The method is presented for very general setup namely for the case when the closedloop system poles are functions of an unknown parameter In most cases the parameter of interest is the system static gain satisfying However any other unknown and vari ID: 27083
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EssenceoftheRootLocusTechniqueInthischapterwestudyamethodfor ndinglocationsofsystempoles.Themethodispresentedforaverygeneralset-up,namelyforthecasewhentheclosed-loopsystempolesarefunctionsofanunknownparameter.Inmostcasestheparameterofinterestisthesystemstaticgainsatisfying.However,anyotherunknownandvariablesystemparameteraffectingpolelocationscanbeusedinsteadof.Themethodisknownastherootlocustechniqueforsolvingpolynomialequationswithconstantorvariableparameters.ItwasoriginallypresentedinEwans(1948,1950).Theimportanceoftherootlocusmethodforcontrolsystemtheoryliesinthefactthatthelocationofthesystempolesdeterminesthesystem291 stabilityandthesystemtransientresponse.Insomecases,thedesiredcontrolsystemperformancecanbeobtainedbychangingonlythesystemstaticgain.Itisknownfromthatthechoiceofthesystemstaticgaindeterminestheerrorsofthesystemsteadystateresponseinthesensethatabiggervalueforimpliessmallervaluesforsteadystateerrors(assumingthatthesystemremainsasymptoticallystable).However,changingcausesthesystemtransientresponseparametersalsotochange.Ifoneisnotabletoachieveallthecontrolsystemrequirementsbychangingonlythestaticgain(theessenceoftherootlocusmethod),onehastodesignadynamiccompensator(controller).ThequestionofdesigningdynamiccompensatorsbyusingtherootlocusmethodwillbeaddressedindetailinChapter8. Therootlocustechniqueallowsadjustmentofthesystempolesbychangingthefeedbacksystemstaticgain.Theclosed-loopfeedbacksystem,ingeneral,canberepresentedbyablockdiagramasgiveninFigure7.1.G(s)H(s) U(s)+-Y(s)Controller - PlantFeedback ElementFigure7.1:Blockdiagramofafeedbackcontrolsystem Thecharacteristicequationoftheaboveclosed-loopsystemisgivenbywhichcanbewrittenaswhererepresentsallstaticgainspresentintheloop.Usuallythestaticgainscomesolelyfromthecontroller-plantelement.Thatis,theplantiscontrolledbychangingthestaticgain,whichcanberepresentedbyablockdiagramgiveninFigure7.2. G1(s)H 1(s)U(s)+-Y(s)ControllerPlantFeedback ElementKFigure7.2:Feedbackcontrolsystemwithastaticcontrollerinthedirectpath Theclosed-looptransferfunctionofthissystemisgivenby \n\n \n \r\r \r \r\r \r Thecorrespondingcharacteristicequationis Thequestiontobeansweredbytherootlocustechniqueis:whatcanbeachievedbychangingthestaticgain,theoretically,fromto?Canwe ndthelocationofsystempolesforallvaluesof?Thepositiveanswertothisquestionledtothedevelopmentoftherootlocustechnique.ItwasdiscoveredbyW.Ewansin1948andwasmathematicallyformulatedin1950inhisfamouspaper(Ewans,1950). Themainideabehindtherootlocustechniqueishiddeninequationwhichisanalgebraicequationinvolvingcomplexnumbers.Itactuallyrepresentstwoequations(forrealandimaginaryparts,orformagnitudesandphaseangles).Inthisbook,wewillconsidertherootlocustechniquefor.Thiswillsimplifyderivationsandmaketherootlocusplotsclearer.Thecomplementaryrootlocusfornegativevaluesofcanbesimilarlyderived.Thisequationproducesthefollowingequationsforthemagnitudes andforthephaseanglesIfwefactoras then,byusingelementaryalgebrawithcomplexnumbers,weget and Thelasttwoequationsarecrucialforthedevelopmentoftherootlocustechnique. Example7.1:Giventheopen-looptransferfunctionThelocationsoftheopen-looppolesandzerosaregiveninFigure7.3.Re!{s}s"1=-2+j2-4d#ca$b-3-2-1j%2j%1Im{s}b&3b&2a'1b&1Figure7.3:Apointinthecomplexplainthatdoesnotlieontherootlocus Takeanypoint(inthecomplexplane.Ifthatpointbelongstotherootlocus,itmustsatisfybothequations(forthemagnitudeandforthephase).Forexample,forthepoint(wehave((((Thus,ifthepoint(belongstotherootlocus,thestaticgainatthatpointmustbeequalto.Itfollowsthatforthepoint(thefollowingmustbesatis ed(((((( whichleadsto))*+,,,,,Wecanconcludethatthepoint)cannotbelongtotherootlocussincethephaseequationisapparentlynotvalid.Thisexampleshowsthatonlyveryselectedpointsfromthecomplexplanecanbelongtotherootlocus. MinumumPhaseSystemsItshouldbeemphasizedthatfromtherootlocusmethoditfollowsthatthesystemshavingunstableopen-loopzerosbecomeunstableforlargevaluesofthestaticgain.Suchkindsofsystemsarecallednonminimumphasesystems,incontrasttominimumphasesystemswhosede nitionisgivenbelow.De nition7.1Systemshavingallopen-loopzerosandpolesintheclosedlefthalfofthecomplexplaneexcludingtheoriginarecalledminimumphasesystems.