Process Control Systems Episode 6 pot

The proportional charac-teristic of a continuous nonlinear controller displays variable damping.. If a linear controller were used to regulate a givenlinear process, a certain proportio

Nonlinear Control Elements I 143

FIG 5.18 Maladjustments in the

z

~3/‘-y-y-program parameters are easy to eltoa great t,j t o o l o n g q too high

diagnose.

Time

2 When e = ez, the on-off output’ drops to 0 percent and the time delay

begins The on-off operator remains in control and preload is sustained

3 At the end of the delay period, transfer is made to the

proportional-plus-reset-plus-derivative controller and preload is replaced with

COII-troller out’put, starting reset uct’ion By this time, the error and its

derivative should both be zero, so the controller output will equal the

preload setting Transfer is therefore ‘Lbumpless.”

The dual-mode system gives the best set-point response attainable

Optimal switching, by definition, is unmatched in the unsteady state,

while t,he linear controller provides the regulation necessary in the steady

state But any control system is only as good as the intelligence with

which it is supplied In the event of maladjustments in the three

param-etcrs el, q, and id, the track of the controlled variable will be imperfect

The value of el will vary directly with the difficulty of the process A s

the process difficulty decreases, the controlled variable is less a function

of load, and hence has more tolerance for inaccuracies in the control

parameters But the degree of performance improvement provided by

dual-mode control also varies directly with process difficulty

The dual-mode system needs six adjustments, which fall into t\vo

independent groups Settings of proportional, reset, and derivative only

pertain t’o t,he steady state, while the program settings are in effect

else-where Consequently, adjusting the dual-mode system is no more

diffi-cult than adjusting two separate controllers R u l e s f o r s e t t i n g t h e

program parameters are self-evident :

1 ,\Inladjustment of el causes overshoot or excessive damping

2 Excess t)ime delay turns the controlled variable downward after the

set point is reached

3 An incorrect preload setting introduces a bump after the

t’ime-delay interval

The effect’s of these mnlndjust~mcnts are graphically demonst’rnted in

Fig 5.18

Recall the specifications which were set, forth at the beginning of the

section 011 dual-mode control *\ Insimum speed has been provided by

the on-off controller The programmed switching crit’icnlly damps the

loop as the set point is approached Offset is climinntcd by reset iu the

144 1 Selecting the Feedback Controller

linear controller FinalIy, noise of magnitude less than el will notactuate the on-off operator and therefore will be no more of a problemthan in a linear system Although complicated and costly, dual-modecontrol cannot be matched for performance

NONLINEAR TWO-MODE CONTROLLERS

It has been demonstrat(ed that a loop whose gain varies inversely withamplitude is prone to limit-cycle Any controller with similar charac-teristics can promote limit cycling in an otherwise linear loop On-offcontrollers are in this category So any nonlinear device that is purposelyinserted into a loop for the sake of engendering stability must have theopposite characteristic: gain increasing with amplitude The onlystabilizing nonlinear devices discussed up to this point have this property-it was manifested as a dead zone in the three-state controller and asthe linear mode in the dual-mode system

It is not difficult to visualize a desirable combination of properties for

a general-purpose nonlinear controller In fact, the characteristics lined for a dual-mode system apply: the controller should have high gain

out-to large signals, low gain out-to small signals, and reset action The variation

of gain with error amplitude can be accomplished continuously orpiecemeal

A Continuous Nonlinear Controller

It is possible to create a cont.roller with a continuous nonlinear functionwhose gain increases with amplitude In contrast to the three-statecontroller, its gain in the region of zero error would be greater than zero,with integrating action to avoid offset But its change in gain withamplitude should be less severe than that of a dual-mode system Thus

it would be more tolerant of inaccuracy in the control parameters.The continuous nonlinear controller could be mathematically described

ex-Nonlinear Control Elements I 145

FIG 5 19 The proportional

charac-teristic of a continuous nonlinear

controller displays variable damping.

-o+Deviation

shape of the parabolic sections shown in Fig 5.19 It is not desirablefor fl to equal zero, since this would render the controller essentiallyinsensitive to small signals, and offset would result A value of p in thevicinity of 0.1 would make the minimum gain of the controller 10/P

A characteristic of this sort produces varying degrees of damping inthe closed loop If a linear controller were used to regulate a givenlinear process, a certain proportional gain could be found which wouldproduce uniform oscillations A straight line represent’ing this gain,labeled “zero damping, ” is superimposed on the curve in Fig 5.19 Ifthe proportional gain of t’he linear controller were halved, the closed loopwould exhibit >a-amplitude damping The controller gain representingji-amplitude damping is also indicated

