Linear Feedback Controls. The Essentials

267 Pages·2013·4.652 MB·English

by  Mark A. Haidekker (Auth.)

1
Introduction to Linear Feedback
Controls
Abstract
Automationandcontrolsdatebackthousandsofyears,andlikelybegunwiththedesire
tokeepwaterlevelsforirrigationconstant.Muchlater,theindustrialrevolutionbrought
aneedformethodsandsystemstoregulatemachinery,forexamplethespeedofasteam
engine. Since about two centuries, engineers have found methods to describe control
systemsmathematically,withtheresultthatthesystembehaviorcouldbemoreaccurately
predictedandcontrolsystemsmoreaccuratelydesigned.Feedbackcontrolsarecontrol
systems where a sensor monitors the property of the system to be controlled, such as
motorspeed,pressure,position,voltage,ortemperature.Commontoallfeedbackcontrol
systemsisthecomparisonofthesensorsignaltoareferencesignal,andtheexistenceof
acontrollerthatinfluencesthesystemtominimizethedeviationbetweenthesensorand
referencesignals.Feedbackcontrolsystemsaredesignedtomeetspecificgoals,suchas
keepingatemperatureorspeedconstant,ortoaccuratelyfollowthereferencesignal.Inthis
chapter,someofthefundamentalprinciplesoffeedbackcontrolsystemsareintroduced,
andsomecommontermsdefined.
Feedback control systems have a long history. The first engineered feedback control
systems,inventednearly2500yearsbeforeourtime,wereintendedtokeepfluidlevels
constant.Oneapplicationwereearlywaterclocks.Withconstanthydrostaticpressure,
theflowrateofwatercanbekeptconstant,andthefilltimeofacontainercanbeused
asatimereference.Acombinationofafloaterandavalveservedasthecontrolunitto
regulatethewaterlevelandthusthehydrostaticpressure.
During the early industrial revolution, the wide adoption of steam engines was
helpedwiththeinventionofthegovernor.Asteamenginesuffersfromlongdelays(for
example,whencoalisaddedtothefurnace),makingthemhardtocontrolmanually.
Thegovernorisanautomatedsystemtocontroltherotationalspeedofasteamengine
inanunsupervisedfashion.Floatregulatorswereagainimportant,thistimetokeepthe
waterlevelinaboilerconstant(Figure1.1).
About200yearsago,engineersmadethetransitionfromintuitively-designedcontrol
systems to mathematically-defined systems. Only with the help of mathematics to
formally describe feedback control systems it became possible to accurately predict
the response of a system. This new paradigm also allowed control systems to grow
more complex. The first use of differential equations to describe a feedback control
systemisattributedtoGeorgeBiddellAiry,whoattemptedtocompensateatelescope’s
positionfortherotationoftheearthwiththehelpofafeedbackcontrolsystem.Airy
alsomadethediscoverythatimproperlydesignedcontrolsmayleadtolargeoscillatory
responses,andthusdescribedtheconceptofinstability.Themathematicaltreatment
LinearFeedbackControls.http://dx.doi.org/10.1016/B978-0-12-405875-0.00001-2
©2013ElsevierInc.Allrightsreserved.
2 LinearFeedbackControls
m
a
e
tS Water
lnlet
Floater
e
v
la
V
Furnace
Figure1.1 Schematicofawater-levelregulatorwithafloater.Assteamisproduced,thewater
levelsinks,andsodoesthefloater.Avalveopensasaconsequence,andnewwaterisallowed
toentertheboiler,causingthefloatertoriseandclosethevalve.
of control systems—notably stability theory—was further advanced by James Clerk
Maxwell,whodiscoveredthecharacteristicequationandfoundthatasystemisstable
ifthecharacteristicequationhasonlyrootswithanegativerealpart.
Anotherimportantstepwasthefrequency-domaindescriptionofsystemsbyJoseph
FourierandPierre-SimondeLaplace.Thefrequency-responsedescriptionofsystems
soon became a mainstream method, driven by progress in the telecommunications.
