控制工程3(英文).ppt

Ch6 The Stability of Linear Feedback SystemsnTheconceptofstabilitynTheRouth-HurwitzstabilitycriterionnTherelativestability.6.1 The concept of stabilityAstablesystemisadynamicsystemwithaboundedoutputtoaboundedinput(BIBO).Theissueofensuringthestabilityofaclosed-loopfeedbacksystemiscentraltocontrolsystemdesign.Anunstableclosed-loopsystemisgenerallyofnopracticalvalue.absolutestability,relativestability.Absolutestability:Wecansaythataclosed-loopfeedbacksystemiseitherstableoritisnotstable.Thistypeofstable/notstablecharacterizationisreferredtoasabsolutestability.Relativestability:Giventhataclosed-loopsystemisstable,wecanfurthercharacterizethedegreeofstability.Thisisreferredtoasrelativestability.6.2TheRouth-Hurwitzstabilitycriterion.where .Anecessaryandsufficientconditionforafeedbacksystemtobestableisthatallthepolesofthesystemtransferfunctionhavenegativerealparts.Anecessarycondition:Allthecoefficientsofthepolynomialmusthavethesamesignandbenonzeroifalltherootsareinleft-handplane(LHP).Thecharacteristicequationiswrittenas.HurwitzandRouthpublishedindependentlyamethodofinvestigatingthestabilityofalinearsystem.Thenumberofrootsofq(s)withpositiverealpartsisequaltothenumberofchangesinsignofthefirstcolumnoftheRoutharray.Routh-Hurwitz stability criterion.CASE1Noelementinthefirstcolumniszero.CASE2Zerointhefirstcolumnwhilesomeotherelementsofrowcontainingazerointhefirstcolumnarenonzero.CASE3Zerosinthefirstcolumn,andotherelementsoftherowcontainingthezeroarealsozero.ConsiderthecharacteristicpolynomialTheRoutharrayis.Case3ConsiderthecharacteristicpolynomialTheRoutharrayisTheauxiliarypolynomial.Designexample：weldingcontrol.6.3Therelativestability nTherelativestabilityofasystemcanbedefinedasthepropertythatismeasuredbytherelativerealpartofeachrootorpairofroots.nAxisshiftandexamples.ConsidercontrolsystemDeterminetherangeofKsatisfyingthestabilityandallpolesM.Step4Therootlocusontherealaxisalwaysliesinasectionoftherealaxistotheleftofanoddnumberofpolesandzeros.Step5Determinethenumberofseparateloci,SL,thenumberofseparatelociisequaltothenumberofpoles.Example7.1Second-ordersystem.Step6 The root loci must be symmetricalwith respect to thehorizontalrealaxiswithangles.Step7 The root loci proceed to the zeros at infinity alongasymptotescenteredatandwithangles.TheselinearasymptotesarecenteredatapointontherealaxisgivenbyTheangleoftheasymptoteswithrespecttotherealaxisis.Example7.2Fourth-ordersystem.Step8 Determine the point at which the locus crosses theimaginaryaxis(ifitdoesso),usingtheRouth-Hurwitzcriterion.TheactualpointatwhichtherootlocuscrossestheimaginaryaxisisreadilyevaluatedbyutilizingtheRouth-HurwitzCriterion.Step9Determinethebreakawaypointontherealaxis(ifany).LetorStep10TheangleoflocusdeparturefromapoleisTheangleoflocusarrivalfromazerois.Step11 Determine the root locations that satisfy the