2.35 Tuning PID Controllerstwanclik.free.fr/electricity/IEPOPDF/1081ch2_35.pdf · 2013-05-18 · ancing. Step changes in set point are the usual disturbance applied to test or tune

414

2.35 Tuning PID Controllers

P. W. MURRILL

(1970)

P. D. SCHNELLE, JR.

(1985)

B. G. LIPTÁK

(1995)

J. GERRY, M. RUEL, F. G. SHINSKEY

(2005)

In order for the reader to fully understand the content andconcepts of this section, it is advisable to first become familiarwith some basic topics. These include gains, time lags and reac-tion curves (Section 2.22), the PID control modes (Section 2.3),feedback and feedforward control (Section 2.9), and relativegain calculations (Section 2.25).

Controllers are designed to eliminate the need for con-tinuous operator attention when controlling a process. In theautomatic mode, the goal is to keep the controlled variable(or process variable) on set point. The controller tuningparameters determine how well the controller achieves thisgoal when in automatic mode.

DISTURBANCES

The purpose of a controller is to keep the controlled variableas close as possible to its set point at all times. How well itachieves this objective depends on the responsiveness of theprocess, its control modes and their tuning, and the size ofthe disturbances and their frequency distribution.

Sources

Disturbances arise from three different sources: set point,load, and noise. Noise is defined as a random disturbancewhose frequency distribution exceeds the bandwidth of thecontrol loop. As such, the controller has no impact on it,other than possibly amplifying it and passing it on to thefinal actuator, which can cause excessive wear and ultimatefailure.

Set point and load changes affect the behavior of thecontrol loop quite differently, owing to the dynamics in theirpath. A controller tuned to follow set point changes tends torespond sluggishly to load variations, and conversely a con-troller tuned to correct disturbances tends to overshoot whenits set point is changed.

Set Point

The set point is the desired value of the controlledvariable and is subject to adjustment by the operator. In acontinuous process plant, most of the control loops operate

as regulators, having a set point that remains unchanged fordays and even months at a time.

Examples of variables held at constant set points aredrum-level and steam temperature of a boiler, most pressureand level variables, pH of process and effluent streams, mostproduct-quality variables, and most temperature loops. Set-point response is of no importance to these loops, but theymust contend with load upsets minute by minute. In fact, theonly loops in a continuous plant that must follow set-pointchanges are flow loops.

Batch plants have frequent transitions between steadystates, some of which require rapid response to set-pointchanges with minimal overshoot. However, some of thesechanges are large enough to saturate the controller, particu-larly at startup. This can cause integral windup, whichrequires special means of prevention to overcome.

The Load

Only pure-batch processes — where no flow intoor out of the process takes place — operate at constant load,and that load is zero. All other processes can expect to encoun-ter variations in load, which are principally changing flow ratesentering and leaving vessels. A liquid-level controller, forexample, manipulates the flow of one liquid stream, whileother streams represent the load. Feedwater flow to a boiler ismanipulated to control drum level and must balance the com-bined flows of steam and blowdown leaving to keep level atset point. The load changes frequently and often unpredictably,but the set point may never change.

In a typical temperature control loop, the load is the flowof heat required to keep temperature constant. Liquid enteringa heat exchanger will require a certain flow of steam to reacha controlled exit temperature. Variations in liquid flow andinlet temperature will change the demand for steam flowmanipulated to keep exit temperature at set point.

Dynamics

The term

process dynamics

can refer to capacitance, inertia,resistance, time constant, dead time or their combinations.There is no dynamics involved with changing the set point,unless intentionally placed there for the purpose of filteringthe set point.

© 2006 by Béla Lipták


415

However, there is

always

dynamics in the load path. Loadvariables are principally the flow rates of streams similar tothose manipulated by the controller. Therefore the dynamicsin their path to the controlled variable are similar — and inmost cases identical — to the dynamics in the loop itself.

Figure 2.35a presents all the essential elements of a con-trol loop, showing its disturbance sources and dynamics.Most frequently, the dynamics are common to both the loaddisturbance and the controller output, meaning that the loadand manipulated streams enter the process at the same point.An example would be the control of composition of a liquidat the exit of a blender, where both the manipulated and loadstreams making up that blend are introduced at a commonentry point.

Less often encountered is the process where the dynam-ics in these two paths differ. An example of this is a shell-and-tube heat exchanger, wherein the temperature of a liquidleaving the tube bundle is controlled by manipulating theflow of steam to the shell. The shell may have more heatcapacity than the tubes, causing the temperature to respondmore slowly to a change in steam flow than to a change inliquid flow. Nonetheless, these two dominant lags will typ-ically not differ greatly.

Step Responses

Step testing is recommended for all control loops where thefrequency content of the disturbance variables is not speci-fied. There are cases of periodic disturbances, and they canpose special problems for control loops that themselves arecapable of resonating at a particular period. They are foundprincipally in cascade loops and in process interactions wherecontrollers manipulate valves in series or in parallel. Theseare considered in other sections of this work. Another perioddisturbance is the cyclic operation of such cleaning devicesas soot-blowers.

For the general case, the step disturbance is the mostdifficult test for the controller in that it contains all frequen-cies, including zero. In fact, the frequency content of the stepis identical to that of integrated white noise — therefore, it isan excellent test for loops subject to random disturbances.

Steps are also quite common in industry, representingconditions caused by sudden startup and shutdown of equip-ment; starting and stopping of multiple burners, pumps andcompressors; and capacity changes of reciprocating compres-sors. If a control loop can respond adequately to a step dis-turbance, then a ramp or exponential disturbance will haveless of an impact on it.

The step is also the easiest test to apply, requiring onlya size estimate, and can be administered manually. Pulsesrequire duration estimates, and doublet pulses require bal-ancing. Step changes in set point are the usual disturbanceapplied to test or tune a loop, even for loops that operate atconstant set point. Figure 2.35b illustrates a step responsewith 1/4 decay ratio.

The usual result of tuning a controller for set pointresponse is to reduce its performance to variations in load.Therefore, the effectiveness of a controller and its tuning asa load regulator need to be determined by simulating a stepload change.

Simulating a Load Change

Some controllers have an adjustable output bias. An acceptablesimulation of load change, when the controller is in automaticand at steady state, is a step change in the value of this bias.The value of the controller output prior to the step is an indi-cation of the current plant load because the loop was in a steadystate. The step in bias in that case moves the controller outputto another value, which disturbs the controlled variable andcauses the controller to integrate back to its previous steady-state output.

Alternatively, controllers that can be transferred “bump-lessly” between manual and automatic modes (most do — all

FIG. 2.35a

Load variables always pass through the dominant dynamic elements.

Load

Process

Set point

SP filler

ControllerLoop

dynamics

Load

dynamics

Noise

Controlled

variable

FIG. 2.35b

Step response curve of a control loop tuned for 1

/

4 decay ratio.

Output

1.0

00

ba

P

a/b = 1/4

Time


416

Control Theory

should) allow simulating a load change by using that feature.This is done by waiting until the loop is at steady state andon set point (zero deviation). At that point switching to the

manual

mode and stepping the output by the desired amountin the desired direction, and immediately (before a deviationdevelops) transferring back to the

automatic

mode. This procedure can be followed for all but the fastest

loops, such as flow loops. For them, a step in set point isacceptable, both because flow loops must follow set-pointchanges, and because for them, set-point tuning gives accept-able load response.

