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ThePIDcontrollercalculationalgorithminvolvesthreeseparateparameterstheProportionaltheIntegralandDerivativevaluesTheProportionalvaluedeterminesthereactiontothecurrenterrortheIntegraldeterminesthereactionbasedonthesumofrecenterrorsandtheDerivativedeterminesthereactiontotherateatwhichtheerrorhasbeenchangingTheweightedsumofthesethreeactionsisusedtoadjusttheprocessviaacontrolelementsuchasthepositionofacontrolvalveorthepowersupplyofaheatingelementBy"
tuning"
thethreeconstantsinthePIDcontrolleralgorithmthePIDcanprovidecontrolactiondesignedforspecificprocessrequirementsTheresponseofthecontrollercanbedescribedintermsoftheresponsivenessofthecontrollertoanerrorthedegreetowhichthecontrollerovershootsthesetpointandthedegreeofsystemoscillationNotethattheuseofthePIDalgorithmforcontroldoesnotguaranteeoptimalcontrolofthesystemorsystem
stability
SomeapplicationsmayrequireusingonlyoneortwomodestoprovidetheappropriatesystemcontrolThisisachievedbysettingthegainofundesiredcontroloutputstozeroAPIDcontrollerwillbecalledaPIPDPorIcontrollerintheabsenceoftherespectivecontrolactionsPIcontrollersareparticularlycommonsincederivativeactionisverysensitivetomeasurementnoiseandtheabsenceofanintegralvaluemaypreventthesystemfromreachingitstargetvalueduetothecontrolaction
AblockdiagramofaPIDcontroller
NoteDuetothediversityofthefieldofcontroltheoryandapplicationmanynamingconventionsfortherelevantvariablesareincommonuse
1Controlloopbasics
AfamiliarexampleofacontrolloopistheactiontakentokeeponesshowerwaterattheidealtemperaturewhichtypicallyinvolvesthemixingoftwoprocessstreamscoldandhotwaterThepersonfeelsthewatertoestimateitstemperatureBasedonthismeasurementtheyperformacontrolactionusethecoldwatertaptoadjusttheprocessThepersonwouldrepeatthisinput-outputcontrolloopadjustingthehotwaterflowuntiltheprocesstemperaturestabilizedatthedesiredvalue
FeelingthewatertemperatureistakingameasurementoftheprocessvalueorprocessvariablePVThedesiredtemperatureiscalledthe
setpointSPTheoutputfromthecontrollerandinputtotheprocessthetappositioniscalledthemanipulatedvariableMVThedifferencebetweenthemeasurementandthesetpointistheerroretoohotortoocoldandbyhowmuchAsacontrolleronedecidesroughlyhowmuchtochangethetappositionMVafteronedeterminesthetemperaturePVandthereforetheerrorThisfirstestimateistheequivalentoftheproportionalactionofaPIDcontrollerTheintegralactionofaPIDcontrollercanbethoughtofasgraduallyadjustingthetemperaturewhenitisalmostrightDerivativeactioncanbethoughtofasnoticingthewatertemperatureisgettinghotterorcolderandhowfastandtakingthatintoaccountwhendecidinghowtoadjustthetapMakingachangethatistoolargewhentheerrorissmallisequivalenttoahighgaincontrollerandwillleadtoovershootIfthecontrollerweretorepeatedlymakechangesthatweretoolargeandrepeatedlyovershootthetargetthiscontrolloopwouldbetermedunstableandtheoutputwouldoscillatearoundthesetpointineitheraconstantgrowingordecayingsinusoidAhumanwouldnotdothisbecauseweareadaptivecontrollerslearningfromtheprocesshistorybutPIDcontrollersdonothavetheabilitytolearnandmustbesetupcorrectlySelectingthecorrectgainsforeffectivecontrolisknownastuningthecontroller
IfacontrollerstartsfromastablestateatzeroerrorPVSPthenfurtherchangesbythecontrollerwillbeinresponsetochangesin
othermeasuredorunmeasuredinputstotheprocessthatimpactontheprocessandhenceonthePVVariablesthatimpactontheprocessotherthantheMVareknownasdisturbancesandgenerallycontrollersareusedtorejectdisturbancesandorimplementsetpointchangesChangesinfeedwatertemperatureconstituteadisturbancetotheshowerprocess
IntheoryacontrollercanbeusedtocontrolanyprocesswhichhasameasurableoutputPVaknownidealvalueforthatoutputSPandaninputtotheprocessMVthatwillaffecttherelevantPVControllersareusedinindustrytoregulatetemperaturepressureflowratechemicalcompositionspeedandpracticallyeveryothervariableforwhichameasurementexistsAutomobilecruisecontrolisanexampleofaprocesswhichutilizesautomatedcontrol