phasecriterionatroot.Thephasecriterionisq=1,2.Step12Determinetheparametervalueataspecificrootusingthemagnituderequirement.Themagnituderequirementatis.Example7.4Fourth-ordersystem.7.3ParameterDesignbytheRootLocusmethodThismethodofparameterdesignusestherootlocusapproachtoselectthevaluesoftheparametersTheeffectofthecoefficienta1maybeascertainedfromtherootlocusequation.7.4SensitivityandtheRootLocusTherootsensitivityofasystemT(s)canbedefinedasthesensitivityofasystemperformancetospecificparameterchangeswehave.7.5Three-term(PID)ControllersThecontrollerprovidesaproportionalterm,anintegrationterm,andaderivativeterm.SummaryInthischapter,wehaveinvestigatedthemovementofthecharacteristicrootsonthes-planeasthesystemparametersarevariedbyutilizingtherootlocusmethod.Therootlocusmethod,agraphicaltechnique,canbeusedtoobtainanapproximatesketchinordertoanalyzetheinitialdesignofasystemanddeterminesuitablealterationsofthesystemstructureandtheparametervalues.Furthermore,weextendedtherootlocusmethodforthedesignofseveralparametersforaclosed-loopcontrolsystem.Thenthesensitivityofthecharacteristicrootswasinvestigatedforundesiredparametervariationsbydefiningarootsensitivitymeasure.AssignmentnE7.4nE7.8.Ch8 Frequency Response MethodsnBasicconceptoffrequencyresponsenFrequencyresponseplotsnDrawingtheBodediagramnPerformancespecificationinthefrequencydomain.8.1 Basic concept of frequency responseThefrequencyresponseofasystemisdefinedasthesteady-stateresponseofthesystemtoasinusoidalinputsignal.Theresultingoutputsignalforalinearsystem,isalsoasinusoidalinthesteadystate;itdiffersfromtheinputwaveformonlyinamplitudeandphaseangle.LetinputTheLaplacetransformationTheoutput undeterminedcoefficient.iscomplexvector.FrequencyCharacteristicsn TransferfunctionandLaplacetransformnFrequencycharacteristicsandFouriertransform.nFrequencycharacteristic,Transferfunctionanddifferentialequationareequivalentinrepresentationofsystem.FrequencycharacteristicandTransferfunction.Computationoffrequencyresponse.8.2FrequencyresponseplotsnPolarplotnBodediagramnNicholschartnFrequencyresponseplotsoftypicalelements.frequency response of an RC filter.Theprimaryadvantageofthelogarithmicplotistheconversionofmultiplicativefactorintoadditivebyvirtueofthedefinitionoflogarithmicgain.Bode diagram of an RC filter.Nichols chart0o180o-180ow0-20dB20dB.Frequencyresponseplotsoftypicalelements n GainnPoleatoriginnZeroatorigin.nPoleontherealaxis(jwT+1)nZeroontherealaxis(jwT+1)nTwocomplexpolesnTwocomplexzeros.Bodediagramofatwin-Tnetwork.8.3 Drawing the Bode diagram.Drawing Bode diagram:(1)(2)Draw the asymptotic approximation of L()in the low frequency range；(3)Change the slope at the break frequency;(4)This approximation can be corrected to the actual magnitude.(1)L(1)La a(w)=20lg(w)=20lgK K 2020 lgwlgw(2)w(2)w1 1，L La a(w)=20lg(w)=20lgK K(3)(3)-20 dB/dec120lgKw.8.4Performancespecificationinthefrequencydomain Attheresonantfrequency,amaximumvalueofthefrequencyresponse,isattained.Thebandwidthisthefrequency,atwhichthefrequencyresponsehasdeclined3dBfromitslow-frequencyvalue.Ingeneral,themagnitudeindicatestherelativestabilityofasystems.Thedesirablefrequency-domainspecificationsareasfollows:1.Relativitysmallresonantmagnitude:,forexample.2.Relativitylargebandwidthssothatthesystemtimeconstantissufficientlysmall.-20-40or-60-20-40wL(w)-60frequency characteristicslowmidhigh.8.5Logmagnitudeandphasediagrams.Designexample:Engravingmachinecontrolsystem.SummarynBasicconceptoffrequencyresponsenFrequencyresponseplotsnDrawingtheBodediagramnPerformancespecificationinthefrequencydomain.AssignmentnE8.1nE8.5nE8.6.