Comparing Set-Point and Load Responses

The steady-state process gain of a flow loop is typicallybetween 1 and 2, as indicated by the controller output beingbetween 50 and 100% when the flow measurement is at fullscale. The proportional gain of a typical flow controller is inthe range of 0.3 to 1.0, with the higher number associatedwith the process that has the lower steady-state gain. There-fore, the proportional loop gain for a typical flow loop is inrange of 0.6 to 1.0. As a result, a step change in set pointwill move the controller output approximately the correctamount to produce the same change in flow, by proportionalaction alone, that gives excellent set-point response.

This is not the case for other loops. Level has the oppositebehavior. To maintain a constant level, the controller mustmatch the vessel’s inflow and outflow precisely. Changingthe set point will cause the controller to change the manip-ulated flow, but only temporarily — when the level reachesthe new set point, the manipulated flow must return to itsoriginal steady-state value.

Therefore,

no

steady-state change in output is requiredfor a level controller to respond to a set-point change. TheIntegrated Error (IE) sustained by a controller following adisturbance varies directly with the change in output betweenits initial and final steady states. In response to a set-pointchange, the level controller has the same initial and finalsteady-state output values and hence sustains zero integratederror.

As a result, the error that is integrated while the level isapproaching the new set point will be matched exactly by anequal area of overshoot. In other words, set-point overshootis unavoidable in a level loop unless set-point filtering isprovided.

Most other processes, such as temperature, pressure, andcomposition, have steady-state gains higher than those of aflow process. But more importantly, they are also dominatedby lags, which allows the use of a higher controller propor-tional gain for tight load regulation. When this high propor-tional gain is multiplied by the process steady-state gain, theresulting loop gain can be as high as 5 to 10 or more.

A set-point step then moves the controller output far morethan required to drive the controlled variable to the new setpoint, producing a large overshoot. To minimize set-pointovershoot, the controller must be detuned, with lower gain

and longer integral time than is optimum for load regulation,or a filter must be applied to the set point.

Figure 2.35c compares responses to steps in set point andload for a process with distributed lag such as a dashedexchanger, distillation column, or stirred tank. The time scaleis normalized to

Σ

τ

, which is the time required for the dis-tributed lag to reach 63.2% of the full response to a step inputin the open loop. It is also the residence time of liquid in astirred tank.

If the PID settings are adjusted to minimize the IntegratedAbsolute Error (IAE) to the set point change, the dashedresponse curve is produced (SP tuning). Note that followinga step change in load, the return to set point is sluggish. Thisis commonly observed with lag-dominant processes. The PIDsettings that produce the minimum-IAE load response, shownin black (no filter), result in a large set-point overshoot,however.

Set-Point Filtering

If optimum load rejection is desired, without the large set-point overshoot that it produces on lag-dominant processes,the proportional response to set-point changes needs to bereduced.

Some PID controllers have the option to eliminate pro-portional action on the set point altogether. This tends toproduce a set-point undershoot, which can significantly dashedthe controller response, and it should never be used in thesecondary controller of a cascade system. (Incidentally,derivative action should never be applied to the set point, asthis always produces overshoot.)

Some controllers can reduce the controller’s proportionalgain when it acts on set point changes, either through the useof a lead-lag filter, or by the use of a specially structuredalgorithm. This adjustment allows separate optimization ofset-point response, after the PID settings have been tuned tooptimize the controller’s load response. The filter used forthe loop whose response is shown in Figure 2.35c has appliedonly half the controller’s proportional gain to the set point.

FIG. 2.35c

Set-point tuning slows load recovery for lag-dominant processes.

Co

ntr

oll

ed v

aria

ble

0

SP filter

No

filterSet

SP tuning

Load

Load

tuning

1 2Time, t/St

3 4



417

OPEN-LOOP TUNING

Applying a step to the process is simple and can be used totune the loop and to obtain a simple model for the process.Two methods are widely used. The first is the “process reac-tion curve,” which is not used to calculate the process modelbut is used to obtain the tuning parameters for rejecting theupsets caused by load changes.

The second method uses the “process model” by obtain-ing a simple process model; the tuning parameters are cal-culated from this model based on either a load rejection ora set point change criterion.

Process Reaction Curve

When a process is at steady state and it is upset by a stepchange, it usually starts to react after a period of time calledthe dead time (Figure 2.35d). After the dead time, most pro-cesses will reach a maximum speed (reaction rate), then thespeed will drop (self-regulating process) or the speed willremain constant (integrating process).

When tuning a loop to remove disturbances caused byload changes, the controller must react at its maximum rateof reaction, and the strength of the reaction will correspondto the maximum speed. Hence, to tune the loop, it is notnecessary to know the process model. It is sufficient to knowthe dead time and the maximum speed to calculate the tuningparameters.

Figure 2.35e illustrates the response of a temperatureloop after a step change in the controller output. This examplewill be used throughout this section to illustrate the differ-

ences between the recommended settings arrived at by usingthe various tuning techniques.

From this curve, it is not possible to determine the pro-cess model since the curve is too short to tell whether thereaction rate remains constant (integrating process) or goesdown (self-regulating process). From such a test, the modelcannot be found but the tuning parameters for load rejectioncan be estimated.

was applied at 10 s and the temperature started to increase

FIG. 2.35d

Reaction curve of a self-regulating process, caused by a step changeof one unit in the controller output. L

r

=

t

d

is dead time, R

r

is reactionrate, and K is process gain.

K

70

65.8

60

50

400 2 4 6 8

80

Lr

LrRr

t0.632

Time (minutes)

Controlled

process variable

(% of full scale)

FIG. 2.35e

Process reaction curve in response of a change in controller output (CO). The process variable (PV) range is 0 to 300 degC and the COrange is 0 to 100%.

130

120Pro

cess

Var

iabl

e

50

48

46

44

42

40

Con

trol

ler

Out

put

40 80

40 80

Time (sec)

23 s

Slope = (132 – 123) deg C

(70 – 40) s


As can be seen from Figure 2.35e, a 10% change in CO

418

Control Theory

23 s seconds later at 33 s. Hence the dead time (

t

d

) is 23 sand the reaction rate (speed) is

2.35(1)

The slope is:

After the dead time (

t

d

) and the reaction rate (

R

r

) havebeen determined, the controller settings are calculated byusing the equations in Table 2.35f.

If a PI controller is to be used for the process that wastested in Figure 2.35e, the values are:

P

=

0.9*10%/(0.1%/s X 23 s)

=

3.9

I

=

3.33 * 23 s

=

76.6 s

=

1.28 minutes.

Ziegler and Nichols recommend using the ratio of thecontroller output divided by the product of the slope and thedead time to calculate the proportional gain. The ideal processhas a small dead time and a small slope, so that the controllercan aggressively manipulate the controller output to bring theprocess back to set point.

The integral and derivative are calculated using the deadtime. The proportional is calculated using the slope and thedead time. If the slope is high, then the controller gain mustbe small because the process is sensitive; it reacts quickly. Ifthe dead time is long, the controller gain must be smallbecause the process response is delayed and therefore thecontroller cannot be aggressive. If one can reduce the slopeand the dead time of any process, it will be easier to control.

One of the advantages of open-loop tuning over theclosed-loop tuning technique is its speed because one doesnot need to wait for several periods of oscillation duringseveral trial-and-error attempts. The other advantage is thatone does not introduce oscillations into the process withunpredictable amplitudes. In open-loop tuning, the userselects the upset that is introduced, and it can be small.

Yet another advantage is that this test can be performed priorto the installation of the control system.

The disadvantages are also multiple. The open-loop testis not as accurate as the closed-loop one because it disregardsthe dynamics of the controller. Another disadvantage is thatthe S-shaped reaction curve and its inflection point are diffi-cult to identify when the measurement is noisy and/or if asmall step change was used.

