40 45

4

2

0

80

75

70

Exit flow increases …

… and tank drains

■ Figure 6. Simulated pumped-tank level control in automatic modeuses a throttling valve to adjust the process variable, the liquid levelin the tank.

LC

Brine FeedFlow, L/min

15.3

DisturbanceFlow, L/min

2.5

TankLevel, m

PVSP

CO

D

4.01

Setpoint, m4.0

ControllerOutput, %

70.0

DischargeFlow, L/min

17.8

CEP June 2008 www.aiche.org/cep 59

Subtracting and solving for Kp* yields:

Graphical modeling of pumped-tank dataComputing integrator gain. The values of the open-loop

data from the pumped-tank simulation in Figure 8 are dis-played in Figure 9. The CO is stepped from 71% down to65%, causing the liquid level (the PV) to rise. The con-troller output is then stepped from 65% up to 75%, causinga downward slope in the liquid level.

The slope of each segment is calculated as the change intank liquid level divided by the change in time. From theplot data, Slope1 is calculated to be 0.13 m/min and Slope2as –0.12 m/min. Using the slopes with their respective COvalues yields the integrator gain, Kp* = –0.025 m/%-min.

Computing dead time. The dead time, θp, is calculated asthe difference in time from when the CO signal was steppedand when the measured PV starts to exhibit a clear responseto that change. From the plot in Figure 10, the pumped-tankdead time is estimated be θp = 1.0 min.

PI control studyNow the controller design and tuning recipe for integrat-

ing processes can be used to design and test a PI controller. Determining bias value, CObias. A commercial controller

is normally put into practice using bumpless transfer —that is, when switching to automatic control, SP is initial-ized to the current value of PV and CObias to the currentvalue of CO. By choosing the current operation as thedesign state at switchover, the controller needs no correc-tive actions and it can smoothly engage.

Controller gain, Kc, and reset time, Ti. The first step inusing the IMC correlations listed in Table 1 is to computeTc, the closed-loop time constant. Tc describes how activethe controller should be in responding to a setpoint changeor in rejecting a disturbance. For integrating processes, thedesign and tuning recipe suggests:

Tc = 3θp = 3 × 1.0 min = 3 min

The PI controller gain, Kc, and reset time, Ti, are computed as:

K

dPVdt

dPVdt

CO COSlope Slope

CO Cp* =

−

−=

−−

2 1

2 1

2 1

2 OO1(7)

dPVdt

K CO t

dPVdt

K CO t

p p

p p

11

22

= −

= −

*

*

( )

( )

θ

θ

and

(6))

■ Figure 8. To perform a manual-mode bump test of the pumped-tankprocess, the controller outputs must be held constant long enough toshow the slope trend in the PV response.

Time, min

CO

, %Le

vel,

m

20 25 30 35 40

5.2

4.8

4.4

4.0

75

70

65

Slope1

Slope2

CO1

CO2

■ Figure 9. The slopes are calculated from bump test data to compute the integrator gain, Kp*.

Time, min

CO

, %Le

vel,

m

20 25 30 35 40

5.2

4.8

4.4

4.0

75

70

65

(27, 5.2)

CO1 = 65 CO2 = 75

(24, 4.8)

(31, 5.2)

(36, 4.6)

■ Figure 10. The difference in time from when the CO signal isstepped and when the measured PV starts to show a clear response to that change provides an estimate of the dead time from the bumptest data.

Time, min

CO

, %Le

vel,

m

20 25 30 35 40

5.2

4.8

4.4

4.0

75

70

65

θP = 1 min

60 www.aiche.org/cep June 2008 CEP

Substituting the Kp*, θp and Tc identified above intothese tuning correlations, we compute:

Recall that the P-only control of an integrating process(Figure 3) can provide a rapid setpoint response with noovershoot until a disturbance changes the balance point ofthe process. As labeled in Figure 11, the PI control set-point response now includes some overshoot.

The benefit of integral action is that when a disturbanceoccurs, a PI controller can reject the upset and return theprocess variable to its setpoint. This is because the con-stant summing of integral action continues to move thecontroller output until the controller error is driven to zero.Thus, PI control requires accepting some overshoot duringsetpoint tracking in exchange for the ability to reject dis-turbances. In many industrial applications, this isconsidered a fair trade.