DuetotheirlonghistorysimplicitywellgroundedtheoryandsimplesetupandmaintenancerequirementsPIDcontrollersarethecontrollersofchoiceformanyoftheseapplications
2PIDcontrollertheory
NoteThissectiondescribestheidealparallelornon-interactingformofthePIDcontrollerForotherformspleaseseetheSection"
AlternativenotationandPIDforms"
ThePIDcontrolschemeisnamedafteritsthreecorrectingtermswhosesumconstitutesthemanipulatedvariableMVHence
wherePoutIoutandDoutarethecontributionstotheoutputfromthe
PIDcontrollerfromeachofthethreetermsasdefinedbelow
21Proportionalterm
TheproportionaltermmakesachangetotheoutputthatisproportionaltothecurrenterrorvalueTheproportionalresponsecanbeadjustedbymultiplyingtheerrorbyaconstantKpcalledtheproportionalgain
Theproportionaltermisgivenby
Where
PoutProportionaloutput
KpProportionalGainatuningparameter
eErrorSPPV
tTimeorinstantaneoustimethepresent
ChangeofresponseforvaryingKpAhighproportionalgainresultsinalargechangeintheoutputforagivenchangeintheerrorIftheproportionalgainistoohighthesystemcanbecomeunstableSeethesectiononLoopTuningIncontrastasmallgainresultsinasmalloutputresponsetoalargeinputerrorandalessresponsiveorsensitivecontrollerIftheproportionalgainistoolowthecontrolactionmaybetoosmallwhenrespondingtosystemdisturbances
IntheabsenceofdisturbancespureproportionalcontrolwillnotsettleatitstargetvaluebutwillretainasteadystateerrorthatisafunctionoftheproportionalgainandtheprocessgainDespitethe
steady-stateoffsetbothtuningtheoryandindustrialpracticeindicatethatitistheproportionaltermthatshouldcontributethebulkoftheoutputchange
comgralterm
ThecontributionfromtheintegraltermisproportionaltoboththemagnitudeoftheerrorandthedurationoftheerrorSummingtheinstantaneouserrorovertimeintegratingtheerrorgivestheaccumulatedoffsetthatshouldhavebeencorrectedpreviouslyTheaccumulatederroristhenmultipliedbytheintegralgainandaddedtothecontrolleroutputThemagnitudeofthecontributionoftheintegraltermtotheoverallcontrolactionisdeterminedbytheintegralgainKi
Theintegraltermisgivenby
ChangeofresponseforvaryingKiWhere
IoutIntegraloutput
KiIntegralGainatuningparameter
τTimeinthepastcontributingtotheintegralresponse
Theintegraltermwhenaddedtotheproportionaltermacceleratesthemovementoftheprocesstowardssetpointandeliminatestheresidualsteady-stateerrorthatoccurswithaproportionalonlycontrollerHoweversincetheintegraltermisrespondingtoaccumulatederrorsfromthepastitcancausethepresentvaluetoovershootthesetpointvalue
crossoverthesetpointandthencreateadeviationintheotherdirectionForfurthernotesregardingintegralgaintuningandcontrollerstabilityseethesectiononlooptuning
23Derivativeterm
TherateofchangeoftheprocesserroriscalculatedbydeterminingtheslopeoftheerrorovertimeieitsfirstderivativewithrespecttotimeandmultiplyingthisrateofchangebythederivativegainKdThemagnitudeofthecontributionofthederivativetermtotheoverallcontrolactionistermedthederivativegainKd
Thederivativetermisgivenby
ChangeofresponseforvaryingKdWhere
DoutDerivativeoutput
KdDerivativeGainatuningparameter
ThederivativetermslowstherateofchangeofthecontrolleroutputandthiseffectismostnoticeableclosetothecontrollersetpointHencederivativecontrolisusedtoreducethemagnitudeoftheovershootproducedbytheintegralcomponentandimprovethecombinedcontroller-processstabilityHoweverdifferentiationofasignalamplifiesnoiseandthusthisterminthecontrollerishighlysensitivetonoiseintheerrortermandcancauseaprocesstobecomeunstableif
thenoiseandthederivativegainaresufficientlylarge
24Summary
TheoutputfromthethreetermstheproportionaltheintegralandthederivativetermsaresummedtocalculatetheoutputofthePIDcontrollerDefiningutasthecontrolleroutputthefinalformofthePIDalgorithmis
andthetuningparametersare
KpProportionalGain-LargerKptypicallymeansfasterresponsesincethe
largertheerrorthelargertheProportionaltermcompensationAnexcessivelylargeproportionalgainwillleadtoprocessinstabilityandoscillation
KiIntegralGain-LargerKiimpliessteadystateerrorsareeliminatedquickerThetrade-offislargerovershootanynegativeerrorintegratedduringtransientresponse