Because of the above considerations, a good approach isto use the open-loop method of tuning in order to obtain thefirst set of initial tuning constants for a loop during startup.Then, refine these settings once the system is operating byretuning the loop using the closed-loop method.

Process Model

There are many ways to use and interpret the dead time (

t

d

)and reaction rate (

R

r

) values obtained from the open-looptuning method. Most open-loop methods are based onapproximating the process reaction curve by a simpler sys-tem. Several techniques are available to obtain a model.

The most common approximation by far is a pure timedelay (dead time) plus a first-order lag. One reason for thepopularity of this approximation is that a real-time delay ofany duration can only be represented by a pure time delaybecause there is yet no other simple and adequate approxi-mation. Theoretically, it is possible to use higher-than-first-order lags plus dead time, but accurate approximations aredifficult to obtain. Thus the real process lag is usually approx-imated by a pure time delay plus a first-order lag. Thisapproximation is easy to obtain, and it is sufficiently accuratefor most purposes.

The process’s dead time is the time period following anupset during which the controlled variable is not yet respond-ing. The time constant is a period between the time when aresponse is first detected and the time when the response hasreached 63.2

% of its final (new steady-state) value. The timeconstant is also the time it would take for the controlledvariable to reach its final value if the initial speed weremaintained.

Bump Tests

Figure 2.35g shows the Ziegler–Nichols pro-cedure to approximate that process reaction curve with a first-order lag plus a time delay. The first step is to draw a straight-line tangent to the process reaction curve at its point ofmaximum rate of ascent (point of infection).

Although this is easy to visualize, it is quite difficult todo in practice. This is one of the main difficulties in thisprocedure, and a considerable number of errors can be intro-duced at this point. The slope of this line is termed thereaction rate

R

r

. The time between the instant when the bumpwas applied and the time at which this line intersects theinitial value of the controlled variable prior to the test is thedead time, or transport time delay

t

d

. Figure 2.35g illustrates the determination of these values

for a one-unit step change (

∆

CO) in the controller output

TABLE 2.35f

Equations for Calculating the Ziegler–Nichols Tuning Parametersfor an Interacting Controller

Type of Controller P (gain) I (minutes/repeat) D (minutes)

P — —

PI 3.33

t

d

—

PID 2

t

d

0.5

t

d

∆COtdRr ∗

0 9. *∆COR tdr ∗

1 2.∆COR tdr ∗

Rtr = ∆

∆PV

∆∆PV C C

s sC

st= −

−= =132 123

70 409

30

9

deg deg degdeeg

deg

%. %/ . %/min

C

C

s

ss

300100

30330

0 1 6 0

∗

= = =



419

(manipulated variable) to a process. If a different-size stepchange in controller output was applied, the value of

τ

d

wouldnot change significantly.

However, the value of

R

r

is essentially directly propor-tional to the magnitude of the change in controller output.Therefore, if a two-unit change in output was used insteadof a one-unit change, the value of

R

r

would be approximatelytwice as large. For this reason the value of

R

r

used in Equation2.35(2) or any other must be the value that would be obtainedfor a one-unit change in controller output.

In addition to the dead time and reaction rate, the valueof the process gain

K

must also be determined as follows:

2.35(2)

There is a second method to determine the pure time delayplus the first-order lag approximation. In order to distinguishbetween these two methods, they will be called Fit 1 (describedin Figure 2.35g) and Fit 2 (described in Figure 2.35h).The only difference between these two is in how the first-order time constant is obtained. In case of Fit 2, the timeconstant of the process is determined as the differencebetween the time when the dead time ends and the time whenthe controlled variable has covered 63.2% of the distancebetween the pre-test steady state and the new one. The deadtime determination by both fits is the same and was alreadydescribed.

Another method to determine the dead time is to measurethe time when the PV moves by 2% of the total change.

The first-order lag time constants are given by:

Fit1:

=

1.5/0.1%/s

=

150 s

(same slope as previous section)

2.35(3)

Fit 2:

=

155 s – 33 s

=

122 s

2.35(4)

In Equation 2.35(4),

t

63.2%

is the time necessary to reach63.2

% of the final value, and

t

0

is the time elapsed betweenthe CO change and the beginning of the PV change. Note thatthe parameters for Fit 1 are based on a single point on theresponse curve, which is the point of maximum rate of ascent.However, the parameters obtained with Fit 2 are based on twoseparate points.

Studies

3

indicate that the open-loop response based onFit 2 always provides an approximation that is as good orbetter than the Fit 1 approximation. A typical curve resultingfrom the above procedure is shown in Figure 2.35i.

The response shown in Figure 2.35i resulted from a ten-unit change in controller output. For different step changes,

K

and

R

r

must be adjusted accordingly. From a curve suchas in Figure 2.35i, a number of parameters can be determined.The controller settings are calculated from the equations inTable 2.23j:

Table 2.35k compares the results obtained in terms of pro-cess gain (

K

), time constant (

τ

(

s

)), dead time (

t

d

(

s

)), and theresulting controller gain (

K

c

)

and integral time setting (

T

i

(min))of a PI controller.

In the following discussion, the process model deter-mined by the second fit will be used to compare against avariety of tuning criteria.

FIG. 2.35g

Maximum slope curve, Fit 1.

170

160

150

140

130

120

50

48

46

44

42

40

400 800

400 800

Pro

cess

var

iab

leC

on

tro

ller

ou

tpu

t

Time (sec)

= 183 − 33 sτ

K = final steady-state change in controlled vaariable (%)change in controller ouput (control uunit)

τ F rK R1 = /

τ F t t1 63 2 0= −. %


420

Control Theory

Comparing the Tuning Methods

One of the earliest methodsfor using the process reaction curve was proposed by Zieglerand Nichols. When using their process reaction curve method,which was described in connection with Figure 2.35e, only

R

r

and

t

d

or

t

0

must be determined. Using these parameters, the

empirical equations to be used to predict controller settings toobtain a decay ratio of 1/4 are given in Table 2.35j in terms of

K

,

t

d

, and

τ

.In developing their equations, Ziegler and Nichols con-

sidered processes that were not “self-regulating.” To illustrate,

FIG. 2.35h

Bump test, Fit 2 curve.

170

160

150

140

130

120

50

48

46

44

42

40

400

400Time (sec)

800

800

Pro

cess

var

iab

leC

on

tro

ller

ou

tpu

t

33 s155 s

120 degC + 0.632 *

45 degC = 148 degC

Process gain = (165−120) degC

(50−40)%

= 4.5 degC = 100*(4.5/300)%%%

= 1.5

FIG. 2.35i

A typical reaction curve using the dead time and time constant obtained by a bump test.

Time (sec)

50

48

46

44

42

40

400 800

800400

Co

ntr

oll

er o

utp

ut

170

160

150

140

130

120

Pro

cess

var

iab

le



421

consider the level control of a tank with a constant rate ofliquid outflow. Assume that the tank is initially operating atconstant level. If a step change is made in the inlet liquid flow,the level in the tank will rise until it overflows. This processis not “self-regulating.”

On the other hand, if the outlet valve opening and outletbackpressure are constant, the rate of liquid removalincreases as the liquid level increases. Hence, in this case,

the level in the tank will rise to some new position but willnot increase indefinitely, and therefore system will be self-regulating. To account for self-regulation, Cohen and Coon

4

introduced an index of self-regulation

µ

defined as:

2.35(5)

Note that this term can also be determined from the processreaction curve. For processes originally considered by Zieglerand Nichols,

µ

equals zero and therefore there is no self-regulation. To account for variations in

µ

, Cohen and Coonsuggested the equations given in Table 2.35l in terms of

t

d

and τ.In case of a proportional control, the requirement that the

decay ratio be 1/4 is sufficient to ensure a unique solution, butfor the case of proportional-plus-reset control, this restraint isnot sufficient.