Around1940,HendrikWadeBodeintroduceddoublelogarithmicplotsofthefrequency
response(todayknownasBode-plots)andthenotionofphaseandgainmarginsasa
metricofrelativestability.Concurrently,progressinautomationandcontrolswasalso
driven by military needs, with two examples being artillery aiming and torpedo and
missileguidance.ThePIDcontrollerwasintroducedin1922byNicolasMinorskyto
improvethesteeringofships.
With the advent of the digital computer came another revolutionary change. The
developmentofdigitalfiltertheoryfoundimmediateapplicationincontroltheoryby
allowing to replace mechanical controls or controls built with analog electronics by
digitalsystems.Thealgorithmictreatmentofcontrolproblemsallowedanewlevelof
flexibility,specificallyfornonlinearsystems.Moderncontrols,thatis,boththenumer-
icaltreatmentofcontrolproblemsandthetheoryofnonlinearcontrolsystemscomple-
mentedclassicalcontroltheory,whichwaslimitedtolinear,time-invariantsystems.
In this book, we will attempt to lay the foundations for understanding classical
controltheory.Wewillassumeallsystemstobelinearandtime-invariant,andmake
linearapproximationsfornonlinearsystems.Bothanaloganddigitalcontrolsystems
arecovered.Thebookprogressesfromthetheoreticalfoundationstomoreandmore
practicalaspects.Theremainderofthischapter(Chapter1)introducesthebasiccon-
ceptsandterminologyoffeedbackcontrols,anditincludesafirstexample:two-point
controlsystems.
IntroductiontoLinearFeedbackControls 3
A brief review of linear systems, their differential equations and the treatment in
theLaplacedomainfollowsinChapters2and3.InChapter3,severalelementallow-
ordersystemsareintroduced.Chapter4complementsChapter3fordigitalsystemsby
introducingsampledsignals,finite-differenceequationsandthez-transform.
Asimplefirst-ordersystemisthenintroducedindetailinChapter5,andtheeffect
offeedbackcontrolisextensivelyanalyzedthroughdifferentialequations.Thesame
analysis is repeated in Chapter 6, but this time the entire analysis takes place in
boththeLaplacedomainandfordigitalcontrollersinthez-domain,thusestablishing
therelationshipbetweentime-domainandLaplace-domaintreatment.Asecond-order
exampleintroducedinChapter9allowstomorecloselyexaminethedynamicresponse
ofasystemandhowitcanbeinfluencedwithfeedbackcontrol.
Subsequent chapters can be seen as introducing tools for the design engineer’s
toolbox:theformaldescriptionoflinearsystemswithblockdiagrams(Chapter7),the
treatmentofnonlinearcomponents(Chapter8),stabilityanalysisanddesign(Chapter
10),frequency-domainmethods(Chapter11),andfinallytheverypowerfulrootlocus
designmethod(Chapter12).Aseparatechapter(Chapter13)coversthePIDcontroller,
whichisoneofthemostcommonlyusedcontrolsystems.
Chapter14isentirelydedicatedtoprovidingpracticalexamplesoffeedbackcon-
trols,rangingfromtemperatureandmotorspeedcontroltomorespecializedapplica-
tions, such as oscillators and phase-locked loops. The importance of Chapter 14 lies
in the translation of the theoretical concepts to representative practical applications.
Theseapplicationsallowtodemonstratehowthemathematicalconceptsofthisbook
relatetopracticaldesigngoals.
ThebookconcludeswithanappendixthatcontainsacomprehensivesetofLaplace-
and z-domain correspondences, an introduction to operational amplifiers as control
elements,andanoverviewofkeycommandsforthesimulationsoftwareScilab.Scilab
(www.scilab.org)isfree,open-sourcesoftware,whichanyreadercanfreelydownload
and install. Furthermore, Scilab is very similar to MATLAB, and readers can easily
translatetheirknowledgetoMATLABifneeded.
