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UniversityCollegeofSoutheastNorway

http://home.hit.no/~hansha

ModelPredictiveControlinLabVIEWHans-PetterHalvorsen,2016.10.31

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PrefaceModelPredictiveControl,orMPC,isanadvancedmethodofprocesscontrolthathasbeeninuseintheprocessindustriessuchaschemicalplantsandoilrefineriessincethe1980s.Modelpredictivecontrollersrelyondynamicmodelsoftheprocess,mostoftenlinearempiricalmodelsobtainedbysystemidentification.

Modelpredictivecontrol(MPC)referstoaclassofcomputercontrolalgorithmsthatutilizeanexplicitprocessmodeltopredictthefutureresponseofaplant.AteachcontrolintervalanMPCalgorithmattemptstooptimizefutureplantbehaviorbycomputingasequenceoffuturemanipulatedvariableadjustments.Thefirstinputintheoptimalsequenceisthensentintotheplant,andtheentirecalculationisrepeatedatsubsequentcontrolintervals.Originallydevelopedtomeetthespecializedcontrolneedsofpowerplantsandpetroleumrefineries,MPCtechnologycannowbefoundinawidevarietyofapplicationareasincludingchemicals,foodprocessing,automotive,andaerospaceapplications.

Programmingtoolslike,e.g.,MATLAB(ModelPredictiveControlToolbox)andLabVIEW(ControlDesignandSimulationModule)hasMPCfunctionality.

DeltaV,whichisaDCS(DistributedControlSystem)systemhasMPCfunctionality(DeltaVPredict/DeltaVPredictPro).

Thesearejustafewexamples,butmentionedherebecausethesetoolsandsystemsareavailbleattheuniversity.

InthisTutorialwewillusethePredictiveControlfunctionalitywhichispartoftheLabVIEWControlDesignandSimulationModule.

ThescopewiththisTutorialisnottogoindepthofthetheorybehindMPC,buttouseandgiveanoverviewoftheMPCimplementationinLabVIEW.

ii Preface

Tutorial:ModelPredictiveControlinLabVIEW

[Figureontitlepage:NationalInstruments,LabVIEWControlDesignUserManual,2008]

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TableofContentsPreface..................................................................................................................i

TableofContents................................................................................................iii

1 IntroductiontoModelPredictiveControl.....................................................4

1.1 Introduction............................................................................................4

1.2 PredictionandControlHorizons.............................................................5

1.3 Model.....................................................................................................7

1.4 CostFunction..........................................................................................8

1.5 Constraints..............................................................................................9

1.6 MPCvs.TraditionalControl(PID).........................................................10

2 LabVIEWControlandSimulationModule...................................................12

3 MPCinLabVIEW..........................................................................................14

3.1 Example1:Simple1.orderModel.......................................................14

3.2 Example2:ModelwithTimeDelay......................................................17

3.3 Example:MultipleInputs......................................................................20

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1 IntroductiontoModelPredictiveControl

1.1 IntroductionTraditionalfeedbackcontrollersoperatebyadjustingcontrolactioninresponsetoachangeintheoutputsetpointofasystem.Modelpredictivecontrol(MPC)isatechniquethatfocusesonconstructingcontrollersthatcanadjustthecontrolactionbeforeachangeintheoutputsetpointactuallyoccurs.Thispredictiveability,whencombinedwithtraditionalfeedbackoperation,enablesacontrollertomakeadjustmentsthataresmootherandclosertotheoptimalcontrolactionvalues.

BelowweseethebasicstructureofMPC:

[Wikipedia]



ModelPredictiveControl(MPC)isacontrolstrategywhichisaspecialcaseoftheoptimalcontroltheorydevelopedinthe1960andlather.MPCconsistsofanoptimizationproblemateachtimeinstants,k.

Themainpointofthisoptimizationproblemistocomputeanewcontrolinputvector, 𝑢",tobefeedtothesystem,andatthesametimetakeprocessconstraintsintoconsideration(e.g.,constraintsonprocessvariables).