Another constraint in addition to the 1/4 decay ratio canbe placed on the response to determine the unique values ofKc and Ti . This second constraint can be to require that thecontrol area of the response be at its minimum, meaning thatthe area between the response curve and the set point be thesmallest. This area is called the error integral or the integralof the error with respect to time.

With the proportional-plus-reset-plus-rate controller(PID), the same problem of not having a unique solutionexists even when the 1/4 decay ratio and the minimum errorintegral constraints are applied. Therefore, a third constraintmust be chosen to obtain a unique solution. Based on thework of Cohen and Coon, it has been suggested that this newconstraint could have a value of 0.5 for the dimensionlessgroup KcKtd /τ. The tuning relations that will result fromapplying these three constraints are given in Table 2.35l. Thismethod has been referred to as the 3C method.5–7

Greg Shinskey suggested a variation to the above, wherethe proportional gain and the integral time are increased.

TABLE 2.35j Ziegler–Nichols’ Recommendations to Obtain the TuningParameters for an Interacting Controller Based on the ReadingsCalculated from a Model


P — —

PI 3.33 td —

PID 2 td 0.5 td

TABLE 2.35kThe Tuning Setting Recommendations for a PI ControllerResulting from the Three Methods of Testing Described

Testing Method Used K τ (s) td (s) Kc Ti (min)

Reaction curve 3.91 1.28

Process model Fit1, ZN 1.5 150 23 3.91 1.28

Process model Fit 2, ZN 1.5 122 23 3.18 1.28

τK td∗

0 9. *τ

K td∗

1 2.τ

K td∗

µ = R Lr r K/

TABLE 2.35lComparison of Equations Recommended by Ziegler–Nichols, Shinskey, Cohen–Coon, and3C for the Determination of the Tuning Settings for PID Controllers

Ziegler–Nichols Shinskey Cohen–Coon 3C

P

P

I

P

I

D

KK tc d= ) −1.0( /τ ( /td τ ) −1.0 ( / ) ..td τ +−1 0 0 333 1 208 0 956. .( /td τ ) −

KK tc d= ) −1.00.9( /τ 0.95( /td τ ) −1.0 0 9 0 0821 0. ( / ) ..td τ +− 0 928 0 946. .( /td τ ) −

Ttidτ

τ= )3.33( / 4.0( /td τ ) 3.33( / ( /

( /

t t

td d

d

τ ττ

) + )+ + )

[ / ]

.

1 11

1 2 20.928( /td τ )−0 583.

KK tc d= ) −1.01.2( /τ 0.855( /td τ ) −1.0 1 35 0 2701 0. ( / ) ..td τ +− 1 370 0 950. ( / ) .td τ −

Ttidτ

τ= )2.0( / 1.6( /td τ ) 2.50( / ( /

( /

t t

td d

d

τ ττ

) + )+ )

[ / ]

.

1 5

1 0 60.740( /td τ )0 738.

Tdtdτ

τ= )0.5( / 0.6( /td τ )0.37( /

( /

t

td

d

ττ)

+ )1 0 2.0.365( /td τ )0 950.


422 Control Theory

Integral Criteria Tuning8

Table 2.35m provides the controller settings that minimizethe respective integral criteria to the ratio td/τ. The settingsdiffer if tuning is based on load (disturbance) changes asopposed to set point changes. Settings based on load changeswill generally be much tighter than those based on set pointchanges. When loops tuned to load changes are subjected toa set point change, a more oscillatory response is observed.

Which Disturbance to Tune for

With tuning parameters calculated for load rejection, theintegral time (Ti) and derivative time (Td) will depend mostlyon the dead time (td) of the process.

In contrast, if the tuning parameters are calculated for aset-point change, the integral time will be longer and thederivative time will be shorter, and they will depend mostlyon the time constant of the process.

The relationship between the controller settings based onintegral criteria and the ratio t0/τ is expressed by the tuningrelationship given in Equation 2.35(6).

2.35(6)

where Y = KKc for proportional mode, τ /Ti for reset mode,Td/τ for rate mode; A, B = constant for given controller andmode; t0, τ = pure delay time and first-order lag time constant.

Hence, using these equations,

2.35(7)

2.35(8)

2.35(9)

Lambda Tuning

Lambda tuning originated from Dahlin in 1968; it is basedon the same IMC theory as MPC,4,5 is model-based, and usesa model inverse and pole-zero cancellation to achieve thedesired closed-loop performance.

Lambda tuning is a method to tune loops based on poleplacement. This method ensures a defined response after aset-point change but is generally too sluggish to properlyreject disturbances. Promoters for this method often claimthat all loops should be tuned on the basis of Lambda tuning.Doing so, the controllers are almost in “idling mode” and

Y At

B

=

0

τ

KA

K

tc

B

=

0

τ

1 0

T

A t

i

B

=

τ τ

T At

d

B

= ∗

ττ0

TABLE 2.35mTuning Settings for Load and Set Point Disturbances

Load Change Set Point Change

A B A B

IAE P 0.902 –0.985

P 0.984 –0.986 0.758 –0.861

I 0.608 –0.707 1.020 –0.323

P 1.435 –0.921 1.086 –0.869

I 0.878 –0.749 0.740 –0.130

D 0.482 1.137 0.348 0.914

ITAE P 0.490 –1.084

P 0.859 –0.977 0.586 –0.916

I 0.674 –0.680 1.030 –0.165

P 1.357 –0.947 0.965 –0.855

I 0.842 –0.738 0.796 –0.147

D 0.381 0.995 0.308 0.929

ISE P 1.411 –0.917

P 1.305 –0.959

I 0.492 –0.739

P 1.495 –0.945

I 1.101 –0.771

D 0.560 1.006

ZN P 1.000 –1.000

P 0.900 –1.000

I 0.333 –1.000

P 1.200 –1.000

I 0.500 –1.000

D 0.500 1.000

CCC P 1.208 –0.956

P 0.928 –0.946

I 1.078 –0.583

P 1.370 –0.950

I 1.351 –0.738

D 0.365 0.950

Shinskey P 1.000 –1.000

P 0.952 –1.000

I 0.250 –1.000

P 0.855 –1.000

I 0.625 –1.000

D 0.600 1.000

4 to 1 decay P 1.235 –0.924

Critical damping (no overshoot,

maximum speed)

P 0.300 –1.000 0.300 –1.000

P 0.600 –1.000 0.350 –1.000

I 0.250 –1.000 Ti =1.16τP 0.950 –1.000 0.600 –1.000

I 0.420 –1.000 Ti = τD 0.420 1.000 0.500 1.000


2.35 Tuning PID Controllers 423

when the process load changes or other disturbance occurs,the time to eliminate this disturbance is quite long for mostprocesses, because the integral time selected equals the pro-cess time constant.

The promoters also suggest the use of a closed-loop timeconstant, which is three times the process time constant.Doing so, the response time in the automatic mode will bethree times longer than in manual. Therefore, the responsetime will be slower in automatic. This is adequate if nodisturbance occurs but if no disturbance occurs, the controlloop is not needed.