1.1 What are Feedback Control Systems?
Afeedbackcontrolsystemcontinuouslymonitorsaprocessandinfluencestheprocess
insuchamannerthatoneormoreprocessparameters(theoutputvariables)staywithin
aprescribedrange.Letusillustratethisdefinitionwithasimpleexample.Assumean
incubator for cell culture as sketched in Figure 1.2. Its interior temperature needs
◦
to be kept at 37 C. To heat the interior, an electric heating coil is provided. The
incubator box with the heating coil can be seen as the process for which feedback
control will be necessary, as will become evident soon. For now, let us connect the
heating coil to a rheostat that allows us to control the heat dissipation of the heating
coil. We can now turn up the heat and try to relate a position of the rheostat to the
interiortemperatureoftheincubator.Aftersomeexperimentation,we’lllikelyfinda
◦
position where the interior temperature is approximately 37 C. Unfortunately, after
eachchangeoftherheostat,wehavetowaitsometimefortheincubatortemperature
4 LinearFeedbackControls
(a) (b)
lom Sensor
tatsoehR lioc reta rtnoCetsys lioc reta
Line in eH Line in eH
Chamber Chamber
Figure1.2 Schematicrepresentationofanincubator.Theinteriorofthechamberissupposed
to be kept at a constant temperature. A rheostat can be used to adjust the heater power (a)
and thus influence the temperature inside the chamber. The temperature inside the chamber
equilibrateswhenenergyintroducedbytheheaterbalancestheenergylossestotheenvironment.
However,theenergylosseschangewhentheoutsidetemperaturechanges,orwhenthedoorto
theincubatorisopened.Thismayrequirereadjustmentoftherheostat.Thesystemin(a)isan
open-loopcontrolsystem.Tokeepthetemperatureinsidethechamberwithintightertolerances,
asensorcanbeprovided(b).Bymeasuringtheactualtemperatureandcomparingittoadesired
temperature, adjustment of the rheostat can be automated. The system in (b) is a feedback
controlsystemwithfeedbackfromthesensortotheheater.
to equilibrate, and the adjustment process is quite tedious. Even worse, equilibrium
dependsontwofactors:(1)theheatdissipationoftheheatercoiland(2)theheatlosses
to the environment. Therefore, a change in the room temperature will also change
theincubator’stemperatureunlesswecompensatebyagainadjustingtherheostat.If
somebody opens theincubator’s door,someheat escapes andthetemperature drops.
◦
Onceagain,itwilltakesometimeuntilequilibriumnear37 Cisreached.
Althoughtherheostatallowsustocontrolthetemperature,itisnotfeedbackcontrol.
Feedback control implies that the controlled variable is continuously monitored and
comparedtothedesiredvalue(calledthesetpoint).Fromthedifferencebetweenthe
controlledvariableandthesetpoint,acorrectiveactioncanbecomputedthatdrivesthe
controlledvariablerapidlytowardthevaluepredeterminedwiththesetpoint.Therefore,
afeedbackcontrolsystemrequiresatleastthefollowingcomponents:
• Aprocess.Theprocessisresponsiblefortheoutputvariable.Furthermore,thepro-
cessprovidesameanstoinfluencetheoutputvariable.IntheexampleinFigure1.2,
theprocessisthechambertogetherwiththeheatingcoil.Theoutputvariableisthe
temperatureinsidethechamber,andtheheatingcoilprovidesthemeanstoinfluence
theoutputvariable.
• A sensor. The sensor continuously measures the output variable and converts the
value of the output variable into a signal that can be further processed, such as a
voltage(inelectriccontrolsystems),aposition(inmechanicalsystems),orapressure
(inpneumaticsystems).
• Asetpoint,ormoreprecisely,ameanstoadjustasetpoint.Thesetpointisrelatedto
theoutputvariable,butithasthesameunitsastheoutputofthesensor.Forexample,
IntroductiontoLinearFeedbackControls 5
Control
deviation Corrective
action
Set- Controlled
Controller Process
point variable
Measurement
Sensor
output
Figure1.3 Blockdiagramschematicofaclosed-loopfeedbacksystem.Intheexampleofthe
incubator,theprocesswouldbetheincubatoritselfwiththeheatercoil,thesensorwouldbe
a temperature sensor that provides a voltage that is proportional to the temperature, and the
controllerissomesortofpowerdriverfortheheatingcoilthatprovidesheatingcurrentwhenthe
temperatureinsidetheincubatordropsbelowthesetpoint.Inalmostallcases,themeasurement
output is subtracted from the setpoint to provide the control deviation. This error signal is
thenusedbytheactualcontrollertogeneratethecorrectiveaction.Thegrayshadedrectangle
highlightsthesubtractionoperation.
ifthecontrolledvariableisatemperature,andthesensorprovidesavoltagethatis
proportionaltothetemperature,thenthesetpointwillbeavoltageaswell.
