heuprightposition.Thesecondpartinvolvesthedesignofacontrollerthatswingsthependulumuptotheunstableequilibrium.Whenthependulumapproachesthelinearizedpoint,thecontrolwillswitchtothestabilizingcontrollerwhichwillbalancethependulumaroundtheuprightposition.ThestatefeedbackcontrollerresponsibleforbalancingthependulumintheuprightpositionisbasedonaLinearQuadraticRegulator(LQR)designusingthelinearizedsystematurallyreturntothisstate.Thestableequilibriumrequiresnocontrolinputtobeachievedand,thus,isuninterestingfromacontrolperspective.Theunstableequilibriumcorrespondstoastateinwhichthependulumpointsstrictlyupwardsand,thus,requiresacontrolforcetomaintainthisposition.Thebasiccontrolobjectiveoftheinvertedpendulumproblemistomaintaintheunstableequilibriumpositionwhenthependuluminitiallystartsinanupright第页position.Thecontrolobjectiveforthisprojectwillfocusonstartingfromthestableequilibriumposition(pendulumpointingdown),swingingituptotheunstableequilibriumposition(pendulumupright),andmaintainingthisstate..MODELLINGAschematicoftheinvertedpendulumisshowninFigure.Figure.InvertedPendulumSetupAcartequippedwithamotorprovideshorizontalmotionofthecartwhilecartposition,p,andjointangle,θ,measurementsaretakenviaaquadratureencoder.Byapplyingthelawofdynamicsontheinvertedpendulumsystem,theequationsofmotionarewheremcisthecartmass,mpisthependulummass,Iistherotationalinertia,listhehalf-lengthofthependulum,Risthemotorarmatureresistance,risthemotorpinionradius,Kmisthemotortorqueconstant,andKgisthegearboxratio.Also,forsimplicity,andnotethattherelationshipbetweenforce,F,andvoltage,V,forthemotoris:第页Letthestatevectorbedefinedas:Finally,welinearizethesystemabouttheunstableequilibrium()T.Notethatθ=correspondstothependulumbeingintheuprightposition.Thelinearizationofthecart-pendulumsystemaroundtheuprightpositionis:WhereFinally,bysubstitutingtheparametervaluesthatcorrespondtotheexperimentalsetup:第页.STABILIZINGCONTROLLERDESIGNThecontrollerdesignapproachforthisprojectisbrokenupintotwocomponents.Thefirstpartinvolvesthedesignofanoptimalstatefeedbackcontrollerforthelinearizedmodelthatwillstabilizethependulumaroundtheuprightposition.Thesecondpartinvolvesthedesignofacontrollerthatswingsthependulumuptotheunstableequilibrium.Whenthependulumapproachesthelinearizedpoint,thecontrolwillswitchtothestabilizingcontrollerwhichwillbalancethependulumaroundtheuprightposition.ThestatefeedbackcontrollerresponsibleforbalancingthependulumintheuprightpositionisbasedonaLinearQuadraticRegulator(LQR)designusingthelinearizedsystem然需要实现决定电压量应用到车电机预期能量状态。该参数V在satV价值决定了可用最大数量控制信号，从而增加最大能源量摆系统。K值决定了有利于最大限度地控制投入使用以达到预期能量状态。这种控制有效地增加钟摆能量到预期值。当作为摆起控制方法，所需值与在其竖直位置摆能量相对应。这将触发开关，使稳定控制器可用于捕捉摆，让其在各不稳定地点趋于平衡点。实验结果最后收集了两个摆起控制方法数据，这些数据是在实验开始后，当摆从最初位置向下到竖直平衡位置然后在不稳定平衡位置附近摆动过程中收集。当电压增益为.伏时，启发式控制器能够更好让摆摆动起来。经过多次调整电压增益，最终实验表明，该控制器能够成功地使在摆动钟摆到竖直位置，保持约时间在平衡状态直立。一个控制器输出过程中启发式控制器实验运行图见图图控制输出启发式控制器图需要注意是，摆起控制器大约需要.