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1、TheInvertedPendulumSystemTheinvertedpendulumsystemisapopulardemonstrationofusingfeedbackcontroltostabilizeanopen-loopunstablesystem.ThefirstsolutiontothisproblemwasdescribedbyRoberge[1]inhisaptlynamedthesis,TheMechanicalSeal."Subsequently,ithasbeenusedinmanybooksandpapersasanexampleo
2、fanunstablesystem.Siebert[2,pages177{182]doesacompleteanalysisofthissystemusingtheRouthCriterion,bymultiplyingoutthecharacteristicequationasapolynomialofsandstudyingthecoefficients.Althoughcorrect,thisapproachisunnecessarilyabstruse.Thissystemistheidealroot-locusanalysisexample.Figure
3、1:GeometryoftheinvertedpendulumsystemConsidertheinvertedpendulumsysteminFigure1.Atapendulumangleoffromvertical,gravityproducesanangularaccelerationequalto,andacartaccelerationofproducesanangularaccelerationofWritingtheseaccelerationsasanequationofmotion,linearizingit,andtakingitsLapla
4、ceTransform,weproducetheplanttransferfunctionG(s),asfollows:wherethetimeconstantisdefinedasThistransferfunctionhasapoleintherighthalf-plane,whichisconsistentwithourexpectationofanunstablesystem.WestartthefeedbackdesignbydrivingthecartwithamotorwithtransferfunctionM(s)anddrivingthemoto
5、rwithavoltageproportionaltotheangle.IncludingthefamiliarmotortransferfunctionFigure2:Root-locusplotofpendulumandmotor,L(s)=M(s)G(s)withtheplantG(s),wegetarootlocuswithonepolethatstaysintherighthalf-plane.Usingnormalizednumbers,wegettheroot-locusplotasisseeninFigure2.Inordertostabilize
6、thesystem,weneedtogetridoftheremainingzeroattheoriginsothatthelocusfromtheplantpoleonthepositiverealaxismovesintothelefthalf-plane.Thusourcompensatormustincludeapoleattheorigin.However,weshouldbalancetheaddedcompensatorpolewithanaddedzero,sothatthenumberofpoleslessthenumberofzerosrema
7、insequaltotwo,leavingtheroot-locusasymptotesat(otherwise,theasymptoteswouldbeandwhicheventuallyleadthepolesintotherighthalf-plane).Thusweuseacompensatorandweassumethat¿M<¿K<¿L.TheblockdiagramofthesystemisshowninFigure3,andtheroot-locusplotbecomesasinFigure4(notethatsincethereisaninver
8、sioni
