Process Control Systems Episode 4 ppt

-kt no flow, there is a static head of 5 lsig, while 10 gpm willraise the pressure to 13 psig; the range of the pressure transmitter is 0 to 25 psip.Estimate what the proportional band o

Analysis of Some Common Loops I 83

FIG 3.9 Agitation reduces the

effective dead time while increasing

the effective time constant.

of Chap 2) Naturally any effort spent in minimizing the samplingtime will be rewarded by bot’h tighter control and faster response

Some analyzers are discont,inuous They produce only one analysis

in a given time interval This characteristic is worthy of much moreattention, because it periodically interrupts t,he control loop Processchromatographs are the principal, but not sole, constituents of thisgroup The response of this kind of control loop will be given extensiveroverage in Chap 4, and methods for coping with it will be presented

A few analyzers exhibit a time lag in addition to the dead t’ime ciated with sample t,ransport Sormally this property is of little conse-quence, except when the process itself consists of nothing but the volume

asso-of a pipeline, whose time const,ant may be less than that, of the analyzer.Those measurements which are fast, are by the same t,oken subject tonoise Conductivity and pH are usually in this category, because theyare fast enough to react to an incompletely mixed solution, or particles

of an immiscible phase

Dead time in sample lines is understandably constant Dead t’ime in apipe carrying the main st,ream varies with flow Dead t,ime w i t h i n astirred tank is slight,ly affected by flow, to the extent of F/F,; in most

systems t,his variation would not be significant The natural period ofthe composition loop would therefore be virtually constant, producing

84 1 Ud n erstanding Feedback Control

constant dynamic gain, except for a process whose dominant element

is a pipeline

Most analyzers are not so far from being linear that they materiallyaffect the gain of the control loop The notable exception is, of course,the pH measurement, whose general properties have already been pre-sented But analyzers are generally given a high order of sensitivity,because of the importance placed on quality control As a result, thegain of a composition-control loop is invariably high Objectively, com-position is not as difficult to control as flow, for example, but the specifica-tions placed on product quality are so stringent that ordinary perform-ance is seldom acceptable The impurity of a product stream leaving afractionator, for example, may be specified at 1.0 f 0.2 percent It isvirtually impossible to regulate flow within + 1 percent in the unsteadystate, yet the composition controller is asked to perform five times as well.This is perhaps the greatest single reason why composition control hasthe distinction of being a problem area Because quality can be measured

to 0.1 percent is apparently reason enough to expect it to be controlled

to the same tolerance

Since the nominal flow F has already been identified as a constant, process

gain is also constant (This is another illustration of the case whereprocess steady-state gain varies with flow, but the time constant does too,

so dynamic gain is invariant Steady-state gain, as calculated above,

is only meaningful at the rated flow F.)

Dimensional gain of the composition process can always be found bywriting a material balance across it If composition of an effluent stream

is controlled by manipulating an influent stream, as in this example, theprocess is linear But if effluent composition is controlled by manipulat-ing the efluent flow, the process is hyperbolic:

x=1

F

d x - X

Analysis of Some Common Loops I 85

(This was already encountered in the temperature-control example wherecoolant temperature was adjusted by manipulating its flow.) E x a m p l e s

of both linear and hyperbolic processes are common in both compositionand temperature applications, because the controlled variable is always afunction of the ratio of one varmble to another If the manipulatedvariable happens to be in the numerator, the process is linear

example.3.6

The process in Fig 3.8 is intended to deliver a solution at a nominal flow

F, of controlled composition Z, by adding a manipulated flow X of

concen-trate to the diluent stream Let the volume of the vessel be 100 gal and thenominal flow 20 gpm If mixing is 95 percent complete, then O.O5V/F will

be the effective dead time in the vessel:

1007d = 0.05 - = 0.25 min

The phase shift of the 3.0-set lag at a period of 2.0 min is

& = -tan-1 ?$!!&z = -9”

A control valve with a 3.0-set lag will contribute another 9” This addedphase shift extends the natural period to approximately

7,, = 2 o180 + 9 + 9 =

1 8 0 2.2 minThe dynamic gain of the process is simply that of the principal time

c o n s t a n t :

2.2 G1 = Gl = 2a4.75 = 0.0737

Dimensional process gain is the percent composition change broughtabout by a change in concentrate flow at the rated throughput:

dx 1

-Z-C

dX F lOO%PO gpm = 5 %lgpm

86 1 Ud n erstanding Feedback Control

Because the process is linear with respect to concentrate flow, a linear valve

i s c h o s e n Let the maximum flow of concentrate be 2 gpm T h e n

G, = 2 gpm/lOO% = 0.02 gpm/%

To illustrate the close tolerances to which product quality is generallyspecified, the analyzer range will be chosen as 4.5 to 5.5 percent, with anormal set point of 5.0 percent The span is 1.0 percent:

