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SystemIdentification andEstimationinLabVIEW

Hans-PetterHalvorsen

SystemIdentificationandEstimationinLabVIEW


Copyright©2017

E-Mail:[email protected]

Web:https://www.halvorsen.blog


i

PrefaceThisTutorialwillgothroughthebasicprinciplesofSystemidentificationandEstimationandhowtoimplementthesetechniquesinLabVIEWandLabVIEWMathScript.

LabVIEWisagraphicalprogramminglanguagecreatedbyNationalInstruments,whileLabVIEWMathScriptisanadd-ontoLabVIEW.LabVIEWMathScripthassimilarsyntaxas,e.g.,MATLAB.LabVIEWMathScriptmaybeusedasaseparatepart(andcanbeconsideredasaminiatureversionofMATLAB)orbeintegratedintothegraphicalLabVIEWcodeusingtheMathScriptNode.

Thefollowingmethodswillbediscussed:

StateEstimation:

• KalmanFilter• Observers

ParameterEstimation:

• LeastSquareMethod(LS)

SystemIdentification

• Sub-spacemethods/Black-Boxmethods• PolynomialModelEstimation:ARX/ARMAXmodelEstimation

Software

YouneedthefollowingsoftwareinthisTutorial:

• LabVIEW• LabVIEWControlDesignandSimulationModule• LabVIEWMathScriptRTModule(LabVIEWMathScript)

“LabVIEWControlDesignandSimulationModule”hasfunctionalityforcreatingKalmanFiltersandObservers,butitalsohasfunctionalityforSystemidentification.

ii

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

TableofContents......................................................................................................................ii

1 IntroductiontoLabVIEWandMathScript.........................................................................5

1.1 LabVIEW....................................................................................................................5

1.2 LabVIEWMathScript..................................................................................................6

2 LabVIEWControlandSimulationModule.........................................................................9

3 ModelCreationinLabVIEW............................................................................................11

3.1 State-spaceModels.................................................................................................12

3.2 Transferfunctions...................................................................................................15

3.2.1 commonlyusedtransferfunctions..................................................................17

4 IntroductiontoSystemIdentificationandEstimation....................................................24

5 StateEstimationwithKalmanFilter................................................................................26

5.1 State-Spacemodel...................................................................................................27

5.2 Observability...........................................................................................................29

5.3 IntroductiontotheStateEstimator........................................................................30

5.4 StateEstimation......................................................................................................36

5.5 LabVIEWKalmanFilterImplementations................................................................38

6 CreateyourownKalmanFilterfromScratch..................................................................44

6.1 TheKalmanFilterAlgorithm....................................................................................44

6.2 Examples.................................................................................................................45

7 OverviewofKalmanFilterVIs.........................................................................................49

7.1 ControlDesignpalette.............................................................................................49

7.1.1 StateFeedbackDesignsubpalette..................................................................49

7.1.2 Implementationsubpalette.............................................................................50

iii TableofContents

Tutorial:SystemIdentificationandEstimationinLabVIEW

7.2 Simulationpalette...................................................................................................51

7.2.1 Estimationsubpalette.....................................................................................51

8 StateEstimationwithObserversinLabVIEW..................................................................54

8.1 State-Spacemodel...................................................................................................54

8.2 Eigenvalues..............................................................................................................56

8.3 ObserverGain..........................................................................................................57

8.4 Observability...........................................................................................................58

8.5 Examples.................................................................................................................59

9 OverviewofObserverfunctions......................................................................................64

9.1 ControlDesignpalette.............................................................................................64

9.1.1 StateFeedbackDesignsubpalette..................................................................64

9.1.2 Implementationsubpalette.............................................................................65

9.2 Simulationpalette...................................................................................................66

9.2.1 Estimationsubpalette.....................................................................................66

10 SystemIdentificationinLabVIEW................................................................................69

10.1 ParameterEstimationwithLeastSquareMethod(LS)...........................................70

10.2 SystemIdentificationusingSub-spacemethods/Black-Boxmethods.....................73

10.3 SystemIdentificationusingPolynomialModelEstimation:ARX/ARMAXmodelEstimation...........................................................................................................................74

10.4 GeneratemodelData..............................................................................................75

10.4.1 Excitationsignals.............................................................................................77

11 OverviewofSystemIdentificationfunctions..............................................................80

12 SystemIdentificationExample....................................................................................85

4

PartI:Introduction

5

1 IntroductiontoLabVIEWandMathScript

InthisTutorialwewilluseLabVIEWandsomeoftheadd-onmodulesavailableforLabVIEW.

• LabVIEW• LabVIEWMathScriptRTModule• LabVIEWControlDesignandSimulationModule

1.1 LabVIEWLabVIEW(shortforLaboratoryVirtualInstrumentationEngineeringWorkbench)isaplatformanddevelopmentenvironmentforavisualprogramminglanguagefromNationalInstruments.Thegraphicallanguageisnamed"G".LabVIEWiscommonlyusedfordataacquisition,instrumentcontrol,andindustrialautomation.Thecodefileshavetheextension“.vi”,whichisanabbreviationfor“VirtualInstrument”.LabVIEWofferslotsofadditionalAdd-onsandToolkits.

FormoreinformationaboutLabVIEW,pleasegotomyBlog:https://www.halvorsen.blog andvisitNationalInstrumentsatwww.ni.com.



1.2 LabVIEWMathScriptMathScriptisahigh-level,text-basedprogramminglanguage.MathScriptincludesmorethan800built-infunctionsandthesyntaxissimilartoMATLAB.Youmayalsocreatecustom-madem-filelikeyoudoinMATLAB.

MathScriptisanadd-onmoduletoLabVIEWbutyoudon’tneedtoknowLabVIEWprogramminginordertouseMathScript.

MathScriptisanadd-onmoduletoLabVIEWbutyoudon’tneedtoknowLabVIEWprogramminginordertouseMathScript.

FormoreinformationaboutMathScript,pleasereadtheTutorial“LabVIEWMathScript”.

HowdoyoustartusingMathScript?

YouneedtoinstallLabVIEWandtheLabVIEWMathScriptRTModule.Whennecessarysoftwareisinstalled,startMathScriptbyopenLabVIEW.IntheGettingStartedwindow,selectTools->MathScriptWindow...:



FormoreinformationaboutMathScript,pleasereadtheTutorial“LabVIEWMathScript”.

MathScriptNode:

YoumayalsouseMathScriptCodedirectlyinsideandcombinedwithyougraphicalLabVIEWcode,forthisyouusethe“MathScriptNode”.Withthe“MathScriptNode”youcancombinegraphicalandtextualcodewithinLabVIEW.Thefigurebelowshowsthe“MathScriptNode”ontheblockdiagram,representedbythebluerectangle.Using“MathScriptNodes”,youcanenter.mfilescripttextdirectlyorimportitfromatextfile.

YoucandefinenamedinputsandoutputsontheMathScriptNodebordertospecifythedatatotransferbetweenthegraphicalLabVIEWenvironmentandthetextualMathScriptcode.

Youcanassociate.mfilescriptvariableswithLabVIEWgraphicalprogramming,bywiringNodeinputsandoutputs.Thenyoucantransferdatabetween.mfilescriptswithyourgraphicalLabVIEWprogramming.Thetextual.mfilescriptscannowaccessfeaturesfromtraditionalLabVIEWgraphicalprogramming.

TheMathScriptNodeisavailablefromLabVIEWfromtheFunctionsPalette:Mathematics→Scripts&Formulas



IfyouclickCtrl+HyougethelpabouttheMathScriptNode:

Click“Detailedhelp”inordertogetmoreinformationabouttheMathScriptNode.

9

2 LabVIEWControlandSimulationModule

LabVIEWhasseveraladditionalmodulesandToolkitsforControlandSimulationpurposes,e.g.,“LabVIEWControlDesignandSimulationModule”,“LabVIEWPIDandFuzzyLogicToolkit”,“LabVIEWSystemIdentificationToolkit”and“LabVIEWSimulationInterfaceToolkit”.LabVIEWMathScriptisalsousefulforControlDesignandSimulation.