• Acontroller.Thecontrollermeasuresthedeviationofthecontrolledvariablefrom
thesetpointandcreatesacorrectiveaction.Thecorrectiveactioniscoupledtothe
inputoftheprocessandusedtodrivetheoutputvariabletowardthesetpoint.
The act of computing a corrective action and feeding it back into the process is
known as closing the loop and establishes the closed-loop feedback control system.
Aclosed-loopfeedbackcontrolsystemthatfollowsourexampleisshownschematically
inFigure1.3.MostfeedbackcontrolsystemsfollowtheexampleinFigure1.3,anditis
importanttonotethatalmostwithoutexceptionthecontrolactionisdeterminedbythe
controllerfromthedifferencebetweenthesetpointandthemeasuredoutputvariable.
Thisdifferenceisreferredtoascontroldeviation.
NotincludedinFigure1.3aredisturbances.Adisturbanceisanyinfluenceotherthan
thecontrolinputthatcausestheprocesstochangeitsoutputvariable.Intheexample
of the incubator, opening the door constitutes a disturbance (transient heat energy
lossthroughtheopendoor),andachangeoftheroomtemperaturealsoconstitutesa
disturbance,becauseitchangestheheatlossfromtheincubatortotheenvironment.
1.2 Some Terminology
We introduced the concept of closed-loop feedback control in Figure 1.3. We now
need to define some terms. Figure 1.4 illustrates the relationship of these terms to a
closed-loopfeedbacksystem.
6 LinearFeedbackControls
D(s)
ε(s) Controller Process
+ Y(s)
+
H(s) G(s)
R(s) — +
Feedforward Path
Feedback Sensor
Path
+
G (s)
FB
+
N(s)
Figure 1.4 General block diagram for a feedback control system. The controlled variable is
Y(s), and the setpoint or reference variable is R(s). A disturbance D(s) can be modeled as
anadditiveinputtotheprocess,andsensornoise N(s)isoftenmodeledasanadditiveinput
tothesensor.NotethatwehavenonchalantlyusedLaplace-domainfunctions,H(s),G(s),and
GFB(s)aswellassignalsR(s),D(s),N(s),andY(s)inplaceoftheirrespectivetime-dependent
functions.Laplace-domaintreatmentofdifferentialequationsiscoveredinChapter3,andthe
majorityoftheexamplesinthisbookaretreatedintheLaplacedomain.
• Process:Alsoreferredtoasplant—theprocessisasystemthathasthecontrolled
variable as its property. The process has some means to influence the controlled
variable.Therefore,theprocesscanbeinterpretedasalinearsystemwithoneoutput
andone(ormore)inputs.
• Sensor:Thesensorisanapparatustomeasurethecontrolledvariableandmakethe
measurement result available to the controller. The sensor itself may have its own
transferfunction,suchasagainordelayfunction.
• Controller:Thecontrollerisadevicethatevaluatesthecontroldeviationandcom-
putesanappropriatecontrolaction.Inmanycases,thecontrolleristheonlypartof
the feedback control system that can be freely designed by the design engineer to
meetthedesigngoalsoftheclosed-loopsystem.
• Disturbance: Any influence other than the control action that influences the con-
trolledvariable.Examplesarevariableloadconditionsinmechanicalsystems,elec-
tromagneticinterference,shearwindsforairplanesetc.Adisturbanced(t)canoften
bemodeledasanadditiveinputtotheprocess.
• Noise:Noiseisabroadbandrandomsignalthatmaybeintroducedinmanyplaces
of a system. Amplifiers typically create noise; dirt or rust in a mechanical system
wouldalsocreatenoise-likerandominfluences.Itisoftenpossibletomodelnoise
asasinglesignaln(t)asanadditiveinputtothesensor.