“Lambda tuning” refers to all tuning methods where thecontrol loop speed of response is a selectable tuning param-eter. The closed loop time constant is referred to as “Lambda”(λ). Therefore, following a set-point change, the PV will reachset point as a first-order system (same type of response as inthe manual mode when the CO is changed).

Lambda tuning has been widely used in the pulp andpaper industry, but control specialists are starting to realizethat it is often too sluggish to handle disturbances. For a first-order plus dead time model

Ti = τ 2.35(10)

2.35(11)

where λ = closed loop time constant; it is recommended touse λ = 3τ.

The performance of lambda tuning is unacceptable forcorrecting upsets caused by load changes if the process timeconstant is larger than the dead time. This is the case withpressure, level, and temperature control applications. Withflow loops, the results are similar to other methods since thetime constant is in the same order of magnitude as the deadtime.

In Table 2.35n, Fit 2 (Figure 2.35h) will be used as thereference to compare the process models found using thedifferent tuning criteria. For load and set-point responses ofthe different tuning techniques, see Figures 2.35o, p, and q.

Adjusting Robustness To remove oscillations in a controlloop, hence to increase the robustness, it is necessary to give

KK

T

tci

d

= ∗+

1λ

TABLE 2.35nThe Tuning Setting Recommendations for a PI ControllerResulting from the Criteria Listed

Criteria Tuning Kc Ti (min)

Load change criteria Ziegler–Nichols 3.18 1.28

CCC 3.00 0.71

Shinskey 3.37 1.53

IAE 3.40 1.03

SP change criteria IAE 2.13 1.16

Lambda 0.21 2.03

FIG. 2.35oLoad responses of the different tuning techniques (example).

200 400 600 800 1000

200 400 600 800 1000

5

15

25

45

Co

ntr

oll

er o

utp

ut

155

145

135

125

115

165

175

185

Pro

cess

var

iab

le

Lambda

ExperTune SP

ExperTune Load

IAE SP

IAEZNCCC

Time (sec)

SSE=16731.6IAE = 1449.67

Shinskey

35


424 Control Theory

up some performance. By reducing the proportional gain ina control loop, the robustness will be increased and the oscil-lations will be reduced or removed.

As a rule of thumb, dividing the proportional gain by afactor of two will eliminate the oscillations; dividing againthe proportional gain by a factor of two will remove most ofthe overshoot. (For more on robustness, see Section 2.26.)

Digital Control Loops

Digital control loops differ from continuous control loops byhaving the continuous controller replaced by a sampler, adiscrete control algorithm calculated by the computer, and ahold device (usually a zero-order hold). In such cases Mooreet al. have shown that the open-loop tuning methods presented

FIG. 2.35pSet-point responses of the different tuning techniques (example).

CCCIAE

200 400 600 800 1000

200 400 600 800 1000

Time (sec)

30

40

50

60

70

80

120

130

140

150

160

170

IAE = 2286.29 SSE = 37317

Co

ntr

oll

er o

utp

ut

Pro

cess

var

iab

le

ZNShinskey

ExperTune Load

IAE (SP)

ExperTune Load Lambda

FIG. 2.35qSet-point responses of the different tuning techniques (example) with a controller where the P is applied only on PV changes.

Co

ntr

oll

er o

utp

ut

160

150

140

130

120

Pro

cess

var

iab

le

64

60

56

52

48

44

40

200 400 600 800 1000

200 400 600 800 1000

CCC

ZN

IAE (L)

IAE (SP)

Lambda

Shinskey

Time (sec)



previously may be used, considering that the dead time usedis the sum of the true process dead time and one-half of thesampling time, as expressed by Equation 2.35(12):

2.35(12)

where T is the sampling time. t′0 is used in the tuning rela-tionships instead of t0. (Section 2.38 deals with the subjectof controller tuning by computer.)

CLOSED-LOOP RESPONSE METHODS

As has been discussed previously, in the open-loop method oftuning, the controller does not even have to be installed in orderfor the controller settings to be determined. When the closed-loop method is used, the controller is in automatic. Describedbelow are the two most common closed-loop methods of tuning,the ultimate method and the damped oscillation method.

Ultimate Method

One of the first methods proposed for tuning controllers wasthe ultimate method, reported by Ziegler and Nichols1 in1942. This method is called “the ultimate method” becauseits use requires the determination of the ultimate gain (sen-sitivity) and the ultimate period. The ultimate gain Ku is themaximum allowable value of gain (for a controller with onlya proportional mode) for which the system is stable. Theultimate period is the response’s period with the gain set atits ultimate value (Figure 2.35r).

In order for a closed loop to display a quarter of theamplitude damping (Figure 2.35b), its loop gain must be at0.5. This means that the product of the gains of all the com-ponents in the loop — composed by the process gain (Gp), thesensor gain (Ks), the transmitter gain (Kt), the controller gain(Kc = 100/PB), and the control valve gain (Kv) — must be at0.5. When the loop is in sustained, undampened oscillation,the gain product of the loop is 1.0 and the amplitude of cyclingis constant (Curve B in Figure 2.35r).

The period when the closed loops oscillate dependsmostly on the amount of dead time in the loop. The period ofoscillation in flow loops is 1 to 3 seconds; for level loops, itis 3 to 30 seconds (sometimes minutes); for pressure loops,5 to 100 seconds, for temperature loops; 0.5 to 20 minutes;and for analytical loops, from 2 minutes to several hours.

When controlled by analog controllers (no dead timeadded by sampling), plain proportional loops oscillate at peri-ods ranging from two to five dead times, PI loops oscillateat periods of three to five dead times, and PID loops at aroundthree dead time periods.

The settings determined by this method will be based onload disturbance rejection and will not be suitable for set-pointchanges. The tuning parameters are in fact calculated on the

basis of the distance where the loop will operate from insta-bility. Settings for load rejection are not too far from instabil-ity, but tuning parameters for set-point change are different,and the process model is needed for their determination.

The optimum integral and derivative settings of control-lers vary with the number of modes in the controller (P = 1,PI = 2 mode, PID = 3 mode) and with the amount of deadtimes in the loop. For noninteracting PI controllers with nonoticeable dead time, one would set the integral (Ti) for about75% of the period of oscillation.

As the dead time-to-time constant ratio rises, the integralsetting becomes a smaller percentage of the oscillationperiod — around 60% when the dead time equals 20% of thetime constant, about 50% when their ratio is at 50%, about33% when they are equal, and about 25% when the dead timeexceeds the time constant.

For noninteracting PID loops with no dead time, onewould set the integral minutes/repeat (I) to a value equal to50 % of the period of oscillation and the derivative time (D)to about 18% of the period. As dead time rises to 20% of thetime constant, (I ) drops to 45% and (D) to 17% of the period.At 50% dead time, (I) = 40% and (D) = 16%. When the deadtime equals the time constant, (I ) = 33% and (D) = 13%;finally, if dead time is twice the time constant, (I ) = 25% and(D) = 12%.

′ = +t t T0 0 2/

FIG. 2.35rUltimate gain is the gain that causes continuous cycling (Curve B)and ultimate period (Pu) is the period of that cycling.

Pu

Curve CCurve B

Curve A

Output

Curve A: unstable, runaway oscillation

Curve B: continuous cycling, marginal stability

Curve C: stable, damped oscillation

Time


426 Control Theory

Tuning Example The same example as was used earlier willbe used to illustrate the “ultimate” closed-loop tuningmethod. The aim of this tuning process is to determine thecontroller gain or proportional band that would cause sus-tained, undampened oscillation (Ku) and to measure the cor-responding period of oscillation, called the ultimate period(Pu). The steps in this tuning sequence are as follows:

1. Set all controller dynamics to zero. In other words, setthe integral to infinite (or maximum) minutes perrepeat or zero (or minimum) repeats per minute andset derivative to zero (or minimum) minutes.