The nonlinear characteristic crosses both these contours of constantdamping Between the intersections are three distinct stability regions

In the region surrounding zero deviation, damping heavier than tude persists, while adjacent to it on both sides are regions of lighterdamping and consequently faster recovery There is st,ill another region

$a-ampli-on each side where damping is less than zero-representing instability.Should a deviation arise large enough to fall into this last area, it will beamplified with each succeeding cycle

To gain a better insight into the response of this nonlinear teristic in a loop with a linear process, the input-output graph of Fig 5.20has been constructed Notice how heavily a small signal is damped.Damped oscillations in a linear loop theoretically go on forever Butwith a nonlinear characteristic of the kind shown, damped oscillationscannot persist beyond one or two cycles On the ot,her hand, a largesignal causes more corrective action than a linear controller, appro-priately damped, could provide A sufficiently larger deviation couldpromote instability, however, so the proportional band of the nonlinearcont>roller must be adjusted for the largest anticipated deviation

charac-As with other nonlinear controllers, set-point response exceeds what isobt’ainable with linear modes This is because set-point changes are

146 1 Selecting the Feedback Controller

FIG 5.20 If the initial deviation is not extreme, it may be damped within one cycle.

m

normally greater and more rapid t’han load disturbances, taking tage of the region of higher gain Load disturbances make their appear-ance as a slow departure of the controlled variable from the set point.Since a linear controller has more gain in the region close about the setpoint, it will generally respond more effectively to small load changes

advan-A comparison of the responses of linear and nonlinear three-mode trollers is shown in Fig 5.21

con-A nonlinear two-mode controller seems generally to outperform a lineartwo-mode controller The nonlinear function provides an extra margin

of stability similar to what can be attained with derivative In caseswhere so much noise is superimposed on the measurement t’hat derivativecannot be used, a nonlinear function can be quit’e valuable

Another feature of the nonlinear controller is its extreme tolerance ofgain changes in the loop Response to upsets of moderate magnitudeappear virtually identical over a proportional band range of 4 : 1 or more.Consequently little care need be given to the set’tings of proportional andreset, save for the possibility of bringing the unstable region too close tothe set point

FIG 5.21 A three-mode nonlinear controller

exhibits better set-point response but poorer

load response than its linear counterpart.

Nonlinear Control Elements I 141

Linear

0’

T i m e , set

60400

T i m e , set

FIG 5.22 The nonlinear two-mode controller

is superior in all respects on a noisy flow loop.

Flow Control

A flow measurement is always accompanied by noise This noise isattenuated somcwhnt by the wide proportional band of the controller andpassed on to the valve If the noise is of any magnitude, the valve may

be stroked suflicicntly to introduce ac+ual changes in flow The nonlinearfunction is an efficient noise filter, in that, it rejects small-amplitudesignals The result is smoother valve motion and a more stable loop.Figure 5.22 shows comparative records for linear and nonlinear control

of a noisy flow loop The nonlinear controller has proven to be quiteeffective on pulsating flows too, where the disturbance is periodic ratherthan random

Level Control

J,evel measurements are often noisy because of splashing and lence In addition, the surface of a liquid tends to resonate hydrau-lically, producing a periodic signal superimposed on the average level.Since the liquid-level process cannot respond fast enough for a change invalve position to dampen these fluctuations, they ought to be disregarded

turbu-by the controller A nonlinear controller does just this, sending a smoothsignal to the valve

It was pointed out in Chap 3 that many tanks with level controls areintended as surge vessels In these applications, tight control is inadmis-sible because it frustrates the purpose of the vessel A wide proportionalband with reset was suggested for control But the nonlinear controller

is, in fact, ideal for this application for two reasons:

1 1Iinor fluctuations in liquid level will not be passed on to the valve,providing smooth delivery of flow

2 lllajor upsets will be met by vigorous corrective action, ensuringthat the upper and lower limits of the vessel will not be violated

This application is often referred to as LLaveraging level control,”because it is desired that the manipulated flow follow the avcragc level in

148 1 Selecting the Feedback Controller

FIG 5.23 The nonlinear two-mode

con-troller prevents minor fluctuations in level

from affecting delivery of flow.

the tank Averaging is really a dynamic process and can be accomplishedwith a suit’able lag But adding a lag would only serve to reduce thespeed of response The nonlinear function, however, provides filt,eringwithout sacrificing speed A typical record of level in a surge vessel andthe corresponding output of its nonlinear controller are presented inFig 5.23

pH Control

The neutralization process has been described as unusually diffkult tocontrol because of the extreme nonlinearitjy of the pH curve Limitcycling (*an he encountered when a linear controller is used, because loopgain varies inversely with deviation This, t,hen, is a natural applicationfor t,he nonlincnr (*ontroller whose gain varies directly with deviation

In fact, any process prone to limit cycbling can benefit by its USC Thenonlinear function in the controller need not be a perfect complement

FIG 5.24 A nonlinear controller can give uniform damping to a pH loop.