• Setpoint:Alsoreferredtoasreferencesignal.Thissignaldeterminestheoperating
point of the system, and, under consideration of the sensor output signal, directly
influencesthecontrolledvariable.
• Control deviation: The control deviation (cid:2)(t) is the time-dependent difference
between setpoint and sensor output. The control deviation is often also referred
toastheerrorvariableorerrorsignal.
IntroductiontoLinearFeedbackControls 7
• Controlaction:Alsotermedcorrectiveaction.Thisistheoutputsignalofthecon-
trollerandservestoactuatetheprocessandthereforetomovethecontrolledvariable
towardthedesiredvalue.
• Controlledvariable:Thisistheoutputoftheprocess.Thedesignengineerspecifies
thecontrolledvariableintheinitialstagesofthedesign.
• Dependentvariable:Anysignalthatisinfluencedbyothersignalsanywhereinthe
entiresystem(suchasthecontroldeviation)isadependentvariable.
• Independentvariable:Anyexternalsignalthatisnotinfluencedbyothersignalsis
anindependentvariable.Examplesarethesetpointr(t)andthedisturbanced(t).
• Feedforwardpath:Thefeedforwardpathincludesallelementsthatdirectlyinflu-
encethecontrolledvariable.
• Feedbackpath:Thefeedbackpathcarriestheinformationaboutthecontrolledvari-
ablebacktothecontroller.Notethatthedefinitionofthefeedforwardandfeedback
pathsisnotrigidandmerelyservesasorientation.
Whenthefeedbackpathisnotconnectedtothefeedforwardpathatthefirstsumma-
tionpoint(andthus(cid:2)(t)=r(t)),thesystemiscalledopen-loopsystem.Theopen-loop
systemisuseful—oftennecessary—tocharacterizetheindividualcomponentsandto
predicttheclosed-loopresponse.Chapters11and12describemethodshowtheclosed-
loopresponsecanbededucedfromtheopen-loopsystem.Oncethefeedbackpathis
connected to the feedforward path, the system becomes a closed-loop system, and
feedbackcontroltakesplace.
1.3 Design of Feedback Control Systems
Feedbackcontrolsystemsmustbedesignedtosuitapredeterminedpurpose.Normally,
onlythecontrollercanbeappropriatelydesigned,whereastheprocessandthesensor
arepredeterminedorconstrained.Feedbackcontrolsystemscanbedesignedtoachieve
specificbehavioroftheoutputvariable,forexample
• Tokeeptheoutputvariablewithinatightlyconstrainedrange,irrespectiveofchanges
intheenvironment.Theincubatorintroducedaboveisagoodexample.
• Torapidlyandaccuratelyfollowachangeinthereferencesignal.Agoodexample
isadiskdrivehead,whereasignalrequestspositioningoftheheadoveraselected
track.Thedriveheadissupposedtoreachthetargettrackasquicklyaspossibleand
thenaccuratelyfollowthetrackforaread/writeoperation.
• Tosuppressasharp,transientdisturbance.Oneexampleisanactivecarsuspension.
Whenthecarrollsoverapothole,thesuspensioncontrolminimizesthemovement
ofthecabin.
• Toreducetheinfluenceofprocesschanges.Theprocessitselfmaysufferfromvari-
ations(forexample,thermalexpansion,accumulationofdirtanddust,mechanical
degradation, changes in fluid levels or pressure, etc.). A feedback control system
cancompensatefortheseinfluencesandkeepthebehaviorofthecontrolledsystem
withinspecifications.
8 LinearFeedbackControls
• Tolinearizenonlinearsystems.Hi-fiaudioequipmentmakesextensiveuseoffeed-
backcontroltoproducehigh-qualityaudiosignalswithlowdistortioninspiteofits
components(transistors),whichhavehighlynonlinearcharacteristics.
