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1、TheInvertedPendulumSystemTheinvertedpendulumsystemisapopulardemonstrationofusingfeedbackcontroltostabilizeanopen-loopunstablesystem.ThefirstsolutiontothisproblemwasdescribedbyRoberge[1]inhisaptlynamedthesis,TheMechanicalSeal."Subsequently,ithasbeenusedinmanybooksandpapersas
2、anexampleofanunstablesystem.Siebert[2,pages177{182]doesacompleteanalysisofthissystemusingtheRouthCriterion,bymultiplyingoutthecharacteristicequationasapolynomialofsandstudyingthecoefficients.Althoughcorrect,thisapproachisunnecessarilyabstruse.Thissystemistheidealroot-locusan
3、alysisexample.Figure1:GeometryoftheinvertedpendulumsystemConsidertheinvertedpendulumsysteminFigure1.Atapendulumangleoffromvertical,gravityproducesanangularaccelerationequalto,andacartaccelerationofproducesanangularaccelerationofWritingtheseaccelerationsasanequationofmotion,l
4、inearizingit,andtakingitsLaplaceTransform,weproducetheplanttransferfunctionG(s),asfollows:wherethetimeconstantisdefinedasThistransferfunctionhasapoleintherighthalf-plane,whichisconsistentwithourexpectationofanunstablesystem.Westartthefeedbackdesignbydrivingthecartwithamotorw
5、ithtransferfunctionM(s)anddrivingthemotorwithavoltageproportionaltotheangle.IncludingthefamiliarmotortransferfunctionFigure2:Root-locusplotofpendulumandmotor,L(s)=M(s)G(s)withtheplantG(s),wegetarootlocuswithonepolethatstaysintherighthalf-plane.Usingnormalizednumbers,wegetthe
6、root-locusplotasisseeninFigure2.Inordertostabilizethesystem,weneedtogetridoftheremainingzeroattheoriginsothatthelocusfromtheplantpoleonthepositiverealaxismovesintothelefthalf-plane.Thusourcompensatormustincludeapoleattheorigin.However,weshouldbalancetheaddedcompensatorpolewi
7、thanaddedzero,sothatthenumberofpoleslessthenumberofzerosremainsequaltotwo,leavingtheroot-locusasymptotesat(otherwise,theasymptoteswouldbeandwhicheventuallyleadthepolesintotherighthalf-plane).Thusweuseacompensatorandweassumethat¿M<¿K<¿L.TheblockdiagramofthesystemisshowninFigu
8、re3,andtheroot-locusplotbecomesasinFigure4(notethatsincethereisaninversioni
