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1、TheInvertedPendulumSystemTheinvertedpendulumsystemisapopulardemonstrationofusingfeedbackcontroltostabilizeanopen-loopunstablesystem.ThefirstsolutiontothisproblemwasdescribedbyRoberge[1]inhisaptlynamedthesis,TheMechanicalSeal."Subsequently,ithasbeenusedinmanybooks
2、andpapersasanexampleofanunstablesystem.Siebert[2,pages177{182]doesacompleteanalysisofthissystemusingtheRouthCriterion,bymultiplyingoutthecharacteristicequationasapolynomialofsandstudyingthecoefficients.Althoughcorrect,thisapproachisunnecessarilyabstruse.Thissystem
3、istheidealroot-locusanalysisexample.Figure1:GeometryoftheinvertedpendulumsystemConsidertheinvertedpendulumsysteminFigure1.Atapendulumangleoffromvertical,gravityproducesanangularaccelerationequalto,andacartaccelerationofproducesanangularaccelerationofWritingtheseac
4、celerationsasanequationofmotion,linearizingit,andtakingitsLaplaceTransform,weproducetheplanttransferfunctionG(s),asfollows:wherethetimeconstantisdefinedasThistransferfunctionhasapoleintherighthalf-plane,whichisconsistentwithourexpectationofanunstablesystem.Westart
5、thefeedbackdesignbydrivingthecartwithamotorwithtransferfunctionM(s)anddrivingthemotorwithavoltageproportionaltotheangle.IncludingthefamiliarmotortransferfunctionFigure2:Root-locusplotofpendulumandmotor,L(s)=M(s)G(s)withtheplantG(s),wegetarootlocuswithonepolethatst
6、aysintherighthalf-plane.Usingnormalizednumbers,wegettheroot-locusplotasisseeninFigure2.Inordertostabilizethesystem,weneedtogetridoftheremainingzeroattheoriginsothatthelocusfromtheplantpoleonthepositiverealaxismovesintothelefthalf-plane.Thusourcompensatormustinclud
7、eapoleattheorigin.However,weshouldbalancetheaddedcompensatorpolewithanaddedzero,sothatthenumberofpoleslessthenumberofzerosremainsequaltotwo,leavingtheroot-locusasymptotesat(otherwise,theasymptoteswouldbeandwhicheventuallyleadthepolesintotherighthalf-plane).Thusweu
8、seacompensatorandweassumethat¿M<¿K<¿L.TheblockdiagramofthesystemisshowninFigure3,andtheroot-locusplotbecomesasinFigure4(notethatsincethereisaninversioni