• LabVIEWControlDesignandSimulationModule• LabVIEWPIDandFuzzyLogicToolkit• LabVIEWSystemIdentificationToolkit• LabVIEWSimulationInterfaceToolkit

BelowweseetheControlDesign&SimulationpaletteinLabVIEW:

InthisTutorialwewillfocusontheVIsusedforParameterandStateestimationandespeciallytheuseofObserversandKalmanFilterforStateestimation.

IfyouwanttolearnmoreaboutSimulation,SimulationLoop,blockdiagramsandPIDcontrol,etc.,IrefertotheTutorial“ControlandSimulationinLabVIEW”ThisTutorialisavailablefromhttps://www.halvorsen.blog.

InthisTutorial,wewillneedthefollowingsubpalettesintheControlDesignandSimulationpalette:

• ControlDesign• SystemIdentification• Simulation

BelowweseetheControlDesignpalette:

10 LabVIEWControlandSimulationModule


BelowweseetheSystemIdentificationpalette:

BelowweseetheSimulationpalette:

Inthenextchapterswewillgoindetailanddescribethedifferentsubpalettesinthesepalettesandexplainthefunctions/SubVIswewillneedforSystemidentificationandEstimation.

11

3 ModelCreationinLabVIEWWhenyouhavefoundthemathematicalmodelforyoursystem,thefirststepistodefine/orcreateyourmodelinLabVIEW.YourmodelcanbeaTransferfunctionoraState-spacemodel.

InLabVIEWandthe“LabVIEWControlDesignandSimulationModule”youcancreatedifferentmodels,suchasState-spacemodelsandtransferfunctions,etc.

IntheControlDesignpalette,wehaveseveralsubpalettesthatdealswithmodels,theseare:

• ModelConstruction• ModelInformation• ModelConversion• ModelInterconnection

BelowwegothroughthedifferentsubpalettesandthemostusedVIsinthesepalettes.

“ModelConstruction”Subpalette:

InthispalettewehaveVIsforcreatingstate-spacemodelsandtransferfunctions.

12 ModelCreationinLabVIEW


→UsetheModelConstructionVIstocreatelinearsystemmodelsandmodifythepropertiesofasystemmodel.YoualsocanusetheModelConstructionVIstosaveasystemmodeltoafile,readasystemmodelfromafile,orobtainavisualrepresentationofamodel.

SomeofthemostusedVIswouldbe:

CDConstructState-SpaceModel.vi

CDConstructTransferFunctionModel.vi

TheseVIsandsomeothersareexplainedbelow.

3.1 State-spaceModelsGiventhefollowingState-spacemodel:

𝑥 = 𝐴𝑥 + 𝐵𝑢

𝑦 = 𝐶𝑥 + 𝐷𝑢

InLabVIEWweusethe“CDConstructState-SpaceModel.vi”tocreateaState-spacemodel:



YoumayusenumericvaluesinthematricesA,B,CandDorsymbolicvaluesbyselectingether“Numeric”or“Symbolic”:

Numeric Symbolic

Example:CreateState-Spacemodel

BlockDiagram:

FrontPanel:



The“CDDrawState-SpaceEquation.vi”canbeusedtoseeagraphicalrepresentationoftheState-spacemodel.

Example:CreateSISO/MIMOState-Spacemodels

SISOModel(SingleInput,SingleOutput):

SIMOModel(SingleInput,MultipleOutput):

MISOModel(MultipleInput,SingleOutput):

MIMOModel(MultipleInput,MultipleOutput):



[EndofExample]

3.2 TransferfunctionsGiventhefollowingTransferfunction:

𝐻 𝑠 =𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟

=𝑏6 + 𝑏7𝑠 + 𝑏8𝑠8 +⋯𝑎6 + 𝑎7𝑠 + 𝑎8𝑠8 +⋯

InLabVIEWweusethe“CDConstructTransferFunctionModel.vi”tocreateaTransferFunction:

Example:TransferFunction

BlockDiagram:



FrontPanel:

[EndofExample]

Example:TransferFunctionwithSymbolicvalues

BlockDiagram:

FrontPanel:



[EndofExample]

3.2.1 commonlyusedtransferfunctions

Forcommonlyusedtransferfunctionswecanusethe“CDConstructSpecialTFModel.vi”:

1.ordersystem:

Thetransferfunctionfora1.ordersystemisasfollows:

𝐻 𝑠 =𝐾

𝑇𝑠 + 1𝑒>?@

Where

𝐾 isthegain

TistheTimeconstant

𝜏 istheTimedelay

Selectthe polymorphicinstanceonthe“CDConstructSpecialTFModel.vi”:



2.ordersystem:

Thetransferfunctionfora2.ordersystemisasfollows:

𝐻 𝑠 =𝐾𝜔68

𝑠8 + 2𝜁𝜔6𝑠 + 𝜔68=

𝐾𝑠𝜔6

8+ 2𝜁 𝑠

𝜔6+ 1

Where

• 𝐾 isthegain• 𝜁 zetaistherelativedampingfactor• 𝜔6[rad/s]istheundampedresonancefrequency.


TimedelayasaPade’approximation:

Time-delaysareverycommonincontrolsystems.TheTransferfunctionofatime-delayis:

𝐻 𝑠 = 𝑒>?@



Insomesituationsitisnecessarytosubstitute 𝑒>?@ withanapproximation,e.g.,thePadé-approximation:

𝑒>?@ ≈1 − 𝑘7𝑠 + 𝑘8𝑠8 + ⋯± 𝑘I𝑠I

1 + 𝑘7𝑠 + 𝑘8𝑠8 + ⋯+ 𝑘I𝑠I


“ModelInformation”Subpalette:

→UsetheModelInformationVIstoobtainorsetparameters,data,andnamesofasystemmodel.Modelinformationincludespropertiessuchasthesystemdelay,systemdimensions,samplingtime,andnamesofinputs,outputs,andstates.

“ModelConversion”Subpalette:



→UsetheModelConversionVIstoconvertasystemmodelfromonerepresentationtoanother,fromacontinuous-timetoadiscrete-timemodel,orfromadiscrete-timetoacontinuous-timemodel.YoualsocanusetheModelConversionVIstoconvertacontroldesignmodelintoasimulationmodelorasimulationmodelintoacontroldesignmodel.

SomeofthemostusedVIsinthe“ModelConversion”subpaletteare:

CDConverttoState-SpaceModel.vi

CDConverttoTransferFunctionModel.vi

CDConvertContinuoustoDiscrete.vi

TheseVIsandsomeothersareexplainedbelow.

ConverttoState-SpaceModels:

Example:ConvertfromTransferfunctiontoState-Spacemodel

BlockDiagram:



FrontPanel:

[EndofExample]

ConverttoTransferFunctions:

Example:ConvertfromState-SpacemodeltoTransferFunction

BlockDiagram:

FrontPanel:



[EndofExample]

ConvertContinuoustoDiscreteModel:

Example:ConvertfromContinuousState-SpacemodeltoDiscreteState-Spacemodel

BlockDiagram:

FrontPanel:



[EndofExample]

“ModelInterconnection”Subpalette:

→UsetheModelInterconnectionVIstoperformdifferenttypesoflinearsysteminterconnections.Youcanbuildalargesystemmodelbyconnectingsmallersystemmodelstogether.

24

4 IntroductiontoSystemIdentificationandEstimation

ThisTutorialwillgothroughthebasicprinciplesofSystemidentificationandEstimationandhowtoimplementthesetechniquesinLabVIEW.

Thefollowingmethodswillbediscussedinthenextchapters:

StateEstimation:

• KalmanFilter• Observers

SystemIdentification:

• ParameterEstimationandtheLeastSquareMethod(LS)• Sub-spacemethods/Black-Boxmethods• PolynomialModelEstimation:ARX/ARMAXmodelEstimation

ThenextchapterswillgothroughthebasictheoryandshowhowitcouldbeimplementedinLabVIEWandMathScript.

