TheRootLocusConceptTheRootLocusProcedureParameterDesignbytheRootLocusMethodDesignExamplesChapter7

TheRootLocusMethod7.1IntroductionTherootlocusmethodwasintroducedbyEvansin1948andhasbeendevelopedandutilizedextensivelyincontrolengineeringpractice.Therootlocustechniqueisagraphicalmethodforsketchingthelocusofrootsinthes-planeasaparameterisvaried.TherootlocustechniquemaybeusedtogreatadvantageinconjunctionwiththeRouth-Hurwitzcriterion.Iftherootlocationsarenotsatisfactory,thenecessaryparameteradjustmentsoftencanbereadilyascertainedfromtherootlocus.7.1IntroductionLocationoftherootsofthecharacteristicequationinthes-plane→performanceofafeedbacksystemChangeoneparameter→adjustthelocationofthecharacteristicroots→achievethedesiredperformanceRootlocusmethod:graphicalmethod,graphofthelocusofrootsasonesystemparametervaries→stability,performanceGeneralRulesforConstructingRootLoci2.Determinetherootlociontherealaxis.Rootlociontherealaxisaredetermined

byopen-looppolesandzeroslyingonit.Thecomplex-conjugatepolesandcomplexconjugate

zerosoftheopen-looptransferfunctionhavenoeffectonthelocationofthe

rootlociontherealaxisbecausetheanglecontributionofapairofcomplex-conjugate

polesorcomplex-conjugatezerosis360°Inconstructingtherootlociontherealaxis,chooseatestpointonit.Ifthetotalnumber

ofrealpolesandrealzerostotherightofthistestpointisodd,thenthispointlies

onarootlocus.TypicalPole–ZeroConfigurationsandCorrespondingRootLoci.7.4AnExampleofaControlSystemAnalysisandDesignUtilizingtheRootLocusMethodSelecttheparameterandthespecificationsofthefeedbacksystem.Obtainamodelandsignal-flowdiagramrepresentingthesystem.SelectthegainKbasedonarootlocusdiagram.Determinethedominantmodeofresponse.Figure7.15Anautomaticself-balancingscaleFigure7.16Modeloftheautomaticself-balancingscale.(a)Signal-flowgraph.(b)Blockdiagram.WhenW(s)=|W|/s,steady-stategainCharacteristicequationFigure7.17RootlocusasKmvaries(onlyupperhalfplaneshown).OnelocusleavesthetwopolesattheoriginandgoestothetwocomplexzerosasKincreases.Theotherlocusistotheleftofthepoleats=?14.7.5ParameterDesignbytheRootLocusMethodCase1:oneparametervariesfromzerotoinfinityIsolatetheparameterofinterestfromthecharacteristicequationastheform1+kF(s).Sketchtherootlocusaskvariesfromzerotoinfinity.Example:Case2:twoparametersvariesIsolateoneparameterofinterestfromthecharacteristicequationastheform1+kF(s).Settheparameterisolatedzero.Sketchtherootlocusasanotherparametervaries.Drawtherootcontours.Example:Example7.5WeldingheadcontrolAweldingheadcontrolsystemSpecifications:Steady-stateerrorforarampinput≤35%ofinputslope.Dampingratioofdominantroots≥0.707;Settlingtimetowithin2%ofthefinalvalue≤3seconds.Adjustableparameters:K1,K2CharacteristicequationSteady-stateDampingratiospecification:θ≤45oSettlingtimespecification:Ts=4/σ≤3s→σ≥1.333Selectparametersusingrootlocusmethod.Setα=K1andβ=K1K2IsolateβLetβ=0,sketchtherootlocusasαvaries.SupposeavalueofK1=α=20,findthecorrespondedpointandtherootlocusasβvaries.Selecttheparameters.Examinethespecifications.7.8DesignExample:LaserManipulatorControlSystemSpecifications:thesteady-stateerrorforaunitrampinput≤0.1stableτ1=0.1s,τ2=0.2sSteady-stateerrorStabilityRoutharrayPerformancespecifications:Stepresponse:Ts≈4/0.51=7.8sOvershoot:72%ConclusionsUnderstandthepowerfulconceptoftherootlocusanditsroleincontrolsystemdesign.Knowhowtosketcharootlocusandalsohowtoobtainacomputer-generatedrootlocusplot.BefamiliarwiththePID