AnMPCalgorithmconsistsof:

• ACostfunction• Constraints• AModeloftheprocess

Thesethingswillbeexplainedindetailbelow.

1.2 PredictionandControlHorizonsPredictionhorizon(𝑵𝒑)-ThenumberofsamplesinthefuturetheMPCcontrollerpredictstheplantoutput.

Controlhorizon(𝑵𝒄)–ThenumberofsampleswithinthepredictionhorizonwheretheMPCcontrollercanaffectthecontrolaction.

Note!

𝑁' ≤ 𝑁)

BelowweseethePredictionandControlHorizons:



[Figure:NationalInstruments,LabVIEWControlDesignUserManual,2008]

Fortime 𝑘 theMPCcontrollerpredictstheplantoutputfortime 𝑘 + 𝑁).Weseefromthefigurethatthecontrolactiondoesnotchangeafterthecontrolhorizonends.

Thefirstinputintheoptimalsequenceisthensentintotheplant,andtheentirecalculationisrepeatedatsubsequentcontrolintervals.ForeachiterationthepredictionhorizonismovingforwardintimeandtheMPCcontrolleragainpredictstheplantoutput.



[Figure:NationalInstruments,LabVIEWControlDesignUserManual,2008]

Predictionhorizon:

AshortpredictionhorizonreducesthelengthoftimeduringwhichtheMPCcontrollerpredictstheplantoutputs.WhenthepredictionhorizonisshorttheMPCcontrollerworksmorelikeatraditionalfeedbackcontroller.

AlongpredictionhorizonincreasesthepredictiveabilityoftheMPCcontroller,buttheperformancepoorerduetoextracalculations.

Controlhorizon:

Ashortcontrolhorizonmeansmorecarefullychangesinthecontrolaction.

Alongcontrolhorizonmeansmoreaggressivechangesinthecontrolaction.

1.3 ModelThemaindrawbackwithMPCisthatamodelfortheprocess,i.e.,amodelwhichdescribestheinputtooutputbehavioroftheprocess,isneeded.Mechanisticmodelsderivedfromconservationlawscanbeused.Usually,howeverinpracticesimplydata-drivenlinearmodelsareused.

InMPCitisassumedthatthemodelisadiscretestate-spacemodeloftheform:



𝑥"-. = 𝐴𝑥" + 𝐵𝑢"

𝑦" = 𝐶𝑥" + 𝐷𝑢"

1.4 CostFunctionThemainideawithMPCisthattheMPCcontrollercalculatesasequenceoffuturecontrolactionssuchthatacostfunctionisminimized.

ThecostfunctionoftenusedinMPCislikethis(alinearquadraticfunction)[NationalInstruments,LabVIEWControlDesignUserManual,2008]:

𝐽 = 𝑦 − 𝑟 8𝑄

:;

"<=

𝑦 − 𝑟 + ∆𝑢8𝑅

:;

"<=

∆𝑢

Where:

𝑁) –Predictionhorizon

𝑟 –Setpoint

𝑦 –Predictedprocessoutput

∆𝑢 –Predictedchangeincontrolvalue, ∆𝑢" = 𝑢" − 𝑢"@.

𝑄 –Outputerrorweightmatrix

𝑅 –Controlweightmatrix

ThisworksforMIMOsystems(MultipleInputandMultipleOutputs)sowearedealingwithvectorsandmatrices.

Forascalarsystemwehave:

𝐽 = 𝑞 𝑦 − 𝑟 B

:;

"<=

+ 𝑟∆𝑢B:;

"<=

Sothebasicproblemistosolve:



𝜕𝐽𝜕𝑢 = 0

Bysolvingthiswegetthefutureoptimalcontrol.

Solving EFEG= 0 isquitecomplexandwillnotbepartofthistutorial,butinthe

figurebelowweseeanillustrationoftheproblem.

1.5 ConstraintsAllphysicalsystemshaveconstraints.Wehavephysicalconstraintslikeactuatorlimits,etc.andwehavesafetyconstraintsliketemperatureandpressurelimits.Finallywehaveperformanceconstraintslikeovershoot,etc.

