毕业设计control of an inverted pendulum外文文献.docx
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毕业设计control of an inverted pendulum外文文献.docx
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毕业设计controlofaninvertedpendulum外文文献
ControlofanInvertedPendulum
JohnnyLam
Abstract:
Thebalancingofaninvertedpendulumbymovingacartalongahorizontaltrackisaclassicproblemintheareaofcontrol.Thispaperwilldescribetwomethodstoswingapendulumattachedtoacartfromaninitialdownwardspositiontoanuprightpositionandmaintainthatstate.Anonlinearheuristiccontrollerandanenergycontrollerhavebeenimplementedinordertoswingthependulumtoanuprightposition.Afterthependulumisswungup,alinearquadraticregulatorstatefeedbackoptimalcontrollerhasbeenimplementedtomaintainthebalancedstate.Theheuristiccontrolleroutputsarepetitivesignalattheappropriatemomentandisfinelytunedforthespecificexperimentalsetup.Theenergycontrolleraddsanappropriateamountofenergyintothependulumsysteminordertoachieveadesiredenergystate.Theoptimalstatefeedbackcontrollerisastabilizingcontrollerbasedonamodellinearizedaroundtheuprightpositionandiseffectivewhenthecart-pendulumsystemisnearthebalancedstate.Thependulumhasbeenswungfromthedownwardspositiontotheuprightpositionusingbothmethodsandtheexperimentalresultsarereported.
1.INTRODUCTION
Theinvertedpendulumsystemisastandardproblemintheareaofcontrolsystems.Theyareoftenusefultodemonstrateconceptsinlinearcontrolsuchasthestabilizationofunstablesystems.Sincethesystemisinherentlynonlinear,ithasalsobeenusefulinillustratingsomeoftheideasinnonlinearcontrol.Inthissystem,aninvertedpendulumisattachedtoacartequippedwithamotorthatdrivesitalongahorizontaltrack.Theuserisabletodictatethepositionandvelocityofthecartthroughthemotorandthetrackrestrictsthecarttomovementinthehorizontaldirection.Sensorsareattachedtothecartandthepivotinordertomeasurethecartpositionandpendulumjointangle,respectively.MeasurementsaretakenwithaquadratureencoderconnectedtoaMultiQ-3generalpurposedataacquisitionandcontrolboard.Matlab/Simulinkisusedtoimplementthecontrollerandanalyzedata.
Theinvertedpendulumsysteminherentlyhastwoequilibria,oneofwhichisstablewhiletheotherisunstable.Thestableequilibriumcorrespondstoastateinwhichthependulumispointingdownwards.Intheabsenceofanycontrolforce,thesystemwillnaturallyreturntothisstate.Thestableequilibriumrequiresnocontrolinputtobeachievedand,thus,isuninterestingfromacontrolperspective.Theunstableequilibriumcorrespondstoastateinwhichthependulumpointsstrictlyupwardsand,thus,requiresacontrolforcetomaintainthisposition.Thebasiccontrolobjectiveoftheinvertedpendulumproblemistomaintaintheunstableequilibriumpositionwhenthependuluminitiallystartsinanuprightposition.Thecontrolobjectiveforthisprojectwillfocusonstartingfromthestableequilibriumposition(pendulumpointingdown),swingingituptotheunstableequilibriumposition(pendulumupright),andmaintainingthisstate.
2.MODELLING
AschematicoftheinvertedpendulumisshowninFigure1.
Figure1.InvertedPendulumSetup
Acartequippedwithamotorprovideshorizontalmotionofthecartwhilecartposition,p,andjointangle,θ,measurementsaretakenviaaquadratureencoder.
Byapplyingthelawofdynamicsontheinvertedpendulumsystem,theequationsofmotionare
wheremcisthecartmass,mpisthependulummass,Iistherotationalinertia,listhehalf-lengthofthependulum,Risthemotorarmatureresistance,risthemotorpinionradius,Kmisthemotortorqueconstant,andKgisthegearboxratio.Also,forsimplicity,
andnotethattherelationshipbetweenforce,F,andvoltage,V,forthemotoris:
Letthestatevectorbedefinedas:
Finally,welinearizethesystemabouttheunstableequilibrium(0000)T.Notethatθ=0correspondstothependulumbeingintheuprightposition.Thelinearizationofthecart-pendulumsystemaroundtheuprightpositionis:
Where
Finally,bysubstitutingtheparametervaluesthatcorrespondtotheexperimentalsetup:
3.STABILIZINGCONTROLLERDESIGN
Thecontrollerdesignapproachforthisprojectisbrokenupintotwocomponents.Thefirstpartinvolvesthedesignofanoptimalstatefeedbackcontrollerforthelinearizedmodelthatwillstabilizethependulumaroundtheuprightposition.Thesecondpartinvolvesthedesignofacontrollerthatswingsthependulumuptotheunstableequilibrium.Whenthependulumapproachesthelinearizedpoint,thecontrolwillswitchtothestabilizingcontrollerwhichwillbalancethependulumaroundtheuprightposition.