25

PartII:Estimation

26

5 StateEstimationwithKalmanFilterKalmanFilterisacommonlyusedmethodtoestimatethevaluesofstatevariablesofadynamicsystemthatisexcitedbystochastic(random)disturbancesandstochastic(random)measurementnoise.

TheKalmanFilterisastateestimatorwhichproducesanoptimalestimateinthesensethatthemeanvalueofthesumoftheestimationerrorsgetsaminimalvalue.

BelowweseeasketchofhowaKalmanFilterisworking:

Theestimator(modelofthesystem)runsinparallelwiththesystem(realsystemormodel).Themeasurement(s)isusedtoupdatetheestimator.

LabVIEWControlDesignandSimulationModulehavelotsoffunctionalityforStateEstimationusingKalmanFilters.Thefunctionalitywillbeexplainedindetailinthenextchapters.

BelowweseetheDiscreteKalmanFilterimplementationinLabVIEW:

TheKalmanFilterfornonlinearmodelsiscalledthe“ExtendedKalmanFilter”.

27 StateEstimationwithKalmanFilter


5.1 State-SpacemodelGiventhecontinuouslinearstatespace-model:



Orgiventhediscretelinearstatespace-model

𝑥JK7 = 𝐴𝑥J + 𝐵𝑢J

𝑦J = 𝐶𝑥J + 𝐷𝑢J

LabVIEW:

InLabVIEWwemayusethe“CDConstructState-SpaceModel.vi”tocreateaState-spacemodel:

Note!IfyouspecifyadiscreteState-spacemodelyouhavetospecifytheSamplingTime.

LabVIEWExample:CreateaState-spacemodel

BlockDiagram:

Thematrices𝐴, 𝐵, 𝐶 and 𝐷 maybedefinedontheFrontPanellikethis:



[EndofExample]

Discretization:

Ifyouhaveacontinuousmodelandwanttoconvertittothediscretemodel,youmayusetheVI“CDConvertContinuoustoDiscrete.vi”inLabVIEW:

LabVIEWFunctionsPalette:ControlDesign&Simulation→ControlDesign→ModelConversion→CDConvertContinuoustoDiscrete.vi

LabVIEWExample:ConvertfromContinuoustoDiscretemodel

Note!WehavetospecifytheSamplingTime.

[EndofExample]

MathScript:

UsethessfunctioninMathScripttodefineyourmodel(ortfifyouhaveatransferfunction).Usethec_to_dfunctiontoconvertacontinuousmodeltoadiscretemodel.

MathScriptExample:



GiventhefollowingState-spacemodel:

𝑥7𝑥8

=0 1

−𝑘𝑚

−𝑐𝑚

N

𝑥7𝑥8 +

01𝑚O

𝑢

𝑦 = 1 0P

𝑥7𝑥8

ThefollowingMathScriptCodecreatesthismodel:

c=1; m=1; k=1; A = [0 1; -k/m -c/m]; B = [0; 1/m]; C = [1 0]; ssmodel = ss(A, B, C)

Ifyouwanttofindthediscretemodelusethec_to_dfunction:

Ts=0.1 % Sampling Time discretemodel = c_to_d(ssmodel, Ts)

[EndofExample]

5.2 ObservabilityAnecessaryconditionfortheKalmanFiltertoworkcorrectlyisthatthesystemforwhichthestatesaretobeestimated,isobservable.Therefore,youshouldcheckforObservabilitybeforeapplyingtheKalmanFilter.

TheObservabilitymatrixisdefinedas:

𝑂 =𝐶𝐶𝐴⋮

𝐶𝐴I>7

Wherenisthesystemorder(numberofstatesintheState-spacemodel).

→Asystemofordernisobservableif 𝑶 isfullrank,meaningtherankof 𝑶 isequalton.

LabVIEW:

TheLabVIEWControlDesignandSimulationModulehaveaVI(ObservabilityMatrix.vi)forfindingtheObservabilitymatrixandcheckifastates-pacemodelisObservable.



LabVIEWFunctionsPalette:ControlDesign&Simulation→ControlDesign→State-SpaceModelAnalysis→CDObservabilityMatrix.vi

Note!InLabVIEW 𝑁 isusedasasymbolfortheObservabilitymatrix.

LabVIEWExample:CheckforObservability

[EndofExample]

MathScript:

InMathScriptyoumayusetheobsvmxfunctiontofindtheObservabilitymatrix.YoumaythenusetherankfunctioninordertofindtherankoftheObservabilitymatrix.

MathScriptExample:

ThefollowingMathScriptCodecheckforObservability:

% Check for Observability: O = obsvmx (discretemodel) r = rank(O)

[EndofExample]

5.3 IntroductiontotheStateEstimatorContinuousModel:



Giventhecontinuouslinearstatespacemodel:

𝑥 = 𝐴𝑥 + 𝐵𝑢 + 𝐺𝑤

𝑦 = 𝐶𝑥 + 𝐷𝑢 + 𝐻𝑣

oringeneral:

𝑥 = 𝑓(𝑥, 𝑢) + 𝐺𝑤

𝑦 = 𝑔(𝑥, 𝑢) + 𝐻𝑣

DiscreteModel:

Giventhediscretelinearstatespacemodel:

𝑥JK7 = 𝐴𝑥J + 𝐵𝑢J + 𝐺𝑤J

𝑦J = 𝐶𝑥J + 𝐷𝑢J + 𝐻𝑣J

oringeneral:

𝑥JK7 = 𝑓(𝑥J, 𝑢J) + 𝐺𝑤J

𝑦J = 𝑔(𝑥J, 𝑢J) + 𝐻𝑣J

Where 𝑣 isuncorrelatedwhiteprocessnoisewithzeromeanandcovariancematrix 𝑄 andwisuncorrelatedwhitemeasurementsnoisewithzeromeanandcovariancematrix,i.e.suchthat

𝑄 = 𝐸{𝑤𝑤`}

𝑅 = 𝐸{𝑣𝑣`}

𝐸{𝑤} istheexpectedvalueormeanoftheprocessnoisevector.

𝐸{𝑣}istheexpectedvalueormeanofthemeasurementnoisevector.

Itisnormaltolet 𝑄 and 𝑅 bediagonalmatrices:

𝑄 =

𝑞77 0 0 00 𝑞88 0 00 0 ⋱ 00 0 0 𝑞II

, 𝑅 =

𝑟77 0 0 00 𝑟88 0 00 0 ⋱ 00 0 0 𝑟ee

𝐺 istheprocessnoisegainmatrix,andyounormallyset 𝐺 equaltotheIdentitymatrix 𝐼:



𝐺 =

1 0 0 00 1 0 00 0 ⋱ 00 0 0 1

𝐻 isthemeasurementnoisegainmatrix,andyounormallyset 𝐻 = 0

StateEstimator:

Thestateestimatorisgivenby:

𝑥JK7 = 𝐴𝑥J + 𝐵𝑢J + 𝐾(𝑦 − 𝑦)

𝑦J = 𝐶𝑥J

Where 𝐾 istheKalmanFilterGain

Itcanbefoundthat:

𝐾 = 𝑋𝐶`𝑄>7

Where 𝑋 isthesolutiontotheRiccatiequation.ItiscommontousethesteadystatesolutionoftheRiccatiequation(AlgebraicRiccatiEquation),i.e., 𝑋 = 0.

Note! 𝑄 and 𝑅 isusedastuning/weightingmatriceswhendesigningtheKalmanFilterGain 𝐾.

BelowweseeaBlockDiagramofa(discrete)KalmanFilter/StateEstimator:



LabVIEW:

InLabVIEWDesignandSimulationModuleweusethe“CDKalmanGain.vi”inordertofindK:

LabVIEWFunctionsPalette:ControlDesign&Simulation→ControlDesign→StateFeedbackDesign→CDKalmanGain.vi

Note!InLabVIEW 𝐿 isusedasasymbolfortheKalmanFilterGainmatrix.