秒，才能达到正常位置。被稳定控制器捕捉到摆在直立位置上点清晰地显示在波形图中。此外，通过摆角，控制器输出到电机车电压在.V和-.V之间交替。从秒开始，在下面位置摆角超过时，控制输出也开始在短短一段连续时间内输出V。因此，钟摆需要另外.秒时间来达到从下面超过位置到第页范围内垂直位置。这个摆角相应情形如图所示。每次摆动摆角略有增加，直至摆接近不稳定平衡。该控制器采用个前摆是来足够接近该稳定控制器垂直位置来抓住它。被稳定控制器激活点清晰地显示在图上。此外，一旦它被激活，在平衡位置周围摆角仍相当稳定。图启发式控制器摆角图能量控制器应用设计参数k=.。同时,由于钟摆系统摩擦力和近似方程(),所需能量抵消了某个略大于值。通过实验可确定适当量。在这些实验中,抵消值可升到E=.。通过对能量控制器重复实验,该控制器至少时间是可靠。一次应用能量控制器实验控制输出结果如图所示。图能源控制器控制输出图重要是要注意到，能源控制需要大约经过秒达到达竖直位置。自从利用极大控制产量来尽快增加系统能量之后，控制产量首先在.v至-.v之间交替（在这个案例中，饱和度第页被定义为在.v）。当摆接近竖直位置时，控制输出系统就开始大幅度下降，因为控制输出是以系统所需能量与价值差为基础。作为启发式控制器，该控制器被激活稳定点在图上是可以清楚地识别。该控制器能源摆角对应图如图所示。需要注意是每个摆杆摆角略有增加。在钟摆工作之前控制器需要摆图能源控制器摆交图靠近竖直位置。人们很容易地看到，当能量控制器成功地使摆摆动到竖直位置时，稳定控制器能够赶上摆，并使其平衡。.结论当摆为了平衡在竖直位置时候，两个摆控制计划就已经开始实施，将其切换到一个稳定控制器。这两种控制器能够成功地将摆动钟摆从一开始向下位置调整到直立位置，并围绕这个平衡点摆动。比起启发式控制系统，能源控制系统更健全，能更成功更可靠地使摆动钟摆到正常位置。有数据表明，能源控制器还比启发式控制器实施更快。能源控制器另一个优点是，它最终还是能够达到竖直位置，即使它超出轨道长度并开始撞到轨道尽头墙上。启发式控制器只是在理论上能实现。另一方面，当车撞击轨道尽头时立即宣告失败了。两种摆起方法仍然需要多个波动才能达到竖直位置，同样也需要一个稳定控制器捕捉到摆竖直位置。总而言之，能源控制器比启发式控制器更容易达到不稳定平衡。已经证明，这两个控制器使摆动钟摆从垂直位置摆向向下位置是一样有效。.参考文献Astrom,K.J.andK.Furuta,“SwingingupaPendulumbyEnergyControl”,Automatica,Vol.,,ECEb/ECECourseWebpages,http://www.ccec.ece.ucsb.edu/people/smith/Eker,J,andK.J.Astrom,“ANonlinearObserverfortheInvertedPendulum”,thIEEEConferenceonControlApplication,Chung,C.C.andJ.Hauser,“NonlinearControlofaSwingingPendulum”,Automatica,Vol.,aturallyreturntothisstate.Thestableequilibriumrequiresnocontrolinputtobeachievedand,thus,isuninterestingfromacontrolperspective.Theunstableequilibriumcorrespondstoastateinwhichthependulumpointsstrictlyupwardsand,thus,requiresacontrolforcetomaintainthisposition.Thebasiccontrolobjectiveoftheinvertedpendulumproblemistomaintaintheunstableequilibriumpositionwhenthependuluminitiallystartsinanupright第页po第页中文字出处:UniversityofCalifornia,SantaBarbara,,:-ControlofanInvertedPendulumJohnnyLamAbstract:Thebalancingofaninvertedpendulumbymovingacartalongahorizontaltrackisaclassicproblemintheareaofcontrol.Thispaperwilldescribetwomethodstoswingapendulumattachedtoacartfromaninitialdownwardspositiontoanuprightpositionandmaintainthatstate.Anonlinearheuristiccontrollerandanenergycontrollerhavebeenimplementedinordertoswingthependulumtoanuprightposition.Afterthependulumisswungup,alinearquadraticregulatorstatefeedbackoptimalcontrollerhasbeenimplementedtomaintainthebalancedstate.Theheuristiccontrolleroutputsarepetitivesignalattheappropriatemomentandisfinelytunedforthespecificexperimentalsetup.Theenergycontrolleraddsanappropriateamountofenergyintothependulumsysteminordertoachieveadesiredenergystate.Theoptimalstatefeedbackcontrollerisastabilizingcontrollerbasedonamodellinearizedaroundtheuprightpositionandiseffectivewhenthecart-pendulumsystemisnearthebalancedstate.Thependulumhasbeenswungfromthedownwardspositiontotheuprightpositionusingbothmethodsandtheexperimentalresultsarereported..INTRODUCTIONTheinvertedpendulumsystemisastandardproblemintheareaofcontrolsystems.Theyareoftenusefultodemonstrateconceptsinlinearcontrolsuchasthestabilizationofunstablesystems.Sincethesystemisinherentlynonlinear,ithasalsobeenusefulinillustratingsomeoftheideasinnonlinearcontrol.Inthissystem,aninvertedpendulumisattachedtoacartequippedwithamotorthatdrivesitalongahorizontaltrack.Theuserisabletodictatethepositionandvelocityofthecartthroughthemotorandthetrackrestrictsthecarttomovementinthehorizontaldirection.Sensorsareattachedtothecartandthepivotinordertomeasurethecartpositionandpendulumjointangle,respectively.MeasurementsaretakenwithaquadratureencoderconnectedtoaMultiQ-generalpurposedataacquisitionandcontrolboard.Matlab/Simulinkisusedtoimplementthecontrollerandanalyzedata.Theinvertedpendulumsysteminherentlyhastwoequilibria,oneofwhichisstablewhiletheotherisunstable.Thestableequilibriumcorrespondstoastateinwhichthependulumispointingdownwards.Intheabsenceofanycontrolforce,thesystemwillnaturallyreturntothisstate.Thestableequilibriumrequiresnocontrolinputtobeachievedand,thus,isuninterestingfromacontrolperspective.Theunstableequilibriumcorrespondstoastateinwhichthependulumpointsstrictlyupwardsand,thus,requiresacontrolforcetomaintainthisposition.Thebasiccontrolobjectiveoftheinvertedpendulumproblemistomaintaintheunstableequilibriumpositionwhenthependuluminitiallystartsinanupright第页position.Thecontrolobjectiveforthisprojectwillfocusonstartingfromthestableequilibriumposition(pendulumpointingdown),swingingituptotheunstableequilibriumposition(pendulumupright),andmaintainingthisstate..MODELLINGAschematicoftheinvertedpendulumisshowninFigure.Figure.InvertedPendulumSetupAcartequippedwithamotorprovideshorizontalmotionofthecartwhilecartposition,p,andjointangle,θ,measurementsaretakenviaaquadratureencoder.Byapplyingthelawofdynamicsontheinvertedpendulumsystem,theequationsofmotionarewheremcisthecartmass,mpisthependulummass,Iistherotationalinertia,listhehalf-lengthofthependulum,Risthemotorarmatureresistance,risthemotorpinionradius,Kmisthemotortorqueconstant,andKgisthegearboxratio.Also,forsimplicity,andnotethattherelationshipbetweenforce,F,andvoltage,V,forthemotoris:第页Letthestatevectorbedefinedas:Finally,welinearizethesystemabouttheunstableequilibrium()T.Notethatθ=correspondstothependulumbeingintheuprightposition.Thelinearizationofthecart-pendulumsystemaroundtheuprightpositionis:WhereFinally,bysubstitutingtheparametervaluesthatcorrespondtotheexperimentalsetup:第页.STABILIZINGCONTROLLERDESIGNThecontrollerdesign        第1页中文4720字出处:UniversityofCalifornia,SantaBarbara,2004,10:24-37ControlofanInvertedPendulumJohnnyLamAbstract:Thebalancingofaninvertedpendulumbymovingacartalongahorizontaltrackisaclassicproblemintheareaofcontrol.Thispaperwilldescribetwomethodstoswingapendulumattachedtoacartfromaninitialdownwardspositiontoanuprightpositionandmaintainthatstate.Anonlinearh

文档预览结束

找同类相似 下载到手机