TABLE 3.3 Properties of Common Loops

P r o p e r t y

Flow andliquidpressure

Gaspressure

Dead time No N o

Capacity Multiple Single

Period l-10 set Zero

Linearity Square Linear

Valve Linear Eq percent

* Applies to liquid pressure

L i q u i dlevel

No

Singlel-10 setLinearIntegrating

A l w a y s5-50 70 10-100 %

S e l d o m Yes

N o EssentialLinear Eq percent

T e m p e r a tureand vaporpressureVariable3-6Min - h r sNonlinearl-2None

C o m position

-C o n s t a n tl-100Min - hrsEitherlo-1,000Often _- loo-l,OOO%Essential

If possibleLinear

Analysis of Some Common Loops I 87

at the same time demonstrating how to identify the significant elements

in a loop Rarely will a flow or level loop need analysis, but when position-control problems arise, this procedure can be of inestimablevalue

com-Much of what has been derived and weighed and discussed in the going pages is summarized in Table 3.3

fore-Nothing that has not been already covered is presented in the table,yet gathering all this information together discloses some interestingfeatures Notice, for example, the similarity between level and flowloops, with respect to both natural period and the presence of noise.Without any doubt, however, each of the five groups above is separateand distinct from the rest

REFERENCES

1 Bradner, M.: Pneumatic Transmission Lag, ISA Paper No 48-4-2

2 Catheron, A R.: Factors in Precise Control of Liquid Flow, ISA Paper No 50-B-2

3 Considine, D M.: “Process Instruments and Controls Handbook,” chap 2,McGraw-Hill Book Company, New York, 1957

4 Esterson, G L., and R E Hamilton: Dynamic Response of a Continuous StirredTank, presented at the Joint Automatic Control Conf., Palo Alto, Calif., June, 1964

P R O B L E M S

3.1 -1 volume booster installed at the inlet to the valve motor of Example 3.2reduces its time constant to 0.5 sec Predict the period of oscillation that Iv-illresult from the change, allowin g- 45” phase lag in the proportional-plus-resetcontroller Calculate the proportional band and reset time for jb-amplitudedamping

3.2 What would be the estimate of the natural period and proportional band

in Prob 3.1 if the dynamic elements were all assumed to be dead time rather thancapacities? Is this a valid approximation? Why?

3.3 Let pressure downst’ream of the valve in Example 3.2 be controlledinstead of flow -kt no flow, there is a static head of 5 l)sig, while 10 gpm willraise the pressure to 13 psig; the range of the pressure transmitter is 0 to 25 psip.Estimate what the proportional band of the controller will be for ;/,-amplitudedamping with the period and reset time used in the example

3.4 A mercury manometer capable of reading f 15-in differential pressure

is used to indicate the flow in a gas stream What is its natural period? HO Wwould it affect the control of flow?

3.5 TO verify the choice of an equal-percentage valve for Example 3.5, culate the process gain and the product of process and valve gain for heat loads

cal-of 5000, 10000, and 15000 Btu/min; assume that the difference between trolled reactor temperature and average coolant temperature varies linearl)with heat transfer rate

con-88 1 Ud n erstanding Feedback Control

3.6 Two fluids are blended in a pipeline 20 ft upstream of where the mixture

is sampled The pipe contains 0.4 gal/ft of length, and the flow rate of the blendvaries from 10 to 80 gpm Dead time in the sample line to the analyzer is 15 sec

A circulating pump is installed to maintain 100 gpm flow through that 20-ftsection of pipe without affecting the throughput Compare the natural periodfor integral control with and without the pump in operation What else doesthe pump provide?

3.7 In the same process, the flow of additive is manipulated through a linearvalve whose maximum flow is 1.2 gpm The range of the analyzer is 0 to 1percent, additive concentration Estimate the proportional band required for

at least ~-amplitude damping if the reset time is set for 60’ phase lag with thepump operating

P A R T &

C H A P T E R 4

No w that the characteristics of typical processes have been

thor-oughly presented, it is possible to look more closely into various means

for controlling them The range of process difficult’y has been seen t’o

vary from zero to several hundred, as measured by the proportional band

needed for damping The very existence of such-a range of control

prob-lems suggests the possibility of a variety of means for their control The

first distinction to be made is between linear and nonlinear control

methods

A linear device is one whose output is directly proportional to its

input(s) and any dynamic function thereof This definition includes not

only proportional !:ontrollers, but those with reset, derivative, lag, dead

tinrePin short, any time function of a linear variable To be sure, a

device is only linear over a specified range A pneumatic controller,

for example, ceases to operate linearly when its output falls to zero or

reaches full supply pressure All linear devices are similarly limited,

and their proper use demands an appreciation of these limitations

9 1

92 1 Selecting the Feedback Controller

In the earlier chapters certain nonlinear characteristics were dealtwith, both in processes and in the measuring devices and valves Anattempt was made in every case to compensate for process nonlinearities