Note!The“KalmanFilterGain.vi”ispolymorphic,dependingonwhatkindofmodel(deterministic/stochasticorcontinuous/discrete)youwiretothisVI,theinputschangesautomaticallyoryoumayusethepolymorphicselectorbelowtheVI:

DeterministicandContinuous:

StochasticandContinuous:

DeterministicandDiscrete:

StochasticandDiscrete:



LabVIEWExample:FindtheKalmanGain

BlockDiagram:

FrontPanel:

[EndofExample]

MathScript:

Usethefunctionskalman,kalman_dorlqetofindtheKalmangainmatrix.

Function Description Example

kalman Calculatestheoptimalsteady-stateKalmangainKthatminimizesthecovarianceofthestateestimationerror.YoucanusethisfunctiontocalculateKforcontinuousanddiscretesystemmodels.

>>[SysKal, K]=kalman(ssmodel, Q, R)



kalman_d Calculatestheoptimalsteady-stateKalmangainKthatminimizesthecovarianceofthestateestimationerror.Theinputsystemandnoisecovariancearebasedonacontinuoussystem.Alloutputsarebasedonadiscretizedsystem,whichisbasedonthesamplerateTs.

[SysKalDisc, K]=kalman_d(ssmodel, Q, R, Ts)

lqe Calculatestheoptimalsteady-stateestimatorgainmatrixKforacontinuousstate-spacemodeldefinedbymatricesAandC.

K=lqe(A,G,C,Q,R)

MathScriptExample:FindKalmanGainusingtheKalmanfunction

MathScriptCodeforfindingthesteadystateKalmanGain:

% Define the State-space model: c=1; m=1; k=1; A = [0 1; -k/m -c/m]; B = [0; 1/m]; C = [1 0]; ssmodel = ss(A, B, C); % Discrete model: Ts=0.1; % Sampling Time discretemodel = c_to_d(ssmodel, Ts); % Check for Observability: O = obsvmx(discretemodel); r = rank(O); % Find Kalman Gain Q=[0.01 0; 0 0.01]; R=[0.01]; [SysKal, K]=kalman(discretemodel, Q, R); K

TheOutputis:

K = 0.64435 0.10759

[EndofExample]

MathScriptExample:FindKalmanGainusingtheKalmanfunction



MathScriptCodeforfindingthesteadystateKalmanGain:

% Define the State-space model: c=1; m=1; k=1; A = [0 1; -k/m -c/m]; B = [0; 1/m]; G=[1 0 ; 0 1]; C = [1 0]; % Find Kalman Gain Q=[0.01 0; 0 0.01]; R=[0.01]; K=lqe(A,G,C,Q,R)

TheOutputis:

K = 0.86121 -0.12916

[EndofExample]

5.4 StateEstimationLabVIEWControlDesignandSimulationModulehavebuilt-infunctionalityforStateEstimationusingtheKalmanFilter.

InthenextsectionwewillcreateourownKalmanFilterStateEstimationalgorithm.

LabVIEW:

TherearedifferentfunctionsandVIsforfindingtheStateEstimationusingtheKalmanFilter.

CDStateEstimator.vi:

LabVIEWFunctionsPalette:ControlDesign&Simulation→ControlDesign→StateFeedbackDesign→CDStateEstimator.vi



BelowweshowanexampleofhowtousetheCDStateEstimator.viinLabVIEW.

LabVIEWExample:StateEstimatorSimulation

Giventhefollowingmodel:

𝑥7𝑥8

= −0.2 0.50 −0.1

N

𝑥7𝑥8 + 0

1O

𝑢

𝑦 = 1 0P

𝑥7𝑥8 + 0

k𝑢

Wewillusethe“CDStateEstimator.vi”inLabVIEW.

BlockDiagrambecomesasfollows:

Note!Wehaveused“CDInitialResponse.vi”forplottingtheresponse.TheVIislocatedintheLabVIEWFunctionsPalette:ControlDesign&Simulation→ControlDesign→TimeResponse→CDInitialResponse.vi

Theresultbecomesasfollows:



Weseetheestimatesaregood.

MathScript:

InMathScriptwemayusethebuilt-inestimatorfunction.

5.5 LabVIEWKalmanFilterImplementationsLabVIEWDesignandSimulationModulehaveseveralbuilt-inversionsoftheKalmanFilter;herewewillinvestigatesomeofthem.

TheControlDesign→ImplementationpaletteinLabVIEW:

Herewehavethe“CDDiscreteKalmanFilter”.

Simulation→EstimationpaletteinLabVIEW:



Herewehaveimplementationsfor:

• ContinuousKalmanFilter• ContinuousExtendedKalmanFilter• DiscreteKalmanFilter• DiscreteExtendedKalmanFilter

Wewillgothroughthe“DiscreteKalmanFilter”indetailandshowsomeexamples.

DiscreteKalmanFilter:

LabVIEWFunctionsPalette:ControlDesign&Simulation→Simulation→Estimation→DiscreteKalmanFilter

Bydefaultyouneedtowiretheinput(𝑢)andoutput(𝑦)vectors:



InordertoConfiguretheblockyouright-clickonitandselect“Configuration…”

IntheConfigurationwindowyoucanenteryourmodelparameters:



Ifyouselect“Terminal”inthe“Parametersource”youmaycreateyourmodelinLabVIEWcodelikethis:

LabVIEWExample:DiscreteKalmanFilter

Giventhefollowinglinearstate-spacemodelofawatertank:

𝑥7𝑥8

= 0 −100 0

N

𝑥7𝑥8 + 0.02

0O

𝑢



𝑦 = 1 0P

𝑥7𝑥8 + 0

k𝑢

Where 𝑥7 isthelevelinthetank,while 𝑥8 istheoutflowofthetank.Onlythelevel 𝑥7 ismeasured.

Step1:Firstwecreatethemodel:

WhereA,B,DandDisdefinedaccordingtothestate-spacemodelabove:

Note!TheDiscreteKalmanFilterfunctioninLabVIEWrequiresastochasticstate-spacemodel,sowehavetocreateastochasticstate-spacemodelorconvertourstate-spacemodelintoastochasticstate-spacemodelasdoneintheLabVIEWcodeabove.

Step2:ThenweusetheDiscreteKalmanFilterfunctioninLabVIEWonourmodel:



TheDiscreteKalmanFilterfunctionalsorequiresaNoisemodel,sowecreateanoisemodelfromour 𝑄 and 𝑅 matricesasdoneintheLabVIEWcodeabove.

Theresultsareasfollows:

Weseetheresultisverygood.

[EndofExample]

44

6 CreateyourownKalmanFilterfromScratch

InthischapterwewillcreateourownKalmanFilterAlgorithmfromscratch.

6.1 TheKalmanFilterAlgorithmLabVIEWDesignandSimulationModulehaveseveralbuilt-inversionsoftheKalmanFilter,butinthischapterwewillcreateourownKalmanFilteralgorithm.

HereisastepbystepKalmanFilteralgorithmwhichcanbedirectlyimplementedinaprogramminglanguage,suchasLabVIEW.Youmay,e.g.,implementitinstandardLabVIEWcodeoraFormulaNodeinLabVIEW.

PreStep:FindthesteadystateKalmanGainK

Kistime-varying,butyounormallyimplementthesteadystateversionofKalmanGainK.Usethe“CDKalmanGain.vi”inLabVIEWoroneofthefunctionskalman,kalman_dorlqeinMathScript.

InitStep:SettheinitialApriori(Predicted)stateestimate

𝑥6 = 𝑥6

Step1:FindMeasurementmodelupdate

𝑦J = 𝑔(𝑥J, 𝑢J)

ForLinearState-spacemodel:


Step2:FindtheEstimatorError

𝑒J = 𝑦J − 𝑦J

Step3:FindtheAposteriori(Corrected)stateestimate

𝑥J = 𝑥J + 𝐾𝑒J

WhereKistheKalmanFilterGain.UsethesteadystateKalmanGainorcalculatethetime-varyingKalmanGain.