Theaboveexamplesillustratesomepossibledesigngoals.Often,thedesigngoals
conflict, and a compromise needs to be found. Rational controller design consists of
severalsteps:
1. Definitionofthecontrolledvariable(s)andcontrolgoals.
2. Characterizationoftheprocess.Theresponseoftheprocesstoanychangeofthe
inputmustbeknown.Ideally,theprocesscanbedescribedbyanequation(usually,
adifferentialequation).Iftheprocessequationisnotknown,itcanbededucedby
measuringtheresponsetospecificinputsignals,suchasastepchangeoftheinput.
3. Characterizationofanynonlinearresponseoftheprocess.Ifaprocessisnotlinear,
afeedbackcontrolsystemcannotbedesignedwiththemethodsofclassicallinear
feedbackcontroltheory.However,whenaprocessexhibitsnonlinearcharacteristics,
itcanoftenbeapproximatedbyalinearsystemneartheoperatingpoint.
4. Design of the sensor. The sensor needs to provide a signal that can be subtracted
fromthesetpointsignalandusedbythecontroller.
5. Designofthecontroller.Thisstepallowthegreatestflexibilityforthedesignengi-
neer.Keydesigncriteriaare:
• Thestability,i.e.,whetherasetpointchangeoradisturbancecancausetheclosed-
loopsystemtorunoutofcontrol.
• The steady-state response of the closed-loop system, i.e., how close the output
variable follows the setpoint, provided that the closed-loop system had enough
timetoequilibrate.
• Thesteady-statedisturbancerejection,i.e.,howwelltheinfluenceofadisturbance
(intheincubatorexample,thedisturbancecouldbeachangeoftheroomtemper-
ature)issuppressedtohaveminimallong-termimpactontheoutputvariable.
• Thedynamicresponse,i.e.,howfasttheclosed-loopsystemrespondstoachange
ofthesetpointandhowfastitrecoversfromadisturbance.
• The integrated tracking error, i.e., how well the system follows a time-varying
setpoint.
6. Testoftheclosed-loopsystem.Sincethemathematicaldescriptionusedtocharac-
terizetheprocessanddesignthecontrollerisoftenanapproximation,theclosed-
loopsystemneedstobetestedunderconditionsthatcanactuallyoccurinpractice.
In many cases, simulation tools can help inthe design and initial testingprocess.
However,evensophisticatedsimulationtoolsarebasedonassumptionsandapprox-
imationsthatmaynotaccuratelyreflecttherealsystem.Moreover,roundingerrors
orevennumericalinstabilities(cf.chaoticsystem,turbulence)maycausesimulation
resultstodeviatesignificantlyfromtheactualsystem.
Figure1.5providessomeinsightastowhatthedesignofafeedbackcontrolsystem
caninvolve.Robots,suchasthepicturedbottlingrobotthatfeedsafillingmachine,need
tobefastandaccurate.Feedbackcontrolsystemsensurethattherobotsegmentsmove
betweentime-varyingsetpoints,suchasthepickuppointandtheconveyorbeltwhere
IntroductiontoLinearFeedbackControls 9
Figure1.5 Examplefortheapplicationoffeedbackcontrolsystemsinrobotics.Therobotinthis
picturefeedsemptybottlesontotheconveyorbeltofafillingmachine.Therobotmovesrapidly
betweenseveralsetpointstopickupandthendepositthebottles.Toachievehighthroughput,
therobotneedstoreachthesetpointsasfastaspossible.However,adeviationfromitspath,
forexamplewhentherobotovershootsitssetpoint,maybreaksomebottlesorevendamagethe
conveyor.
the bottles are deposited. It is desirable that the robot reaches its setpoints as fast as
possibletoensurehighthroughput.However,therobotcannotbeallowedtoovershoot
itstarget,eventhoughadefinedovershootmayactuallyspeeduptheprocess.Inthefirst
example(Figure1.3),weonlyconsideredthesteady-statebehavior,thatis,theincubator
temperatureatequilibrium.Therobotexampleshowsusthatthedynamicbehavior(also
called the transient response) can be equally important in the design considerations.
