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1、TheInvertedPendulumSystemTheinvertedpendulumsystemisapopulardemonstrationofusingfeedbackcontroltostabilizeanopen-loopunstablesystem.ThefirstsolutiontothisproblemwasdescribedbyRoberge[1]inhisaptlynamedthesis,TheMechanicalSeal."Subsequently,ithasbeenusedinmanybooksandpapersasanex
2、ampleofanunstablesystem.Siebert[2,pages177{182]doesacompleteanalysisofthissystemusingtheRouthCriterion,bymultiplyingoutthecharacteristicequationasapolynomialofsandstudyingthecoefficients.Althoughcorrect,thisapproachisunnecessarilyabstruse.Thissystemistheidealroot-locusanalysisex
3、ample.Figure1:GeometryoftheinvertedpendulumsystemConsidertheinvertedpendulumsysteminFigure1.Atapendulumangleoffromvertical,gravityproducesanangularaccelerationequalto,andacartaccelerationofproducesanangularaccelerationofWritingtheseaccelerationsasanequationofmotion,linearizingit
4、,andtakingitsLaplaceTransform,weproducetheplanttransferfunctionG(s),asfollows:wherethetimeconstantisdefinedasThistransferfunctionhasapoleintherighthalf-plane,whichisconsistentwithourexpectationofanunstablesystem.Westartthefeedbackdesignbydrivingthecartwithamotorwithtransferfunct
5、ionM(s)anddrivingthemotorwithavoltageproportionaltotheangle.IncludingthefamiliarmotortransferfunctionFigure2:Root-locusplotofpendulumandmotor,L(s)=M(s)G(s)withtheplantG(s),wegetarootlocuswithonepolethatstaysintherighthalf-plane.Usingnormalizednumbers,wegettheroot-locusplotasisse
6、eninFigure2.Inordertostabilizethesystem,weneedtogetridoftheremainingzeroattheoriginsothatthelocusfromtheplantpoleonthepositiverealaxismovesintothelefthalf-plane.Thusourcompensatormustincludeapoleattheorigin.However,weshouldbalancetheaddedcompensatorpolewithanaddedzero,sothatthen
7、umberofpoleslessthenumberofzerosremainsequaltotwo,leavingtheroot-locusasymptotesat(otherwise,theasymptoteswouldbeandwhicheventuallyleadthepolesintotherighthalf-plane).Thusweuseacompensatorandweassumethat¿M<¿K<¿L.TheblockdiagramofthesystemisshowninFigure3,andtheroot-locusplotbeco
8、mesasinFigure4(notethatsincethereisaninversioni