Step4:FindtheApriori(Predicted)stateestimateupdate

𝑥JK7 = 𝑓(𝑥J, 𝑢J)



ForLinearState-spacemodel:


Step1-4goesinsidealoopinyourprogram.

Thisisthealgorithmwewillimplementintheexamplebelow.

6.2 ExamplesLabVIEWExample:KalmanFilteralgorithm


𝑥7𝑥8

= 0 −100 0

N

𝑥7𝑥8 + 0.02

0O

𝑢

𝑦 = 1 0P

𝑥7𝑥8 + 0

k𝑢


FirstwehavetofindthesteadystateKalmanFilterGainandcheckforObservability:

Thenweruntherealprocess(orsimulatedmodel)inparallelwiththeKalmanFilterinordertofindestimatesfor 𝑥7 and 𝑥8:



InthiscasewehaveusedaSimulationLoop,butaWhileLoopwilldothesame.

Blocks/SubVIs:

Realprocess/Simulatedprocess:

Hereweeitherhaveamodelofthesystemorread/writedatafromtherealprocessusingaDAQcard,e.g.,USB-6008fromNationalInstruments.

ImplementationoftheKalmanFilterAlgorithm:

TheBlockDiagramisasfollows:



Thisisageneralimplementationandwillworkforalllineardiscretesystems.

Theresultsareasfollows:



[EndofExample]

49

7 OverviewofKalmanFilterVIsInLabVIEWthereareseveralVIsandfunctionsusedforKalmanFilterimplementations.

7.1 ControlDesignPaletteInthe“ControlDesign”palettewefindsubpalettesfor“StateFeedbackDesign”and“Implementation”:

7.1.1 StateFeedbackDesignsubpalette

Inthe“StateFeedbackDesign”subpalettewefindVIsforcalculationtheKalmanGain,etc.

→UsetheStateFeedbackDesignVIstocalculatecontrollerandobservergainsforclosed-loopstatefeedbackcontrolortoestimateastate-spacemodel.YoualsocanuseState

50 OverviewofKalmanFilter


FeedbackDesignVIstoconfigureandteststate-spacecontrollersandstateestimatorsintimedomains.

KalmanFilterGainVI:

7.1.2 Implementationsubpalette

Inthe“Implementation”subpalettewefindVIsforimplementingadiscreteObserverandadiscreteKalmanFilter.

→UsetheImplementationVIsandfunctionstosimulatethedynamicresponseofadiscretesystemmodel,deployadiscretemodeltoareal-timetarget,implementadiscreteKalmanfilter,andimplementcurrentandpredictiveobservers.

DiscreteKalmanFilter:



7.2 SimulationPaletteInthe“Simulation”palettewefindthe“Estimation“subpalette:

7.2.1 Estimationsubpalette

Inthe“Estimation”palettewefindVIsforimplementingacontinuous/discreteKalmanFilter.



→UsetheEstimationfunctionstoestimatethestatesofastate-spacesystem.Thestate-spacesystemcanbedeterministicorstochastic,continuousordiscrete,linearornonlinear,andcompletelyorpartiallyobservable.

ContinuousKalmanFilterVIs:



DiscreteKalmanFilterVIs:

54

8 StateEstimationwithObserversinLabVIEW

ObserversareanalternativetotheKalmanFilter.AnObserverisanalgorithmforestimatingthestatevariablesinasystembasedonamodelofthesystem.ObservershavethesamestructureasaKalmanFilter.

InObserversyouspecifyhowfastandstableyouwanttheestimatestoconvergetotherealvalues,i.e.,youspecifytheeigenvaluesofthesystem.BasedontheeigenvaluesyouwillfindtheObservergainKthatisusedtoupdatetheestimates.

OnesimplewaytofindtheeigenvaluesistousetheButterwortheigenvaluesfromtheButterworthpolynomial.WhenwehavefoundtheeigenvalueswecanthenusetheAckermaninordertofindtheObservergain.

LabVIEWControlDesignandSimulationModulehavelotsoffunctionalityforStateEstimationusingObservers.Thefunctionalitywillbeexplainedindetailinthenextchapters.

8.1 State-SpacemodelGiventhecontinuouslinearstatespace-model:



Orgiventhediscretelinearstatespace-model





LabVIEW:

InLabVIEWwemayusethe“CDConstructState-SpaceModel.vi”tocreateaState-spacemodel:

Note!IfyouspecifyadiscreteState-spacemodelyouhavetospecifytheSamplingTime.

LabVIEWExample:CreateaState-spacemodel

BlockDiagram:

Thematrices𝐴, 𝐵, 𝐶 and 𝐷 maybedefinedontheFrontPanellikethis:

[EndofExample]



8.2 EigenvaluesOnesimplewaytofindtheeigenvaluesistousetheButterwortheigenvaluesfromtheButterworthpolynomial.

ButterwortPolynomial:

TheButterworthPolynomialisdefinedas:

𝐵I 𝑠 = 𝑎I𝑠I +…+ 𝑎8𝑠8 + 𝑎7𝑠 + 1

where 𝑎6 = 1, 𝑎7, 𝑎8, … , 𝑎I arethecoefficientsintheButterworthPolynomial.

Herewewillusea2.orderButterworthPolynomial,whichisdefinedas:

𝐵8 𝑠 = 𝑎8𝑠8 + 𝑎7𝑠 + 1

where 𝑎6 = 1, 𝑎7 = 2𝑇, 𝑎8 = 𝑇8.

Thisgives:

𝐵8 𝑠 = 𝑇8𝑠8 + 2𝑇𝑠 + 1

wheretheparameter 𝑇 isusedtodefinedthespeedoftheresponseaccordingto:

𝑇e ≈ 𝑛𝑇

where 𝑇e isdefinedastheObserverresponsetimewherethestepresponsereach63%ofthesteadystatevalueoftheresponse.

→Sowewilluse 𝑇e asthetuningparameterfortheObserver.

LabVIEW:

InLabVIEWwecanusethe“PolynomialRoots.vi”tofindtherootsbasedontheButterworthPolynomial

LabVIEWFunctionsPalette:Mathematics→Polynomial→PolynomialRoots.vi



BelowweseetheMathematicsandthePolynomialpalettesinLabVIEW.

MathScript:

InMathScriptwecanusetherootsfunctioninordertofindtheeigenvaluesbasedonagivenpolynomial.

8.3 ObserverGainLabVIEW:

InLabVIEWwecanusethe“CDAckerman.vi”tofindtheObservergainbasedonsomegiveneigenvalues(foundfromtheButterwortPolynomial).

LabVIEWFunctionsPalette:ControlDesign&Simulation→ControlDesign→StateFeedbackDesign→CDAckerman.vi



MathScript:

InMathScriptwecanusetheackerfunctioninordertofindtheObservergainbasedonsomegiveneigenvalues(foundfromtheButterwortPolynomial).

8.4 ObservabilityAnecessaryconditionfortheObservertoworkcorrectlyisthatthesystemforwhichthestatesaretobeestimated,isobservable.Therefore,youshouldcheckforObservabilitybeforeapplyingtheObserver.

TheObservabilitymatrixisdefinedas:

𝑂 =𝐶𝐶𝐴⋮

𝐶𝐴I>7

Wherenisthesystemorder(numberofstatesintheState-spacemodel).

→Asystemofordernisobservableif 𝑶 isfullrank,meaningtherankof 𝑶 isequalton.

LabVIEW:

TheLabVIEWControlDesignandSimulationModulehaveaVI(ObservabilityMatrix.vi)forfindingtheObservabilitymatrixandcheckifastates-pacemodelisObservable.

LabVIEWFunctionsPalette:ControlDesign&Simulation→ControlDesign→State-SpaceModelAnalysis→CDObservabilityMatrix.vi



Note!InLabVIEW 𝑁 isusedasasymbolfortheObservabilitymatrix.

LabVIEWExample:CheckforObservability

[EndofExample]

MathScript:

InMathScriptyoumayusetheobsvmxfunctiontofindtheObservabilitymatrix.YoumaythenusetherankfunctioninordertofindtherankoftheObservabilitymatrix.