Simplified,wecancharacterizethecoreoffeedbackcontroldesignasfollows:
Feedbackcontroldesignallowstoinfluenceaprocesswithanundesirabletrans-
ferfunctionbymeansofacontrollersuchthatthecombined(i.e.,controlledor
closed-loop)systemhasadesirabletransferfunction.
1.4 Two-Point Control
Two-pointcontrolisanonlinearfeedbackcontrolmethodthatisbrieflycoveredhere
becauseofitsubiquity.Roomthermostats,ovens,refrigerators,andmanyotherevery-
dayitemscontaintwo-pointcontrolsystems.Two-pointcontrolimpliesthatacorrective
actioniseitherturnedonoroff.Atypicalexampleisaroomthermostatthatcontrols
aheaterfurnace.Itismorecost-efficienttodesignafurnacethatiseitheronoroff,as
comparedtoafurnacewithcontrolledheatingpower(suchasvariableflameheight).
In this example, the closed-loop system with two-point control has therefore a tem-
perature sensor and a two-point controller, i.e., a controller that turns the corrective
action—thefurnace—eitheronoroff.
10 LinearFeedbackControls
(a) (b)
On
x
x - Δ x x +Δ x
0 0
On Off On Off On
Off
Figure1.6 Controlcurveofatwo-pointcontroller(a)andapossibletimecourseofthecontrolled
variable(b).Thecontrollerexhibitshysteresis,thatis,itturnsthecorrectiveaction(heating)on
iftheinputvariable x ismorethan(cid:3)x belowthesetpoint x0 (hereplacedintheorigin)and
keepsitonuntilxexceedsx0bytheoffset(cid:3)x,whereuponitturnsthecorrectiveactionoff.The
correctiveactionstaysoffuntiltheinputvariableagaindropsbelowx0−(cid:3)x.Thisswitching
behaviorisindicatedbythearrowsin(a)thatmarkthedirectionofthepath.Ifsuchacontrolleris
usedtocontrolaprocess,forexamplearoomwithaheater,thetemperature(controlledvariable)
fluctuatesaroundthesetpointwithanamplitudeof2(cid:3)x.
Atwo-pointcontrolsystemneedstobedesignedwithdifferenton-andoff-points
to prevent undefined rapid switching at the setpoint. The difference in trip points,
depending on whether the trip point is approached from above or below, is referred
toashysteresis.Figure1.6ashowsthecharacteristiccurveofatwo-pointswitch.The
value of the input variable where the switch turns off is lower than the value where
it turns on. The center of the hysteresis curve is adjustable and serves as setpoint.
Whenusedintheexampleofafurnace,thecontrolledvariable,i.e.,thetemperature,
fluctuatesbetweenthetwotrippointsofthetwo-pointcontrol.Apossibletimecourse
ofthetemperaturedeviationfromthesetpointisshowninFigure1.6b.
In two-point control systems, the width of the hysteresis determines the balance
between setpoint accuracy and process concerns. The narrower the hysteresis curve,
themoreoftentheprocesswillswitch,causingstart-upstresstotheprocess.Ifthestart
phaseoftheprocessisassociatedwithcost(stress,wear,higherenergyconsumption),
thecostneedstobalancedwithaccuracyneeds.Tostriketheoptimumbalanceisthe
dutyofthedesignengineer.
Extensionsofthetwo-pointprinciplearestraightforward.Forexample,thedirection
ofaheatpumpcanbereversedwithanelectricalcontrolsignal.Withthisadditional
signal, the heat pump can therefore be set to alternatively heat or cool. To use this
feature, the controller needs to be extended into a three- or four-point switch with
differenthysteresiscurvesforheatingandcooling(Figure1.7).
Two-pointcontrolsystemsobeythesamemathematicalmodelsthatapplyforlinear
systems,buttheon-phaseandtheoff-phaseoftheprocessneedtobehandledseparately.
◦
Letusexaminetheexampleofaroomthermostatwitha1 hysteresis.Weassumethat
theroomtemperatureincreaseslinearlywhentheheaterison,anddecreaseslinearly
whentheheaterisoff,albeitwithdifferentrates:
(cid:2)
k ·t whenheaterison
(cid:3)T(t)= 1 (1.1)
−k ·t whenheaterisoff
2