2. Set the gain or proportional band to some arbitraryvalue near the expected setting (if known) or at Kc = 1(PB = 100%) if no better information is available.

3. Let the process stabilize. Once the PV is stable, intro-duce an upset. The simplest way to do that is to movethe set point up or down by a safe amount (for exam-ple, move it by 2% for half a minute) and then returnit to its original value.

The result will be an upset in the PV resembling thecharacteristics of curve A, B, or C in Figure 2.35r. If theresponse is undampened (curve A), the gain (or proportional)setting is too high (proportional narrow); inversely, if theresponse is damped (curve C), the gain (or proportional)setting is too low (proportional wide). Therefore, if theresponse resembles curve A, the controller gain is increased;if it resembles curve C, the gain is reduced, and the test isrepeated until curve B is obtained.

After one or more trials, the state of sustained, undamp-ened oscillation will be obtained (curve B), and at that pointthe test is finished. (Make sure that the oscillation is a sinu-soidal and not a limit cycle.) Next, read the proportional gainthat caused the sustained oscillation. This is called the “ulti-mate gain” (Ku), and the corresponding period is the ultimateperiod of oscillation (Pu).

Once the values of Ku and Pu are known, one might usethe recommendations of Ziegler–Nichols (Table 2.35s) or therecommendations that were described earlier, which also con-sider the dead time-to-time constant ratio. No one tuning isperfect, and experienced process control engineers do comeup with their own “fudge factors” based on experience.

In order to use the ultimate gain and the ultimate periodto obtain the controller settings for proportional controllers,Ziegler and Nichols correlated the decay ratio vs. gainexpressed as a fraction of the ultimate gain for several sys-tems. From the results they concluded that if the controllergain is set to equal one-half of the ultimate gain, it will oftengive a decay ratio of 1/4, i.e.,

2.35(13)

By analogous reasoning and testing, the equations inTable 2.35s were found to also give reasonably good settingsfor noninteracting two- and three-mode controllers. Again itshould be noted that these equations are empirical and excep-tions abound. For the same example as before, Figure 2.35tillustrates the ultimate cycling response.

The ultimate gain and period obtained from Figure 2.35tare: Ku = 7.75 and Pu = 87 s.

Hence the recommended tuning settings for the processthat was used in the example are: Kp = 3.49 and Ti = 1.21minutes.

There are a few exceptions to the tuning proceduredescribed here because in some cases, decreasing the gainmakes the process more unstable. In these cases, the “ultimate”method will not give good settings. Usually in cases of thistype, the system is stable at high and low values of gain butunstable at intermediate values. Thus, the ultimate gain forsystems of this type has a different meaning. To use the ulti-mate method for these cases, the lower value of the ultimategain is sought.

Advantages and Disadvantages The main advantage of theclosed-loop tuning method is that it considers the dynamicsof all system components and therefore gives accurate resultsat the load where the test is performed. Another advantageis that the readings of Ku and Pu are easy to read and theperiod of oscillation can be accurately read even if the mea-surement is noisy.

TABLE 2.35sTuning Parameters Based on the Measurement of Ku and Pu

Recommended by Ziegler–Nichols for a NoninteractingController


P 0.5 Ku — —

PI 0.45 Ku Pu/1.2 —

PID 0.6 Ku Pu/2 Pu/8

FIG. 2.35tUltimate cycling response of the same process that was tested inFigure 2.35e.

145

125

300

300 600

600

Pu

55

35

Co

ntr

oll

er o

utp

ut

Pro

cess

var

iab

le

Time (sec)

Kc uK u= =0 5 2. ( )PB PB



The disadvantages of the closed-loop tuning method arethat when tuning unknown processes, the amplitudes ofundampened oscillations can become excessive (unsafe) andthe test can take a long time to perform. One can see that whentuning a slow process (period of oscillation of over an hour),it can take a long time before a state of sustained, undampenedoscillation is achieved through this trial-and-error technique.For these reasons, other tuning techniques have also beendeveloped and some of them are described below.

Damped Oscillation Method

Harriott has proposed a slight modification of the previousprocedure. For some processes, it is not feasible to allowsustained oscillations and therefore, the ultimate method can-not be used. In this modification of the ultimate method, thegain (proportional control only) is adjusted, using steps anal-ogous to those used in the ultimate method, until a responsecurve with 1/4 of the decay ratio is obtained. However, withthis tuning method, it is necessary to note only the period Pof the response.

Again it should be noted that the equations in Table 2.35uare empirical and exceptions abound.

Example for Damped Cycling From Figure 2.35v the gainand period are found to be Kc1/4 = 7.75 and P = 87 s. Hence therecommended tuning settings for the PI controller are Kp = 3.49and Ti = 1.21 minutes.

After these modes are set, the sensitivity is again adjusteduntil a response curve with 1/4 of a decay ratio is obtained.

This method usually requires about the same amount of workas the ultimate method since it is often necessary to experi-mentally adjust the value of the gain to obtain a decay ratioof 1/4. It is also possible to use this method to use a differentdecay ratio criterion.

Advantages and Disadvantages In general, there are twomajor disadvantages to the ultimate and damped oscillationmethods. First, both are essentially trial-and-error methods,since several values of gain must be tested before the ultimategain or the gain to give a 1/4 decay ratio are to be determined.To make one test, especially at values near the desired gain,it is often necessary to wait for the completion of severaloscillations before it can be determined whether the trialvalue of gain is the desired one.

Second, while one loop is being tested in this manner, itsoutput may affect several other loops, thus possibly upsettingan entire unit. While all tuning methods require that somechanges be made in the control loop, other techniques requireonly one and not several tests, unlike the closed-loop methods.

Also, if the tuning parameters are too aggressive, theexpected response can be obtained by increasing the propor-tional band (or decreasing the proportional gain). The integraland derivative settings probably need to be modified. Theproportional gain has to be reduced to 3.5 to have a quarter-of-amplitude decay.

COMPARISON OF CLOSED AND OPEN LOOP

Table 2.35w provides a comparison of open-loop and closed-loop results for the process example used in Figure 2.35e.

FREQUENCY RESPONSE METHODS

Frequency response methods for tuning controllers involvefirst determining the frequency response of the process,which is a process characteristic. From this, tuning can bedeveloped. Frequency response methods (FRM) may haveseveral advantages over other methods: These are:

1. FRM require only one process bump to identify the pro-cess. The bump can be a change in automatic or manual,

TABLE 2.35uHarriott Tuning Parameters for a Noninteracting ControllerCalculated from Kc1/4 Obtained to Reach a Quarter-of-Amplitude Decay


P Kc1/4 — —

PID adjusted P/1.5 P/6

FIG. 2.35vDamped cycling for the example.

150

140

130

120

80

70

60

50

40

30

Pro

cess

var

iab

leC

on

tro

ller

ou

tpu

t

300 600TABLE 2.35wComparison of Closed and Open Loop Test Results

Type of Tuning Test Kc Ti (min)

Open loop Reaction curve 3.91 1.28

Model Fit 2 3.18 1.28

Closed loop Closed loop, ultimate cycling 3.49 1.21

Closed loop, damped cycling 3.5 1.20


428 Control Theory

and be either a pulse, step, or other type of bump. A set-point change provides excellent data from FRM.

2. FRM do not require any prior knowledge of the pro-cess dead time or time constant. With the other timeresponse methods, one often needs a dead time esti-mate and a time constant estimate.