MathScriptExample:

ThefollowingMathScriptCodecheckforObservability:

% Check for Observability: O = obsvmx (discretemodel) r = rank(O)

[EndofExample]

8.5 ExamplesHerewewillshowimplementationsofanObserverinLabVIEWandMathScript.


𝑥7𝑥8

= 0 −100 0

N

𝑥7𝑥8 + 0.02

0O

𝑢

𝑦 = 1 0P

𝑥7𝑥8 + 0

k𝑢




LabVIEWExample:ObserverGain

BelowweseetheBlockDiagraminLabVIEWforcalculatingtheObservergain:

Weusedthe“PolynomialRoots.vi”inordertofindthepolesasspecifiedintheButterworthPolynomial.

Weusea2.orderButterworthPolynomial:

𝐵8 𝑠 = 𝑇8𝑠8 + 2𝑇𝑠 + 1

wheretheparameter 𝑇 isusedtodefinedthespeedoftheresponseaccordingto:

𝑇e ≈ 𝑛𝑇 ↔ 𝑇 =𝑇e𝑛

Intheexampleweset 𝑇e = 2𝑠 and 𝑛 = 2 intheexample.

Thisgives:

𝐵8 𝑠 = 𝑠8 + 1.41𝑠 + 1

Sothecoefficientsinthepolynomialareasfollows:

Thenwehaveusedthe“CDAkerman.vi”tofindtheObservergain.

Theresultbecomes:



[EndofExample]

LabVIEWExample:ObserverEstimator

LabVIEWhaveseveralbuilt-inObserverfunctions,e.g.,the“CDContinuousObserver.vi”wewilluseinthisexample.BelowweseetheBlockDiagramfortheObserver:

Theresultisasfollows:



[EndofExample]

MathScriptExample:ObserverGain

HerewewilluseMathScriptinordertofindtheObservergainforthesamesystemasabove.

TheCodeisasfollows:

% Define the State-space model: A = [0 -10; 0 0]; B = [0.02; 0]; C = [1 0]; D=[0]; ssmodel = ss(A, B, C,D); % Check for Observability: O = obsvmx (ssmodel); r = rank(O); %Butterwort Polynomial: B2=[1, 1.41, 1]; p=roots(B2); % Find Observer Gain K = ackermann(ssmodel, p, 'L')

Theresultis:



K = 1.41 -0.1

[EndofExample]

64

9 OverviewofObserverfunctionsObserversareverysimilartoKalmanfilters.Inobserverstheestimatorgainiscalculatedfromspecifiedeigenvaluesorpolesoftheestimatorerrordynamics(inotherwords:howfastyouwanttheestimationerrortoconvergetorealstates).

InLabVIEWthereareseveralVIsandfunctionsusedforObserverimplementations.

9.1 ControlDesignpaletteInthe“ControlDesign”palettewefindsubpalettesfor“StateFeedbackDesign”and“Implementation”:

9.1.1 StateFeedbackDesignsubpalette

Inthe“StateFeedbackDesign”subpalettewefindVIsforcalculationtheObserverGain,etc.

65 OverviewofObserverfunctions


→UsetheStateFeedbackDesignVIstocalculatecontrollerandobservergainsforclosed-loopstatefeedbackcontrolortoestimateastate-spacemodel.YoualsocanuseStateFeedbackDesignVIstoconfigureandteststate-spacecontrollersandstateestimatorsintimedomains.

AckermannVI:

9.1.2 Implementationsubpalette

Inthe“Implementation”subpalettewefindVIsforimplementingadiscreteObserverandadiscreteKalmanFilter.

→UsetheImplementationVIsandfunctionstosimulatethedynamicresponseofadiscretesystemmodel,deployadiscretemodeltoareal-timetarget,implementadiscreteKalmanfilter,andimplementcurrentandpredictiveobservers.

DiscreteObserver:



9.2 SimulationpaletteInthe“Simulation”palettewefindthe“Estimation“sub-palette:

9.2.1 Estimationsubpalette

Inthe“Estimation”palettewefindVIsforimplementingcontinuous/discreteObserversandKalmanFilter.



→UsetheEstimationfunctionstoestimatethestatesofastate-spacesystem.Thestate-spacesystemcanbedeterministicorstochastic,continuousordiscrete,linearornonlinear,andcompletelyorpartiallyobservable.

ContinuousObserverVI:

DiscreteObserverVI:

68

PartIII:SystemIdentification

69

10 SystemIdentificationinLabVIEWThemodelcanbeinformofdifferentialequationsdevelopedfromphysicalprinciplesorfromtransferfunctionmodels,whichcanberegardedas“black-box”-modelswhichexpressestheinput-outputpropertyofthesystem.Someoftheparametersofthemodelcanhaveunknownoruncertainvalues,forexampleaheattransfercoefficientinathermalprocessorthetime-constantinatransferfunctionmodel.Wecantrytoestimatesuchparametersfrommeasurementstakenduringexperimentsonthesystem.

Herewewilldiscuss:

• ParameterEstimationandtheLeastSquareMethod(LS)• Sub-spacemethods/Black-Boxmethods• PolynomialModelEstimation:ARX/ARMAXmodelEstimation

InLabVIEWwecanusethe“SystemIdentificationPalette”.

The“SystemIdentification”paletteinLabVIEW:

InthenextchapterswewillusethedifferentfunctionalityavailableintheSystemIdentificationToolkit.

70 SystemIdentificationinLabVIEW


10.1 ParameterEstimationwithLeastSquareMethod(LS)

ParameterEstimationusingtheLeastSquareMethod(LS)isusedtofindamodelwithunknownphysicalparametersinamathematicalmodel.

TheLeastsquaremethodcanbewrittenas:

𝑌 = Φ𝜃

Where

𝜽 istheunknownparametervector

𝑌 istheknownmeasurementvector

Φ istheknownregressionmatrix

Thesolutionfor 𝜃 maybefoundas:

𝜃 = Φ>7𝑌

Itcanbefoundthattheleastsquaresolutionfor 𝑌 = Φ𝑌 is:

𝜃st = (ΦuΦ)>7ΦuY

ImplementationinMathScript/MATLAB:

theta=inv(phi’*phi)* phi’*Y

orsimply:

theta=phi\Y

InLabVIEWwecanusetheblocks(“AxB.vi”,“TransposeMatrix.vi”,“InverseMatrix.vi”)inthe“LinearAlgebra”(locatedintheMathematicspalette)paletteinordertofintheLeastSquaresolution:



Wecanalsousethe“SolveLinearEquations.vi”:

Example:

Giventhefollowingmodel:

𝑦 𝑢 = 𝑎𝑢 + 𝑏

Thefollowingvaluesarefoundfromexperiments:

𝑦 1 = 0.8

𝑦 2 = 3.0

𝑦 3 = 4.0

Wewillfindtheunknowns 𝑎 and 𝑏 usingtheLeastSquare(LS)methodinMathScript/LabVIEW.

Wehavethat:

𝑌 = Φ𝜃

Where

𝜃 istheunknownparametervector

𝑌 istheknownmeasurementvector



Φ istheknownregressionmatrix

Thesolutionfor 𝜃 maybefoundas(if Φ isaquadraticmatrix):

𝜃 = Φ>7𝑌

Itcanbefoundthattheleastsquaresolutionfor 𝑌 = Φ𝜃 is:


Weget:

0.8 = 𝑎 ∙ 1 + 𝑏

3.0 = 𝑎 ∙ 2 + 𝑏

4.0 = 𝑎 ∙ 3 + 𝑏

Thisbecomes:

0.83.04.0z

=1 12 13 1{

𝑎𝑏|

MathScript:

Wedefine 𝑌 and Φ inMathScriptandfind 𝜃 by:

phi = [1 1; 2 1; 3 1]; Y = [0.8 3.0 4.0]'; theta = inv(phi'*phi)* phi'*Y %or simply by theta=phi\Y

Theanswerbecomes:

theta = 1.6

-0.6

i.e.:

𝑎 = 1.6

𝑏 = −0.6

Thisgives:



𝑦 𝑢 = 1.6𝑢 − 0.6

LabVIEW:

BlockDiagram:

FrontPanel:

Wecanalsousethe“SolveLinearEquations.vi”directly:

[EndofExample]

10.2 SystemIdentificationusingSub-spacemethods/Black-Boxmethods

Sub-spacemethods/Black-Boxmethodsisusedtofindamodelwithnon-physicalparameters.