Nonlinear Control Elements I 149

a reasonable fit can be made

The input-output graph of Fig 5.24 shows how a constant loop gain

is achieved

A Discontinuous Nonlinear Controller

The nonlinear function shown in Fig 5.19 can be approximated bythree straight lines The center is essentially a dead zone where little

or no control action takes place This function is not difficult to duce into a linear controller; it involves sending the controlled variable

intro-to the set-point input through high and low limits Within the limits,there is no error signal; elsewhere an error is developed as the differenw

b e t w e e n the measurement a n d the neartir limit ITigure 5.25 describesthe arrangement of the instrument and its proportional function Pro-portional, rcsct, high, and low limits are adjustable

This nonlirwnr coiltrollcr is often used in nvernging lcvcl npplic*ations

Its dead zone is also a vnluablr feature in the pH-caontrol system dcwribcd

in Chap 10

P R O B L E M S

5.1 I lincnr proress is found to be undmn~wd unrltr proportional c*ontrol with

a hand of 20 Iwrwrlt I\-hat will hal)lwn if the band is reduced to 10 Iwrccnt ;

to 5 percent?

5.2 I thernml ~~roccss with IO-SW dcstl tinw rind n Gnin lag is to be cooled

with refrigerant sul)l)licd front a solrnoiti val\-c If the ~nl\-c is left on, the ptrnturc falls to O’F; \vhtln it, is off, thr trnllwrnture riws to GOOF bktimatc theIwriod nnrl nliil)litudc of the limit q.c*lc if thcx on-off controllfr wcrc Iwrfcct

tcm-5.3 ‘I’hc o n - o f f cwntrollcr used f o r the, ~~roc’css i n Prob 5.2 wtunlly hns :Ldifkwntial gal) of 2°F lCstinmt(~ the lwriotl and :m~plitudr of the limit ryclr,taking the difftrcntial gal) into account

150 1 Selecting the Feedback Controller

5.4 A lever is driven by a bidirectional constant-speed motor to a positiondetermined by a three-state controller The motor has a speed of 10 percent

of full stroke per second, and an inertial time constant of 1.0 sec Differentialgap in the controller is 2 percent of full stroke How wide does the dead zonehave to be to prevent limit cycling? What would be the period of the cycling?5.5 A batch chemical reactor is to be brought up to operating temperaturewith a dual-mode system Full controller output supplies heat through a hot-water valve, while zero output opens a cold-mater valve fully; at 50 percent out-

p u t , b o t h v a l v e s a r e c l o s e d While full heating is applied, the temperature of thebatch rises at, l”F/min; the time constant of the jacket is estimated at 3 min, andthe total dead time of the system is 2 min The normal load is equivalent to

30 percent, of controller output Estimate the required values for the threeadjustments in the optimal switching program

5.6 h given linear process is undamped with a proportional band setting of

50 percent for a linear controller If a continuous nonlinear controller is usedwith a linearity setting of /3 = 0.2, how narrow can the proportional band beset and still tolerate an error of 20 percent?

FACULTAD DE 1NGENIERIA

U D E G

This chapter deals with situations where a single variable is

manipu-lated to satisfy the specification of a certain combination of controlled

variables In any system with a single manipulated variable, only one

controlled variable is capable of independent specification To put it

in other words, there can be only one independent set point at any given

time This, however, does not exclude the incorporation of several

con-trolled variables, as long as their combination contains but one degree

of freedom

Thus we encounter the cascade control system, where the final element

is manipulated through an intermediate or secondary controlled variable

whose value is dependent on the primary In ratio control systems, a

specification is set on a designated mathematical combination of two or

more measured variables Selective control embodies the logical

assign-ment of the final eleassign-ment to whichever controlled variable (of several)

is in danger of violating its specified limits Finally, adaption is the act

of automatically modifying a controller to satisfy a combination of

func-1 5 3

154 1 Multiple-loop Systems

tions of a controlled variable The common denominator in all thesesituations is the manipulation of a single final element through more thanone control loop

CASCADE CONTROL

The output of one controller may be used to manipulate the set point

of another The two controllers are then said to be cascaded, one uponthe other Each controller will have its own measurement input, butonly the primary controller can have an independent set point andonly the secondary controller has an output to the process The manipu-lated variable, the secondary controller, and its measurement constitute

a closed loop within the primary loop Figure 6.1 shows the configuration.The principal advantages of cascade control are these:

