Essence of the Root Locus echnique In this chapter we study method for nding locations of system poles - Description

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 Download Pdf

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Essence of the Root Locus echnique In this chapter we study method for nding locations of system poles

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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

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Essence of the Root Locus echnique In this chapter we study method for nding locations of system poles






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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)H1(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 andforthephaseanglesIfwefactorasthen,byusingelementaryalgebrawithcomplexnumbers,weget andThelasttwoequationsarecrucialforthedevelopmentoftherootlocustechnique. 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.