InMPCyounormallydefinetheseconstraints:

Constraintsintheoutputs:

𝑦HIJ ≤ 𝑦 ≤ 𝑦HKL



Constraintsintheinputs:

∆𝑢HIJ ≤ ∆𝑢 ≤ ∆𝑢HKL

𝑢HIJ ≤ 𝑢 ≤ 𝑢HKL

Note! ∆𝑢" = 𝑢" − 𝑢"@.

TheMPCcontrollertakesalltheseconstraintsintoconsiderationwhencalculatingthefuturecontrols.

1.6 MPCvs.TraditionalControl(PID)MPCisoftenusedinadditiontotraditionalcontrollikePID.InlargeplantsMPCisnotareplacementfortraditionalPID,butusedinadditiontoPIDcontrollers.PIDcontrollersareusedassingle-loopcontrollers,whileMPCisusedasanoverallsystem.PIDhandlesonlyasingleinputandasingleoutput(SISOsystems),whileMPCisamoreadvancedmethodofprocesscontrolusedforMIMOsystems(MultipleInputs,multipleOutputs).

TraditionalControl(PID) MPC

• Noknowledgeaboutconstraints

• Setpointfarfromconstraints• Notoptimalprocess

operation• SISOsystems

• Constraintsincludedinthedesign

• Setpointcanbeclosertoconstraints

• Improvedprocessoperation• MIMOsystems• Amathematicalmodelisneeded



• Amathematicalmodelisnotneeded

ThemodelsusedinMPCaregenerallyintendedtorepresentthebehaviorofcomplexdynamicalsystems.TheadditionalcomplexityoftheMPCcontrolalgorithmisnotgenerallyneededtoprovideadequatecontrolofsimplesystems,whichareoftencontrolledwellbyPIDcontrollers.CommondynamiccharacteristicsthataredifficultforPIDcontrollersincludelargetimedelaysandhigh-orderdynamics.

AnotheradvantageofMPCisthatcrosscouplinginmultipleinputandmultipleoutput(MIMO)systemsaretakenintoconsiderationinanoptimalway.MPCisasimplemethodforcontrollingMIMOsystems.

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2 LabVIEWControlandSimulationModule

TheMPCfunctionalityinLabVIEWispartofthe“ControlDesignandSimulationModule”.

The“ControlDesignandSimulation”paletteinLabVIEW:

The“ControlDesign”paletteinLabVIEW:

The“PredictiveControl”paletteinLabVIEW:

13 LabVIEWControlandSimulationModule


HereisashortdescriptionofthedifferentVIs:

Youusethe“CDCreateMPCController”VItocreateanMPCcontroller.ThisVIbasestheMPCcontrolleronastate-spacemodeloftheplantthatyouprovide.

The“CDImplementMPCController”isusedtocalculatethecontrolvaluesforeachsamplingtimeandisnormallyimplementedinaloop,e.g.,aWhileLoop.

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3 MPCinLabVIEWInthischapterwewillusetheVisinthe“PredictiveControl”paletteininsomeexample.

3.1 Example1:Simple1.orderModelGiventhefollowingsystem:

𝑥 = −1𝑇 𝑥 + 𝐾𝑢

Where

𝑇 isthetimeconstantforthesystem

𝐾 is,e.g.,thepumpgain

Weset 𝑇 = 5𝑠 and 𝐾 = 2

FrontPanel:

15 MPCinLabVIEW


Results:

SetpointProfile:

ConstraintsandWeighting:

16 MPCinLabVIEW


BlockDiagram:

Wecandividethesolutioninto2differentparts:

InitializationtheMPCController:

Thisissomethingwedoonlyoncewhenwestarttheprogram.Weusethe“CDCreateMPCController.vi”.

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RuntheController:

Thisoperationisperformedeachsample,andthisisnormallyexecutedinsidealoop,e.g.aWhileLoop.Weusethe“CDImplementMPCController.vi”.