Asub-spacemethods/Black-BoxmethodestimatesalineardiscreteState-spacemodelontheform:





LabVIEWoffersfunctionalityforthis.Inthe“ParametricModelEstimation”palettewefindthe“SIEstimateState-SpaceModel.vi”whichcanbeusedforsub-spaceidentification.

“ParametricModelEstimation”palette:

ThisVIestimatestheparametersofastate-spacemodelforanunknownsystem.

10.3 SystemIdentificationusingPolynomialModelEstimation:ARX/ARMAXmodelEstimation

LabVIEWoffersVIsforARX/ARMAXmodelestimationinthe“ParametricModelEstimation”palette.



ForARXmodelswecanuse“SIEstimateARXmodel”:

ForARMAXmodelswecanuse“SIEstimateARMAXmodel”:

10.4 GeneratemodelDataInordertofindamodelweneedtogeneratedatabasedontherealprocess.Thestimulus(exitation)signalandtheresponsesignalwillthenbeinputtothefunctions/VIs(algorithms)inLabVIEWthatyouwillusetomodelyourprocess.

BelowweexplainhowwedothisinLabVIEW.

Datalogging:



UseLabVIEWforexcitingtheprocessandloggingsignals.Useopen-loopexperiments(nofeedbackcontrolsystem).YoucanusetheWritetoMeasurementFilefunctionontheFileI/OpaletteinLabVIEWforwritingdatatotextfiles(usetheLVMdatafileformat,nottheTDMSfileformatwhichgivebinaryfiles).

IntheFileI/OpaletteinLabVIEWwehavelotsoffunctionalityforwritingandreadingfiles.

Belowweseethe“FileI/O”paletteinLabVIEW:

InthisTutorialwewillfocusonthe“WriteToMeasurementFile”and“ReadFromMeasurementFile”.

The“WriteToMeasurementFile”and“ReadFromMeasurementFile”isso-called“ExpressVIs”.WhenyoudragtheseVI’stotheBlockDiagram,aconfigurationdialogpops-upimmediately,likethis:



Inthisconfigurationdialogyousetfilename,filetype,etc.

Notethatthese“ExpressVIs”havenoBlockDiagram.

10.4.1 Excitationsignals

Itisimportanttohaveagoodexcitationsignal,youcanusedifferentexcitationsignals,suchas:

• APRBSsignal(PseudoRandomBinarySignal) • AChirpSignal• AUp-downsignal

LabVIEW:

InLabVIEWyoucanusesomeofthefunctionsintheSignalGenerationpalette:



LabVIEWFunctionsPalette:SignalProcessing→SignalGeneration

PRBSSignalAPRBSsignallookslikethis:

LabVIEW:

InLabVIEWyoucanusethe“SIGeneratePseudo-RandomBinarySequence.vi”function.

LabVIEWFunctionsPalette:ControlDesign&Simulation→Systemidentification→Utilities→SIGeneratePseudo-RandomBinarySequence.vi



ChirpSignalAChirpsignallookslikethis:

LabVIEW:

InLabVIEWyoucanusethe“ChirpPattern.vi”function.

LabVIEWFunctionsPalette:SignalProcessing→SignalGeneration→ChirpPattern.vi

80

11 OverviewofSystemIdentificationfunctions

InLabVIEWwecanusetheSystemIdentificationToolkit.

The“SystemIdentification”paletteinLabVIEW:

→UsetheSystemIdentificationVIstocreateandestimatemathematicalmodelsofdynamicsystems.YoucanusetheVIstoestimateaccuratemodelsofsystemsbasedonobservedinput-outputdata.

The“SystemIdentification”paletteinLabVIEWhasthefollowingsubpalettes:

Icon Name Description

Preprocessing

DataPreprocessing

UsetheDataPreprocessingVIstopreprocesstherawdatathatyouacquiredfromanunknownsystem.



Parametric

ParametricModelEstimation

UsetheParametricModelEstimationVIstoestimateaparametricmathematicalmodelforanunknown,linear,time-invariantsystem.

Frequency

Frequency-DomainModelEstimation

UsetheFrequency-DomainModelEstimationVIstoestimatethefrequencyresponsefunction(FRF)andtoidentifyatransferfunction(TF)orastate-space(SS)modelofanunknownsystem.

Grey-Box

PartiallyKnownModelEstimation

UsethePartiallyKnownModelEstimationVIstocreateandestimatepartiallyknownmodelsfortheplantinasystem.

Recursive

RecursiveModelEstimation

UsetheRecursiveModelEstimationVIstorecursivelyestimatetheparametricmathematicalmodelforanunknownsystem.

Nonparametric

NonparametricModelEstimation

UsetheNonparametricModelEstimationVIstoestimatetheimpulseresponseorfrequencyresponseofanunknown,linear,time-invariantsystemfromaninputandcorrespondingoutputsignal.

Validation

ModelValidation

UsetheModelValidationVIstoanalyzeandvalidateasystemmodel.

Analysis

ModelAnalysis

UsetheModelAnalysisVIstoperformaBode,Nyquist,orpole-zeroanalysisofasystemmodelandtocomputethestandarddeviationoftheresults.

Conversion

ModelConversion

UsetheModelConversionVIstoconvertmodelscreatedintheLabVIEWSystemIdentificationToolkitintomodelsyoucanusewiththeLabVIEWControlDesignandSimulationModule.YoucanconvertanAR,ARX,ARMAX,output-error,Box-Jenkins,general-linear,orstate-spacemodelintoatransferfunction,zero-pole-gain,orstate-spacemodel.Youalsocanconverta



continuousmodeltoadiscretemodelorconvertadiscretemodeltoacontinuousmodel.

Management

ModelManagement

UsetheModelManagementVIstoaccessinformationaboutthesystemmodel.Modelinformationincludespropertiessuchasthesystemtype,samplingrate,systemdimensions,noisecovariance,andsoon.

Utilities

Utilities

UsetheUtilitiesVIstoperformmiscellaneoustasksondataorthesystemmodel,includingproducingdatasamples,displayingmodelequations,mergingmodels,andsoon.

The“DataPreprocessing”paletteinLabVIEW:

Someimportantfunctionsinthe“DataPreprocessing”paletteare:



The“ParametricModelEstimation”paletteinLabVIEW:

Someimportantfunctionsinthe“ParametricModelEstimation”paletteare:

The“ParametricModelEstimation”paletteinLabVIEWhassubpalettefor“PolynomialModelEstimation”:



→UsethePolynomialModelEstimationVIstoestimateanAR,ARX,ARMAX,Box-Jenkins,oroutput-errormodelforanunknown,linear,time-invariantsystem.

Someimportantfunctionsinthe“PolynomialModelEstimation”paletteare:

85

12 SystemIdentificationExampleWewanttoidentifythemodelofagivensystem.

Wehavefoundthemodeltobe:

𝑥 = −1𝑇𝑥 + 𝐾𝑢(𝑡 − 𝜏)

where

𝑇 isthetimeconstant

𝐾 isthesystemgain,e.g.pumpgain

𝜏 isthetime-delay

→Wewanttofindthemodelparameters 𝑇, 𝐾, 𝜏 usingtheLeastSquaremethod.WewilluseLabVIEWandMathScript.

Setthesystemontheform 𝒚 = 𝛗𝜽

Solutions:

Weget:

𝑥�= 𝑥 𝑢(𝑡 − 𝜏)

�

−1𝑇𝐾|

i.e.