3. FRM do not require any prior knowledge of the pro-cess structure. Time response methods often requirethe user to have such model structure knowledge, i.e.,whether it is first or second order or whether it is anintegrator. For FRM-based tuning none of this isrequired; only the process data are needed.

Obtaining the Frequency Response

The process frequency response is a graph of amplitude ratioand phase vs. oscillation or sine wave frequency. If one injectsa sine wave into a linear process at the controller output, thePV will also display a sine wave. The output (PV) sine wavewill probably be of smaller height relative to the input andwill be shifted in time.

The ratio of the heights is the amplitude ratio at thefrequency of the input sine wave. The shift in time is thephase shift or phase lag. A time shift resulting in the trough

of the output when aligned with the crest of the input isgenerally thought to be 180 degrees out of phase or 180degrees of phase lag.

By applying a variety of sine wave inputs to a process,one can obtain a table of amplitude ratios and phase lagsdependent on the sine wave frequency. If one plots these, theresult will be the frequency response of the process.

Using Fourier analysis computer software programs onecan calculate the process frequency response from a bump,pulse, or any other signal that applies sufficient excitation tothe controller output. In the evaluation both the CO and PVtrends are used. The data provided for these programs shouldstart from a settled state, experience a quick change, and endsettled.

Any one of the responses in Figures 2.35e, g, h, i, o, p, q,and v would provide adequate data for frequency response–based testing. Figure 2.35x shows the typical frequencyresponse arrived at by the use of computer software.

PID Tuning Based on Frequency Response

In most processes, both the amplitude ratio and the phaseangle will decrease with increasing frequencies. Assuming

FIG. 2.35xTypical model (solid line) and actual (dash line) process frequency response.

−1.25

−23.75

0

−180

−360

Am

pli

tud

e ra

tio

(d

B)

Ph

ase

ang

le (

deg

)

10−1 100 101

10−1 100 101

Frequency (radians/sec)



that the combined phase and amplitude ratio decreases withfrequency when the process and the controller frequencyresponses are combined, the following general stability ruleapplies: A control system will be unstable if the open-loopfrequency response has an amplitude ratio that is larger thanone when the phase lag is 180 degrees.

To provide proper tuning, a margin of safety in the gainand phase is desired. Tuning constants are therefore adjustedto result in the highest gain at all frequencies and yet achievea certain margin of safety or stability. This is best accom-plished using computer software.

FINE TUNING

The performance of a controller can be tested by simulatinga load change in the closed loop as described at the beginningof this section. This can be especially useful if the initialtuning is not satisfactory or if the process characteristics havechanged. It is simply a trial-and-error method of recognizingand approaching an optimum — or otherwise desirable —load response.

Optimum Load Response

As described earlier, the optimum load response is generallyconsidered to be that which has a minimum IAE, in that itcombines minimum peak deviation with low integrated errorand short settling time. The second curve from the bottom ineach of Figures 2.35y and z represents a minimum-IAE loadresponse for a distributed lag under PI control.

This second curve from the bottom has a symmetricalfirst peak, low overshoot, and effective damping. The timescale of these curves is normalized to the 63.2% open-loopstep response of the distributed lag, identified as Στ. On eitherside of the optimum curve in both figures are other responsecurves, which were produced by changing one of the con-troller settings.

In Figure 2.35y, the proportional band of the controlleris the parameter being adjusted. Note that increasing theproportional band increases the height of the first peak andalso its damping; but along with the increased dampingcomes a loss of symmetry. In Figure 2.35z, it is the integraltime of the controller that is being adjusted. Note that it hasno effect at all on the first peak but determines the locationof the second, i.e., the overshoot or undershoot of the processvariable’s deviation.

The integrated error IE of a standard PID controller variesdirectly with the product of its proportional band and integraltime. While increasing either of these settings improvesdamping, it also increases IE in direct proportion, and there-fore it costs performance. Increasing both settings above theiroptimum compounds this effect.

Effect of Load Dynamics

All of the tuning rules described in this section to optimizeload regulation will apply only to loops in which the dynam-ics in the load path are identical to those in the path of thecontroller output. Yet Figure 2.35a shows the possibility thatthe dynamics are different in the two paths, for example inthe case of heat exchangers.

Any dead time in the load path, or variation thereof, willhave no effect on the load response curve, simply delayingit by more or less time. The dominant lag in the load path iswhat determines the shape of the leading edge of the responsecurve.

As a demonstration, a distributed lag in a PI control loopwas simulated to assess the effects of variations in loaddynamics. Three different load-response curves appear inFigure. 2.35aa representing ratios of Στq /Στm varying from0.5 to 2, where subscripts q and m identify the load andmanipulated-variable paths, respectively.

In all three cases, the PI controller has been tuned to min-imize IAE following a simulated load change, simulated bystepping the controller output, which produces the center curverepresented by Στq/Στm = 1. However, when a true load step

FIG. 2.35yThe proportional setting affects the height of the first peak and itssymmetry.

Time, t/St

Popt × 3

Popt × 1.5

Popt

Popt ÷ 1.5

Dev

iati

on

0

0 1 2 3 4

FIG. 2.35zThe integral setting primarily affects the location of the second peak.

Dev

iati

on

lopt ÷ 1.5

Time, t/St

lopt

lopt × 1.5

lopt × 3

0

0 1 2 3 4


430 Control Theory

passes through different dynamics than in the controller-outputpath, the resulting response curve is no longer minimum-IAE.

As might be expected, faster dynamics in the load pathproduce a faster rising deviation, which therefore peaks at ahigher value before the controller can overcome it. The largerpeak is then followed by a large overshoot. By contrast,slower dynamics in the load path reduce the peak deviation,followed by an undershoot. The IAE for the curve whereΣτq /Στm = 0.5 is actually only 6% higher than the minimum,but for the case where Στq /Στm = 2, the IAE is actually 53%above the minimum, indicating substantial need for correc-tion. The appearance of the exaggerated overshoot and theundershoot indicate that those responses are no longer min-imum-IAE, and that the integral time needs adjusting. Retun-ing the controllers for minimum-IAE response in the pres-ence of different load dynamics resulted in the developmentof the correction factors plotted in Figure 2.3bb.

To use these factors, one should first tune the controller tominimize IAE following a load change simulated by steppingthe controller output. After that the appropriate correction factorsshould be found for the expected ratio of load-to-manipulatedlag Στq/Στm.

The integral time will need correcting regardless of thecontroller used—it must be multiplied by a factor repre-sented by the top curve for a PID controller and the middlecurve for a PI controller. If the controller is a noninteracting(ideal, parallel) PID, then its proportional band should bemultiplied by the correction factor, which is indicated by thedashed line. PI and interacting (series) PID controllers do notrequire a correction to the proportional setting, and no cor-rection to the derivative setting is required for any controller.

Symbols, Abbreviations

CO Controller outputFRM Frequency response methodGp Process gainIAE Integral of absolute errorIE Integral errorISE Integral of squared errorITAE Integral of absolute error × timeITSE Integral of squared error × timeKp Proportional gain for a PID controllerKu Ultimate controller gainPV Process variable or measurementRr Reaction rate, slopeRRT Relative response time, the time to remove most

of a disturbanceSP Set point� (Tau) Process time constant (seconds)

FIG. 2.35aaLoad dynamics affect the shape of the response curve.

Σtq/Σtm = 0.5

1

2

0

0

1 2 3 4

Time, t/Σt

Dev

iati

on

FIG. 2.35bbTo obtain minimum-IAE response, integral time and possibly the proportional band may require correction for load dynamics.