[EndofExample]

3.2 Example2:ModelwithTimeDelayGiventhefollowingsystem:

𝑥 = −1𝑇 𝑥 + 𝐾𝑢(𝑡 − 𝜏)

Where

𝑇 isthetimeconstantforthesystem

𝐾 is,e.g.,thepumpgain

18 MPCinLabVIEW


𝜏 isthetimedelay

Weset 𝑇 = 5𝑠, 𝐾 = 2 and 𝜏 = 3𝑠

TheMPCalgorithmrequiresthatthemodelisalinearstate-spacemodel,butthetimedelaycausesproblems.

Asolutioncouldbetotransformthedifferentialequationwehavetoatransferfunction.Thenwecanusebuilt-infunctionsinLabVIEWtoconvertittoalinearstate-spacemodel.

WeuseLaplaceonthedifferentialequationabove:

𝑠𝑥(𝑠) = −1𝑇 𝑥(𝑠) + 𝐾𝑢(𝑠)𝑒

@YZ

Note!WeusethefollowingLaplacetransformation:

𝐹 𝑠 𝑒@YZ ⟺ 𝑓(𝑡 − 𝜏)

𝑠𝐹(𝑠) ⟺ 𝑓(𝑡)

Thisgives:

𝑠𝑥 𝑠 +1𝑇 𝑥 𝑠 = 𝐾𝑢(𝑠)𝑒@YZ

Next:

𝑥 𝑠 𝑠 +1𝑇 = 𝐾𝑢(𝑠)𝑒@YZ

Next:

𝑥 𝑠𝑢(𝑠) =

𝐾

𝑠 + 1𝑇𝑒@YZ

Finally:

𝐻 𝑠 =𝑥 𝑠𝑢(𝑠) =

𝐾𝑇𝑇𝑠 + 1 𝑒

@YZ =𝐾`a`𝑇𝑠 + 1 𝑒

@YZ

Withvalues(𝑇 = 5, 𝐾 = 2, 𝜏 = 3):

19 MPCinLabVIEW


𝐻 𝑠 =𝑥 𝑠𝑢(𝑠) =

105𝑠 + 1 𝑒

@bZ

LabVIEWapplication:

Wecanusethe“CDConstructSpecialTFModel.vi”inordertocreatethetransferfunction.Thenweusemiscellaneousfunctionsinordertoendupwithadiscretestate-spacemodelthathandlesthetimedelay(additionalstatesareadded).

Where

Restofthecodeissimilartopreviousexample,exceptthatwehavebeenusingastatemachineinordertoimplementthecode.

Belowweseethefrontpanel:

20 MPCinLabVIEW


[EndofExample]

3.3 Example:MultipleInputsInthisexamplewewilluseMPConaMISOsystemwith2inputsand1output.

21 MPCinLabVIEW


Wedefinethesetpointprofilethesystemshouldfollowinadvance(thefuturesetpointisknown)andseehowtheMPCcontrollerworksinordertofollowthesetpoint.

Weseethatthecontrollerstartstoreactbeforethereferenceactuallychanges,whichisatypicallyfeaturefortheMPCcontroller.

HereweseethemaindifferencebetweenaMPCcontrollerandamoretraditionalPIDcontroller.AnothermaindifferencebetweenMPCandPIDisthatMPCcanhandleMIMO(MultipleInputs,MultipleOutputs)systems,whilePIDisusedforSISOsystems(SingleInput,SingleOutput).

BlockDiagram:

Belowweseetheblockdiagramfortheprogram:

22 MPCinLabVIEW


Wecandividethesolutioninto2differentparts:

InitializationtheMPCController:

Thisissomethingwedoonlyoncewhenwestarttheprogram.Weusethe“CDCreateMPCController.vi”.

RuntheController:

Thisoperationisperformedeachsample,andthisisnormallyexecutedinsidealoop,e.g.aWhileLoop.Weusethe“CDImplementMPCController.vi”.

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[EndofExample]

Hans-PetterHalvorsen,M.Sc.

E-mail:[email protected]

Blog:http://home.hit.no/~hansha/

UniversityCollegeofSoutheastNorway

www.usn.no

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