𝜃 = −1𝑇𝐾

Inordertofind 𝜃 usingtheLeastSquaremethodweneedtologinputandoutputdata.Thismeansweneedtodiscretizethesystem.

WeuseasimpleEulerforwardmethod:

𝑥 ≈𝑥JK7 − 𝑥J

𝑇@

𝑇@ isthesamplingtime.

86 SystemIdentificationExample


Thisgives:

𝑥JK7 − 𝑥J𝑇@�

= 𝑥J 𝑢J> ?�̀

�

−1𝑇𝐾|

Let’sassume 𝜏 = 3𝑠 (whichcanbefoundfromasimplestepresponseontherealsystem),wethenget(withsamplingtime 𝑇@ = 0.1):


= 𝑥J 𝑢J>�6�

−1𝑇𝐾|

Note!In3secondswelog30pointswithdatausingsamplingtime 𝑻𝒔 = 𝟎. 𝟏!!!

Giventhefollowingloggingdata(thedataisjustforillustrationandnotrealistic):

𝒌 𝒖 𝒚

1 0.9 3

2 1.0 4

3 1.1 5

4 1.2 6

5 1.3 7

6 1.4 8

7 1.5 9

Weusethefollowingsamplingtime: 𝑻𝒔 = 𝟏𝒔

Fromasimplestepresponse,wehavefoundthetime-delaytobe: 𝜏 = 3𝑠.

→Setthesystemontheform 𝒀 = 𝚽𝛉

Solutions:

Withtime-delay 𝜏 = 3𝑠and 𝑇@ = 1𝑠 weget:




= 𝑥J 𝑢J>��

−1𝑇𝐾|

Usingthegivendatasetwecansetontheform 𝑌 = Φθ:

7 − 68 − 79 − 8

=6 0.97 1.08 1.1

−1𝑇𝐾

i.e.:

111z

=6 0.97 1.08 1.1

{

−1𝑇𝐾|

Note!Weneedtomakesurethedimensionsarecorrect.

→Wefindthemodelparameters(𝛉)usingMathScript

MathScriptgives:

clear, clc Y = [1, 1, 1]'; phi = [6, 0.9; 7, 1.0; 8, 1.1]; theta = phi\Y %or theta = inv(phi'*phi)*phi'*Y T = -1/theta(1) K = theta(2)

MathScriptrespondswiththefollowinganswers:

theta =

-0.3333

3.3333

T =

3

K =

3.3333



i.e.,themodelparametersbecome:

𝑇 = 3, 𝐾 =103, 𝜏 = 3

Whichgivesthefollowingmodell:

𝑥 = −1𝑇𝑥 + 𝐾𝑢(𝑡 − 𝜏)

Withvalues:

𝑥 = −13𝑥 +

103(𝑡 − 3)

ImplementthemodelinLabVIEW.Use 𝑻 = 𝟓, 𝑲 = 𝟐, 𝝉 = 𝟑 andsimulatethesystem.Plotthestepresponseforthesystem.

Model:

𝑥 = −1𝑇𝑥 + 𝐾𝑢(𝑡 − 𝜏)

Where 𝑇 = 5, 𝐾 = 2, 𝜏 = 3

Solutions:

Weimplementthemodelusinga“SimulationSubsystem”andusetheavailableblocksintheControlandDesignModule.

Themodelmaybeimplementedasfollows:



Wesimulatethesystemusingthefollowingprogram:

BlockDiagram:

FrontPanel:

Wedoasimplestepresponse:



Findthetransferfunctionforthesystem:

𝐻 𝑠 =𝑥(𝑠)𝑢(𝑠)

Model:

𝑥 = −1𝑇𝑥 + 𝐾𝑢(𝑡 − 𝜏)

PlotthestepresponseforthetransferfunctioninMathScript.Compareanddiscusstheresultsfromprevioustask.

Solutions:

Weusethedifferentialequation:

𝑥 = −1𝑇𝑥 + 𝐾𝑢(𝑡 − 𝜏)

Laplacetransformationgives:



𝑠𝑥(𝑠) = −1𝑇𝑥(𝑠) + 𝐾𝑢(𝑠)𝑒>?@

Note!WeusethefollowingLaplacetransformation:

𝐹 𝑠 𝑒>?@ ⟺ 𝑓(𝑡 − 𝜏)

𝑠𝐹(𝑠) ⟺ 𝑓(𝑡)

Thenweget:

𝑠𝑥 𝑠 +1𝑇𝑥 𝑠 = 𝐾𝑢(𝑠)𝑒>?@

and:

𝑥 𝑠 𝑠 +1𝑇

= 𝐾𝑢(𝑠)𝑒>?@

and:

𝑥 𝑠𝑢(𝑠)

=𝐾

𝑠 + 1𝑇𝑒>?@

Finally:

𝐻 𝑠 =𝑥 𝑠𝑢(𝑠)

=𝐾𝑇

𝑇𝑠 + 1𝑒>?@ =

𝐾��𝑇𝑠 + 1

𝑒>?@

Withvalues(𝑇 = 5, 𝐾 = 2, 𝜏 = 3):

𝐻 𝑠 =𝑥 𝑠𝑢(𝑠)

=10

5𝑠 + 1𝑒>�@

MathScript:

WeimplementthetransferfunctioninMathScriptandperformastepresponse.Weusethestep()function.

clear, clc s=tf('s'); K=2; T=5; H1=tf(K*T/(T*s+1)); delay=3; H2=set(H1,'inputdelay',delay); step(H2)



% You may also use: figure(2) H = sys_order1(K*T, T, delay) step(H)

Thestepresponsebecomes:

→Weseethatthestepresponseisthesameasintheprevioustask.

Loginputandoutputdatabasedonthemodel.

Model:

𝑥 = −1𝑇𝑥 + 𝐾𝑢(𝑡 − 𝜏)

Where 𝑇 = 5, 𝐾 = 2, 𝜏 = 3

Solutions:

Wesavethedatausing“WriteToMeasurementFile”inLabVIEW.

Basedonasimplestepresponsewecanfindthetime-delay 𝜏.

Weusethesameapplicationasinaprevioustask:



Findthemodelparameters (𝑻 og 𝑲)usingLeastSquareinLabVIEWbasedontheloggeddata.

Note!Theanswersshouldbe 𝑇 ≈ 5 and 𝐾 ≈ 2.

Model:

𝑥 = −1𝑇𝑥 + 𝐾𝑢(𝑡 − 𝜏)

Where 𝑇 = 5, 𝐾 = 2, 𝜏 = 3

Solutions:

Itisagoodideatosplityourprogramintodifferentlogicalpartsusing,e.g.,SubVIsinLabVIEW.

Thedifferentparts/stepscould,e.g.,be:

1. GetLoggedDatafromFilea. Input:FileNameb. Outputs: 𝑢 and 𝑦 (𝑇��)

2. Transformthedataandstackdatainto 𝑌 and Φ



a. Inputs: 𝑢 and 𝑦 (𝑇��)b. Outputs: 𝑌 and Φ

3. FindtheLeastSquaresolution 𝜃st = (ΦuΦ)>7ΦuYa. Input: 𝑌 and Φb. Output: 𝜃(𝜃�, 𝐾�)

LabVIEWcode:

BlockDiagram:

FrontPanel:



→Asyouseetheresultis 𝑇 = 5 and 𝐾 = 2 (asexpected).

ThedifferentSubVI’sdothefollowing:

1.“OpenMeasurementDatafromFile.vi”

ThisSubVIopenstheloggeddatafromfile(createdinaprevioustask).

BlockDiagram:



2.“CreateLSMatricesfromLoggedDatawithTimeDelay.vi”

ThisSubVI“stack”dataontheform:


= 𝑥J 𝑢J> ?�̀

�

−1𝑇𝐾|

BlockDiagram:

3.“FindLSSolution.vi”

ThisSubVIfindtheLSsolution:


BlockDiagram:

system identification and estimation in labview · 6 introduction to labview and mathscript...

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