so as to obtain constant loop gain This assures uniformity of ance under all conditions of operation In general, compensation iseffected external to the controller, leaving the controller as a lineardevice

perform-But within the domain of linear controllers, a variety of dynamicelements exists Each dynamic element, such as reset or derivative, hascertain undesirable properties along with those which are beneficial Athorough understanding of the assets and liabilities of each control mode

is prerequisite to their intelligent selection

PERFORMANCE CRITERIA

If selection between various control configurations is to be made, somebasis must be established for their comparison For example, a givenprocess may be controlled in a number of ways One way will be betterthan the others from the standpoint of performance, i.e., how it responds

to a set-point or load change The three load-response curves in Fig 1.13show the performance of three different controllers in the same process.The shape of the load-response curve depends to a considerable degree

on the type of control action used and the settings of the parametersinvolved Furthermore, the penalty ascribed to a typical response curve

is determined by the specifications of the process Several means ofweighing the response curve suggest themselves:

1 Integrated error: Since the error (T - c) can be either positive ornegative, an integrated error of zero could be obtained in a continuouslyoscillating loop Integrated error is therefore not, of itself, a measure

4 Integrated square error (ISE): The instantaneous error is firstsquared and then summed (integrated) Squaring prevents a negativeerror from canceling a positive one (as does absolute value) and alsoweighs large errors more heavily than small ones

5 Root mean square (rms) error: This index is the standard deviation

of t,he error If the error reduces to zero with time, so does the rms

of error magnitude and IAE.

For the case where the response curve lies wholly on one side of zeroerror, the integrated error equals the IAE But this is not the limit ofusefulness of the integrated error, for it represents the average errorthat has existed over a particular time span The average error orintegrated error is a valid basis for comparing response curves with equaldamping, like those comparisons shown in Fig 1.13 By specifying thedamping, the objection raised in number 1 above is overruled Inte-grated error will therefore be used as a performance index throughoutthe balance of the book, and in every case >i-amplitude damping will

be meant, unless otherwise indicated

Sensitivity of a Process to Disturbances

The choice of integrated error as a performance index has a very tical aspect, in that it can be readily calculated from controller settings

prac-In a proportional-plus-reset controller,

Prior to a load change, at time t 1, the output will be stationary at a levelVL~, and the error will be zero After the transient from a load change hassubsided, i.e., at time tz, the output will come to rest at a new level ~122,

at which the error will again be zero Then, subtract,ing the two outputs,

Reducing the last expression yields

(4.1)

Let the integrated error resulting from the load change AH be nated B:

desig-I!: = I;f’e dt

94 1 Selecting the Feedback Controller

Then the load respouse of a giver1 control loop cau be assessed on thebasis of integrated error per unit load change:

E’ PR

-

=-AT?% 100

Again this is integrated error and not ME; the damping of the loop must

be assured before this ndex can be used

The load-response criteriou E/A/H depends ou the proportioual baudand reset t,irne which, in t,urn, depend on the characteristics of the plant.This is another way of illustrutiug the difficulty of coutrol which wasdescribed in Chap 1 If the proportional baud can be made to approachzero because of he ease with which the process can be contcolled, or thereset t,irue because of its speed of response, E/AM will approach zero.The integrated error will be found useful iu evaluating not ouly thedifficulty of a process, but also the effectiveness of the means used inits control

Error Magnitude

The magnitude of an error is a fuuctiou of how fast the load changetakes place If the load change is very gradual, several orders of nmg-nitude longer in duration thau the reset time, it may produce no nieasur-able error magnitude; the ntcgmted error, however, does not depend

on rate of change of load An instantaneous load chauge will be

coun-tered by proportional control action, aud derivative, if used If theproportional-plus-reset control equation is written in the differeutial forni,

dm

A plot of e versus t,he rate of change of load dw/dt can be constructedfrom it (see Fig 4.1) The maximum value of e is limited by proportionalaction to P A~,/100. Derivative nctiou can reduce the effect of a rapidload change principally by allowing a reduction in the proportioual baudsetting

PAm

FIG 4.1 The magnitude of the error

is a function of the rate of change of load as well as its magnitude.

Linear Controllers 95 I

TWO- AND THREE-MODE CONTROLLERS

Although the primary functions of proportional, derivative, and resethave already been introduced, many of their features remain to be defined.The discussion will be restricted to the commonly available controllers,i.e., proportional-plus-derivative, proportional-plus-reset, and propor-tional-plus-reset-plus-derivative (Reset controllers are rarely used inprocess work and are not available as standard items from most manu-facturers Derivative by itself is not recognized as a controlling mode.)