1 Disturbances arising within the secondary loop are corrected by thesecondary controller before they can influence the primary variable

2 Phase lag existing in the secondary part of the process is reducedmeasurably by the secondary loop This improves the speed of response

of the primary loop

3 Gain variations in the secondary part of the process are overcomewithin its own loop

4 The secondary loop permits an exact manipulation of the flow ofmass or energy by the primary controller

Cascade control is of great value where high performance is mandatory

in the face of random disturbances or where the secondary part of theprocess contains an undue amount of phase shift For example, a second-ary loop should be closed around an integrating element whenever prac-ticable, to overcome its inherent 90” lag On the other hand, flow is used

as the secondary variable whenever disturbances in line pressure must

be prevented from affecting the prime variable

Secondary

Controller controller Primary

FIG 6.1 Cascade control resolves the process

into two parts, each within a closed loop.

Improved Control through Multiple Loops

FIG 6.2 The primary controller sees a closed loop as a part of the process.

It must be recognized, however, that cascade control cannot beemployed unless a suitable intermediate variable can be measured.Many processes are so arranged that they cannot be readily broken apart

in this way

Properties of the Inner Loop

The secondary or inner loop confronts the primary controller as a newtype of dynamic element The inner loop can be represented as a singleblock, the diagram of Fig 6.1 being resolved into the simpler configura-tion shown in Fig 6.2

Heretofore the dynamic properties of a closed loop were of little cern The controller was simply adjusted for a damping which satisfiedcertain transient response specifications Moreover there was only oneperiod of oscillation to be considered

con-But each loop has its own natural period and, as may be expected, theperiod of the primary loop is to a great extent determined by that of thesecondary Consequently the gain and phase of the secondary loop,whose natural period will be designated T,~, must be known for any value

of the primary period 701, since the latter is dependent on the former.The dynamic properties of the open secondary loop can be convertedinto its closed-loop characteristics by solving for the response of CP withrespect to 1’2 Refer to the block diagram in Fig 6.3

Let g, and g, be vectors representing gain and phase of the processand the controller, respectively Then

FIG 6.3 The input to the secondary

loop is r2, its output is 0.

156 1 Multiple-loop Systems

The vector gain of the closed secondary loop will be designated g02: it isthe ratio of output c2 to input r2 The vector consists of a scalar gainGo2 and a phase angle c#+,~

c2 EC&P go2 = T, = 1 + g,g, (6.1)The product g,g, is the open-loop vector If the inner loop has beenadjusted for f/4-amplitude damping, its open-loop gain will be 0.5 at theperiod of oscillation But the phase lag at the period of oscillation is180”, which makes the gain vector 0.5, L-180”, or -0.5 T h e closed-loop vector go2 at the natural period is then

This indicates that an infinitesimal change in 1’2 would change c2 enough

so that it would never return to equilibrium, and indeed this is the case

To find the gain and phase characteristics of a loop away from itsnatural period, the vector equation for the inner loop must be solved forvarious values of input period 701 This entails first finding the gain and

‘phase of the open loop, g,g, This vector must then be added to thevector 1.0, LO” to form the denominator of the equation Then theclosed-loop gain is the quotient of the magnitude of the two vectors,and its phase is the difference between their phase angles

example 6.1

A typical example is that of a closed loop cont,aining dead time, an grating capacity, and a proportional controller adjusted for >i-amplitudedamping The natural period is known to he 7,,2 = 4rd2. The open-loopgain is 0.5 at 701 = 70C and varies directly as TV, The open-loop I)hase

inte-is -90” for the integrating element, with an additional -360r&r01 or-907,,JTo1 for the dead time Then

g,g, = 0.5 2, L -90” - 90 rs?J

From this information, closed-loop gain and phase arc plotted in Fig 6.4.The primary loop will contain certain eIements of the process in addi-tion t’o the secondary loop These elements can be expected to con-tribute phase lag of 90” or more Therefore the area of greatest interest

Improved Control through Multiple Loops I 157

’ -180"

1 2 3 4 5 6 7 8 9 10

FIG 6.4 Gain and phase of a typical damped loop.

in the response of the secondary loop will be where its phase lag is less

than 90” Notice that gain and phase of the closed loop go in opposite

directions in this region This was not true of the common open-loop

elements-capacity and dead time Therefore trouble will be

encoun-tered in the primary loop as 701 approaches 702

But the closed-loop characteristics have three very important

advan-tages over the corresponding open-loop characteristics, at relatively high

values of T,~:

1 The gain of the closed loop approaches 1.0, which is not only less

than the gain of the open loop, but is not subject to variation

2 The phase of the closed loop is less than that of the secondary