control of an inverted pendulum外文文献本科论文.docx

control of an inverted pendulum外文文献本科论文.docx

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controlofaninvertedpendulum外文文献本科论文

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.Ina