Limitations of Derivative

It has been pointed out that perfect derivative action is not available

in conventional controllers Derivative gain is limited to about 10, andthe maximum available phase lead is in the vicinity of 45” In effect,derivative is therefore accompanied by a lag, whose time constant is

><c the value of the derivative time

In most controllers, derivative act,s on the output Physically, it is alag introduced in the feedback path around the controller amplifier.Therefore, if he output of the controller is constant, no derivative actionwill take place no matter what the controlled variable may be doing.This situation occurs whenever the controller’s output has reached one

of its limits The output of a pneumatic controller, for example, can go

as low as zero or as high as 20 psi (the supply pressure), although therange of the control valve-hence the proportional band-is 3 to 15 psi

Ko derivative action will take place, then, until the controlled variableapproaches the proportional band as in Fig 4.2 In the discussion ontwo-capacity processes, this property placed a limitation on the width

of the proportional band required for critical damping

In most controllers, derivative action does not distinguish betweenmeasurement and set point The purpose of derivative is to speed theresponse of the closed loop, but the set point lies outside the loop Thecontrolled variable cannot change instantaneously, because of the lagsinherent in the process But it is normal to introduce set-point changesinstantaneously, which derivative action amplifies into gross output

FIG 4.2 Derivative action begins

when the controller comes out of

saturation.

96 1 Selecting the Feedback Controller

Derivative on error or output Derivative on measurement

FIG 4.3 Derivative makes a controller

hgper-sensitive to set-point changes.

fluctuations Consequently, properly adjusted derivative acting uponthe error or the output is hypersensitive to set-point variations Ideally,derivative should act on neither the output nor the error, but on themeasurement alone Figure 4.3 shows the effect of derivative acting onthe set point

Response can be improved markedly by introducing a lag in the point circuit Figure 4.4 indicates the sort of results that can be obtainedwith such an arrangement

set-Set-point changes are often introduced automatically, by the output

of another controller in cascade (see Chap 6) The controller whose setpoint is adjusted in this way cannot tolerate derivative, especially if theprimary (adjusting) controller has it

Reset “Windup”

Whenever a sustained deviation is imposed upon a controller containingreset, its output will eventually be driven off scale This will happenwhenever the loop is opened, as in the case of plant shutdown or transfer

to manual control If the measurement has been held below the setpoint, the controller will be integrating so as to raise it When the loop

is closed again, the measurement will be driven above the set point andthe controller must integrate back down again to the normal output.When a process is shut down by the closing of hand valves, reset actionbegins to force the proportional band of the controller upward to its

FIG 4.4 Introducing a lag in the set-point circuit can eliminate overshoot and reduce settling time.

I

Linear Controllers 97 I

limit, trying to raise the measurement The controller then finds thateven 100 percent output will not bring the deviation to zero, whereas, innormal operation, 50 percent may have been enough Figure 4.5 showsthe proportional band at its limits during shutdown, such that no controlaction can begin again until the set point is crossed If derivative acts

on the controller output, it is powerless to help the situation, because

it is disabled as long as the measurement is outside the proportional band.Notice that the proportional band is actually beyond the set point.Reset forces it up to the full available supply pressure (or voltage), which

is always well in excess of 100 percent output In a pneumatic controller,supply pressure is normally 20 psi During prolonged shutdown, pres-sure in the reset bellows will reach 20 psi If the’error were suddenlyreduced to zero under this condition, controller output would be 20 psi,equal to the pressure in the reset bellows (In the case of a reverse-actingcontroller, or when the error is sustained in the other direction, resetpressure will fall to 0 psi, which is 3 psi below 0 percent output.)

With regard to batch processes which must be started up several times

a day, this problem is serious Particularly demanding is temperaturecontrol of a batch chemical reactor, where overshoot is intolerable T h esituation can be improved to some degree with a controller whose deriva-

t i v e a c t s u p o n t h e i n p u t ’ Many new controllers incorporate thisfeature

A logical solution to the reset-windup problem is to add enough gence to the controller to make it aware of a shutdown condition This

intelli-is done by placing in the controller’s reset circuit a switch energized bythe output Whenever the output exceeds 100 percent, the switch dis-ables the reset circuit, leaving a proportional (or proportional-plus-deriva-tive) controller In the absence of automatic reset action, a bias musttake its place Because this bias equals the output of the controller atzero deviation, it is ordinarily adjusted in relation to the expected processload For this reason it is sometimes called the “preload” setting

If the preload setting is too high, response is similar to that without the

“antiwindup” switch (Fig 4.5) With too little preload, throttlingbegins prematurely and the controller must bring the measurement to the

FIG 4.5 Control action does not begin until after the set point is crossed, therefore overshoot is inevitable.