K Tc i=+

+= = + =

10 025

2 3 13 1

2 3 1 72– .( )

( )( )–18 m/% miin

KK

T

TT Tc

p

c p

c pi c= ⋅

+

+=

1 222* ( )

θ

θ ++θ p (8)

Process Control

Literature Cited1. Cooper, D. J., ed., “Practical Process Control,”

www.controlguru.com (2008).2. Rice, R., and D. J. Cooper, “A Rule-Based Design

Method ology for the Control of Non-Self-RegulatingProcesses,” Proc. ISA Expo 2004, ISA CD Vol. 454,TP04ISA076 (2004).

3. Arbogast, J. E., and D. J. Cooper, “Extension ofIMC Tuning Correlations for Non-Self-Regulating(Integrating) Processes,” ISA Transactions, 46,pp. 303 (2007).

■ Figure 11. A PI controller provides setpoint tracking and disturbance rejection.

Accept some PV overshoot …

inlet flow disturbance

TimeSample Time, T = 1 s

CO

, %D

, %PV

and

SP,

%

10 20 30 40 50 60 70 80

5.04.84.64.4

807060

4321

… to get disturbance rejection

CEP

Level control in a distillation column reflux drum

At the top of a petroleum-refinery distillation column(below, top), vapor enters a condenser and flows as liq-

uid into a reflux drum. The liquid then exits the drum andeither returns to the column as reflux or exits the unit as dis-tillate. The control strategy design for the column is to main-tain a fixed distillate flow and adjust the level of the refluxdrum through manipulation of the reflux flowrate returning tothe top of the column.

Distillation columns are very sensitive unit operationswith very slow response times (long time constants). If thelevel controller is tuned aggressively for tight setpoint track-ing, large and rapid reflux flow changes could dramaticallyimpact column efficiency and stability. Thus, the reflux drumneeds to be tuned for conservative control actions whilemaintaining the level constraints.

Using the tuning procedure outlined in this article, thereflux drum level not only tracks closer to setpoint — it doesso with 95% less controller output movement.

■ Using the tuning recipe for reflux drum (top) level control improves theperformance (bottom) with 95% less controller output movement.

30

Leve

l PV

/ SP,

%

40

50

60

Time, h

Ref

lux

Flow

, %

2030405060708090

Lower Constraint

Upper Constraint

Aggressively Tuned PI Controller Conservatively Tuned PI Controller

Reflux Drum Level Performance

0 4 8 12 16 20 24 28

Reflux Drum

Distillate ValveReflux Valve

DistillationColumn

LIC

FICL

FICD

Condenser

Glossary and Nomenclature

CO = controller output signalCObias = controller bias or null valueDLO = design level of operatione(t) = current controller error, defined as SP – PVFOPDT = first-order-plus-dead-time modelIMC = internal model controlKc = controller gain, a tuning parameterKp* = integrator gainPV = measured process variableSP = setpointSVK = single-valve keggingT = sample timeTc = closed-loop time constantTi = reset time, a tuning parameterTp = overall process time constantθp = process dead time

CEP June 2008 www.aiche.org/cep 61

ROBERT RICE, PhD, is director of solutions engineering at Control Station, Inc., aprovider of process control solutions (One Technology Dr., Tolland, CT 06084;Phone: (860) 872-2920 x101; E-mail: bob.rice@controlstation.com; Website: www.controlstation.com). He has extensive field experience in bothregulatory and advanced controls and has published papers on a wide arrayof topics associated with automatic process control, including multi-variableprocess control and model predictive control. He has led the developmentand support of LOOP-PRO Product Suite, a PID diagnostic and optimizationtoolkit, and is a trainer for the company’s portfolio of practical process controltraining workshops. Prior to joining Control Station, he was an engineer withPPG Industries. He received his BS in chemical engineering from VirginiaPolytechnic Institute and State Univ. and both his MS and PhD in chemicalengineering from the Univ. of Connecticut.

DOUGLAS J. COOPER, PhD, is founder and chief technology officer of ControlStation, Inc. (Phone: (860) 872-2920; E-mail:doug.cooper@controlstation.com) and a professor of chemical, materials andbiomolecular engineering at the Univ. of Connecticut. He is also the authorand editor of controlguru.com, an e-book of industry best practices forimproving process control. He is a recognized specialist in the fields ofadvanced process modeling, monitoring and control; intelligent technologiesand adaptive process control; and software tools for process control systemanalysis, tuning and training. Prior to forming Control Station, he heldresearch positions with Arthur D. Little and Chevron. He received his BS inchemical engineering from the Univ. of Massachusetts, Amherst, MS inchemical engineering from the Univ. of Michigan, and PhD in chemicalengineering from the Univ. of Colorado.