10

1

0.10.1 1 10

PID

Integral corr.

PI

Prop. corr.

Nonint. PID

Σtq/Σtm



t0(td) Process dead time (seconds)Ti Integral time (s) -for a PID controllerTd Derivative time (s) for a PID controller�F PV filter time constanttu Ultimate period

Bibliography

Astrom, K.J., and Hagglund, T., Automatic Tuning of PID Controllers,Research Triangle Park, NC: ISA Press, 1988.

Bjornsen, B.G., A Fluid Amplifier Pneumatic Controller, Control Engineering,June 1965.

Box, G.E.P., and Jenkins, G.M., Time Series Forecasting and Control, SanFrancisco: Holden-Day, 1970.

Caldwell, W.I., Coon, G.A., and Zoss, L.M., Frequency Response for ProcessControl, New York: McGraw-Hill, 1959.

Campbell, D.P., Process Dynamics, New York: John Wiley & Sons, 1958.Chen, C.T., Introduction to Linear System Theory, New York: Holt, Rinehart

and Winston, 1970.Corripio, A.B., Tuning of Industrial Control Systems, Research Triangle

Park, NC: ISA Press, 1990.Gerry, J.P., How to Control Processes with Large Dead Times, Control

Engineering, March 1998. Gerry, J.P., PID settings sometimes fail to make the leap, Intech, November

1999.Gerry, J.P., Tune Loops for Load Upsets VS Setpoint Changes, CONTROL

Magazine, September 1991.Higham, J.D., Single-Term Control of First- and Second-Order Processes with

Dead Time, Control, February 1968.Hougen, J.O., Experiences and Experiments with Process Dynamics, Chemical

Engineering Process Monograph Series, Vol. 60, No. 4, 1964.Howell, J., Rost, D., and Gerry, J.P., PC Software Tunes Plant Startup, InTech,

June 1988.Kraus, T.W., and Myron, T.J., Self-Tuning PID Controller Uses Pattern

Recognition Approach, Control Engineering, June 1984.Larson, R.E., et al., State Estimation in Power Systems, Part I and Part II,

IEEE Trans. on Power Apparatus and Systems, Vol. PAS 89, No. 3,March 1970.

Lelic, M.A., and Zarrop, M.B., Generalized Pole-Placement Self-TuningController, Int. J. Control, 46, 1987, pp. 569–607.

Lipták, B.G., How to Set Process Controllers, Chemical Engineering,November 1964.

Lopez, A.M., et al., Controller Tuning Relationships Based on InstrumentationTechnology, Control Engineering, November 1967.

Luyben, W.L., Process Modeling, Simulation and Control for ChemicalEngineers, 2nd edition, New York: McGraw-Hill, 1990.

Mehra, R.K., Identification of Stochastic Linear Dynamic Systems UsingKalman Filter Representation, AIAA Journal, January 1971.

Mehra, R.K., On-Line Identification of Linear Dynamic Systems with Applica-tions to Kalman Filtering, IEEE Trans. on Automatic Control, Vol. AC 16,No. 1, February 1971.

Miller, J.A., et al., A Comparison of Open-Loop Techniques for Tuning Con-trollers, Control Engineering, December 1967.

Murrill, P.W., Automatic Control of Processes, Scranton, PA: InternationalTextbook Co., 1967.

Raven, F., Automatic Control Engineering, New York: McGraw-Hill, 1961.Rovira, A.A., Ph.D. dissertation, Louisiana State University, Baton Rouge,

LA, 1969.

Rovira, A.A., Murrill, P.W., and Smith, C.L., Tuning Controllers for Set-PointChanges, Instruments and Control Systems, December 1969.

Ruel, M., Loop Optimization: before You Tune, Control Magazine, Vol. 12,No. 03 (March 1999), pp. 63–67.

Ruel, M., Loop Optimization: Troubleshooting, Control Magazine, Vol. 12,No. 04, (April 1999), pp. 64–69.

Ruel, M., Loop Optimization: How to Tune a Loop, Control Magazine, Vol. 12,No. 05 (May 1999), pp. 83–86.

Ruel, M., Instrument Engineers' Handbook, chap. 5.9, Plantwide ControlLoop Optimization, 3rd ed., Bela G. Liptak, Ed., Boca Raton, FL:CRC Press, 2002, 40 pp.

Shinskey, F.G., Process Control Systems, 4th ed., New York: McGraw-Hill,1996.

Shinskey, F.G., Process Control Systems, 4th ed., New York: McGraw-Hill,1996.

Shinskey, F.G., Interaction between Control Loops, Instruments and ControlSystems, May and June 1976.

Salmon, D.M., and Kokotovic, P.V., Design of Feedback Controllers for Non-linear Plants, IEEE Trans. on Automatic Control, Vol. AC 14, No. 3,1969,

Sastry, S., and Bodson, M., Adaptive Control: Stability, Convergence andRobustness, Englewood Cliffs, NJ: Prentice Hall, 1989.

Seborg, D., Edgar, E., and Mellichamp, D., Process Control, New York:Wiley, 1990.

Seborg, D.E., Edgar, T.F., and Shah, S.L., Adaptive Control Strategies forProcess Control, AI Chem. E. Journal, 32, 1986, pp. 881–913.

Smith, C.A., and Corripio, A.B., Principles and Practice of Automatic ProcessControl, New York: John Wiley & Sons, 1985.

Smith, C.L., Controller Tuning That Works, Instruments and Control Systems,September 1976.

Smith, C.L., and Murrill, P.W., A More Precise Method for the Tuning ofControllers, ISA Journal, May 1996.

Smith, C.L., Controllers—Set Them Right, Hydrocarbon Processing andPetroleum Refiner, February 1966.

Smith, O.J.M., Close Centrol of Loops with Dead Time, Chemical EngineeringProgress, May 1957.

Sood, M., Tuning Proportional-Integral Controllers for Random LoadChanges, M.S. thesis, Department of Chemical Engineering, Universityof Mississippi, University, MS, 1972.

Sood, M., and Huddleston, H.T., Optimal Control Settings for Random Dis-turbances, Instrumentation Technology, March 1973.

Stephanopoulos, G., Chemical Process Control: An Introduction to Theoryand Practice, Englewood Cliffs, NJ: Prentice Hall, 1984.

Suchanti, N., Tuning Controllers for Interacting Processes, Instruments andControl Systems, April 1973.

Wade, H.L., High-Capability Single-Station Controllers: A Survey, InTech,September 1988.

Wells, C.H., Application of Modern Estimation and Identification Techniquesto Chemical Processes, AIChE Journal, 1971.

Wells, C.H., and Mehra, R.K., Dynamic Modeling and Estimation of Carbonin a Basic Oxygen Furnace, paper presented at 3rd Int. IFAC/IFIPSConf. on the Use of Digital Computers in Process Control, Helsinki,Finland, June 1971.

Wellstead, P.E., and Zarrop, M.B., Self-Tuning Systems, New York: Wiley,1991.

Widrow, B., and Stearns, S.D., Adaptive Signal Processing, EnglewoodCliffs, NJ: Prentice Hall, 1985.

Ziegler, J.G., and Nichols, N.B., Optimal Settings for Automatic Controllers,ASME Trans., November 1942.

Ziegler, J.G., and Nichols, N.B., Optimal Settings for Controllers, ISA Journal,August 1964.


Documents

2.35 Tuning PID Controllerstwanclik.free.fr/electricity/IEPOPDF/1081ch2_35.pdf · 2013-05-18 · ancing. Step changes in set point are the usual disturbance applied to test or tune