ThestatefeedbackcontrollerresponsibleforbalancingthependulumintheuprightpositionisbasedonaLinearQuadraticRegulator(LQR)designusingthelinearizedsystem.InaLQRdesign,thegainmatrixKforalinearstatefeedbackcontrollawu=-Kxisfoundbyminimizingaquadraticcostfunctionoftheform
whereQandRareweightingparametersthatpenalizecertainstatesorcontrolinputs.
Theweightingparameterschoseninthedesignoftheoptimalstatefeedbackcontrollerare:
Basedonthisdesign,thecontrollergainmatrixforthelinearizedsystemis:
ByusingthisKandthecontrollawu=-Kx,thesystemisstabilizedaroundthelinearizedpoint(pendulumupright).Sincethiscontrollawisbasedonthelinearizedsystem,thestatefeedbackoptimalcontrollerisonlyeffectivewhenthependulumisneartheuprightposition.
4.STATEESTIMATION
Fortheinvertedpendulumexperimentalsetup,notallthestatevariablesareavailableformeasurement.Infact,onlythecartposition,p,andthependulumangle,θ,aredirectlymeasured.Thismeansthatthecartvelocityandthependulumangularvelocityarenotimmediatelyavailableforuseinanycontrolschemesbeyondjuststabilization.Thus,anobserverisreliedupontosupplyaccurateestimationsofthestatesatallcart-pendulumpositions.
Alinearfullstateobservercanbeimplementedbasedonthelinearizedsystemderivedearlier.Thisobserverissimpleindesignandprovidesaccurateestimationofallthestatesaroundthelinearizedpoint.Theobserverisimplementedbyusingaduplicateofthelinearizedsystemdynamicsandaddinginacorrectiontermthatissimplyagainontheerrorintheestimates.TheobservergainmatrixisdeterminedbyanLQRdesignsimilartothatusedtodeterminethegainoftheoptimalstatefeedbackstabilizingcontroller.Inthiscase,theweightingparametersarechosentobe:
Basedonthisdesign,theobservergainmatrixis:
Sincethelinearfullstateobserverisbasedonthelinearizedsystem,itisonlyeffectiveinestimatingthestatevariableswhenthecart-pendulumsystemisneartheuprightposition.Thus,alow-passfilteredderivativeisusedtoestimatethetwounmeasuredstates,cartvelocityandpendulumangularvelocity,whenthesystemisnotclosetotheunstableequilibrium.Thismethodapproximatesthecartvelocityandpendulumangularvelocitybyusingafinitedifferenceandthenpassingitthroughalow-passfilter.Thefollowingfilterischosenforthisestimationmethod:
Theproblemswithsuchamethodarethatitintroducessomedelayandhasagainthatisslightlylessthanone.Thestateestimatesobtainedfromthefilteredderivative,however,arereasonablyaccuratefortheswing-upcontrollersimplementedinthispaper.
5.SWING-UPCONTROLLERDESIGN
Twodifferentcontrolschemeswereimplementedtoswingthependulumfromthedownwardspositiontotheuprightposition.Thefirstisaheuristiccontrollerthatprovidesaconstantvoltageintheappropriatedirectionand,thus,drivesthecartbackandforthalongthetrackrepeatedly.Itwillrepeatthisactionuntilthependulumiscloseenoughtotheuprightpositionsuchthatthestabilizingcontrollercanbetriggeredtomaintainthisbalancedstate.Thesecondschemeisanenergycontrollerthatregulatestheamountofenergyinthependulum.Thiscontrollerinputsenergyintothecart-pendulumsystemuntilitattainstheenergystatethatcorrespondstothependulumintheuprightposition.Similartotheheuristiccontrolmethod,theenergycontrolmethodwillalsoswitchtothestabilizingcontrollerwhenthependulumisclosetotheuprightposition.Theswitchthattriggersthestabilizingcontrollerinbothcasesisactivatedwhenthependulumiswithin5°oftheuprightpositionandtheangularvelocityisslowerthan2.5radianspersecond.
HeuristicController
Theheuristiccontrollerisalogic-basedcontroldesignthatdeterminesthedirectionandthemomentintimethecartshouldmovedependingonthestateofthesystem.Aspecificvoltagegainisappliedtothecartmotorbasedonresultsfromrepeatedexperimentation.Thiscontrollerwillmakethecartdriveforwardorbackwheneverthependulumcrossesthedownwardspositionanddependingonthedirectionthatthependulumisswingingwhenitreachesthedownwardsposition.
Thelogic-basedcontroldesigniscompletelydependentonthependulumangle,oneoftheavailablemeasuredstatevariables.Thecontrolschemewillchangethedirectionofthecartmovementwheneverthependulumanglecrossesthedownwardsposition.Sincethiscontroldesignisbasedsolelyonthependulumangle,thedownwardspositionistheoptimalmomentintimetoaddenergytothependulumbymovingthecartintheappropriatedirection.Thedirectionthecartmovesistheoppositesignofthependulumangleimmediatelyafteritcrossesthedownwardsposition.Whenthedirectionofthecartmovementisdetermined,aconstantvoltagegainisappliedtothecartmotorinthatsamedirectionuntilthependulumre
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