控制工程3(英文)

.,Ch6TheStabilityofLinearFeedbackSystems,TheconceptofstabilityTheRouth-HurwitzstabilitycriterionTherelativestability,.,6.1Theconceptofstability,Astablesystemisadynamicsystemwithaboundedoutputtoaboundedinput(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-Hurwitzstabilitycriterion,.,CASE1Noelementinthefirstcolumniszero.,CASE2Zerointhefirstcolumnwhilesomeotherelementsofrowcontainingazerointhefirstcolumnarenonzero.,CASE3Zerosinthefirstcolumn,andotherelementsoftherowcontainingthezeroarealsozero.,.,Considerthecharacteristicpolynomial,TheRoutharrayis,.,Case3,Considerthecharacteristicpolynomial,TheRoutharrayis,Theauxiliarypolynomial,.,.,Designexample：weldingcontrol,.,6.3Therelativestability,Therelativestabilityofasystemcanbedefinedasthepropertythatismeasuredbytherelativerealpartofeachrootorpairofroots.Axisshiftandexamples,.,.,Considercontrolsystem,DeterminetherangeofKsatisfyingthestabilityandallpolesM.Step4Therootlocusontherealaxisalwaysliesinasectionoftherealaxistotheleftofanoddnumberofpolesandzeros.Step5Determinethenumberofseparateloci,SL,thenumberofseparatelociisequaltothenumberofpoles.,.,Example7.1Second-ordersystem,.,Step6Therootlocimustbesymmetricalwithrespecttothehorizontalrealaxiswithangles.Step7Therootlociproceedtothezerosatinfinityalongasymptotescenteredatandwithangles.TheselinearasymptotesarecenteredatapointontherealaxisgivenbyTheangleoftheasymptoteswithrespecttotherealaxisis,.,Example7.2Fourth-ordersystem,.,.,Step8Determinethepointatwhichthelocuscrossestheimaginaryaxis(ifitdoesso),usingtheRouth-Hurwitzcriterion.TheactualpointatwhichtherootlocuscrossestheimaginaryaxisisreadilyevaluatedbyutilizingtheRouth-HurwitzCriterion.Step9Determinethebreakawaypointontherealaxis(ifany).LetorStep10TheangleoflocusdeparturefromapoleisTheangleoflocusarrivalfromazerois,.,.,.,.,.,Step11Determinetherootlocationsthatsatisfythephasecriterionatroot.Thephasecriterionisq=1,2.Step12Determinetheparametervalueataspecificrootusingthemagnituderequirement.Themagnituderequirementatis,.,Example7.4Fourth-ordersystem,.,.,7.3ParameterDesignbytheRootLocusmethod,Thismethodofparameterdesignusestherootlocusapproachtoselectthevaluesoftheparameters,Theeffectofthecoefficienta1maybeascertainedfromtherootlocusequation,.,.,.,.,.,.,7.4SensitivityandtheRootLocus,TherootsensitivityofasystemT(s)canbedefinedas,thesensitivityofasystemperformancetospecificparameterchanges,wehave,.,.,.,.,.,7.5Three-term(PID)Controllers,Thecontrollerprovidesaproportionalterm,anintegrationterm,andaderivativeterm,.,.,.,.,.,.,Summary,Inthischapter,wehaveinvestigatedthemovementofthecharacteristicrootsonthes-planeasthesystemparametersarevariedbyutilizingtherootlocusmethod.Therootlocusmethod,agraphicaltechnique,canbeusedtoobtainanapproximatesketchinordertoanalyzetheinitialdesignofasystemanddeterminesuitablealterationsofthesystemstructureandtheparametervalues.Furthermore,weextendedtherootlocusmethodforthedesignofseveralparametersforaclosed-loopcontrolsystem.Thenthesensitivityofthecharacteristicrootswasinvestigatedforundesiredparametervariationsbydefiningarootsensitivitymeasure.,.,Assignment,E7.4E7.8,.,Ch8FrequencyResponseMethods,BasicconceptoffrequencyresponseFrequencyresponseplotsDrawingtheBodediagramPerformancespecificationinthefrequencydomain,.,8.1Basicconceptoffrequencyresponse,Thefrequencyresponseofasystemisdefinedasthesteady-stateresponseofthesystemtoasinusoidalinputsignal.Theresultingoutputsignalforalinearsystem,isalsoasinusoidalinthesteadystate;itdiffersfromtheinputwaveformonlyinamplitudeandphaseangle.,.,Letinput,TheLaplacetransformation,Theoutput,undeterminedcoefficient,.,.,iscomplexvector,.,FrequencyCharacteristics,TransferfunctionandLaplacetransformFrequencycharacteristicsandFouriertransform,.,Frequencycharacteristic,Transferfunctionanddifferentialequationareequivalentinrepresentationofsystem.,.,FrequencycharacteristicandTransferfunction,.,Computationoffrequencyresponse,.,8.2Frequencyresponseplots,PolarplotBodediagramNicholschartFrequencyresponseplotsoftypicalelements,.,.,frequencyresponseofanRCfilter,.,.,.,Theprimaryadvantageofthelogarithmicplotistheconversionofmultiplicativefactorintoadditivebyvirtueofthedefinitionoflogarithmicgain,.,BodediagramofanRCfilter,.,.,Nicholschart,0o,180o,-180o,w,0,-20dB,20dB,.,Frequencyresponseplotsoftypicalelements,GainPoleatoriginZeroatorigin,.,Poleontherealaxis(jwT+1)Zeroontherealaxis(jwT+1)TwocomplexpolesTwocomplexzeros,.,.,.,.,.,Bodediagramofatwin-Tnetwork,.,.,8.3DrawingtheBodediagram,.,.,.,.,.,.,DrawingBodediagram:(1)(2)DrawtheasymptoticapproximationofL()inthelowfrequencyrange；(3)Changetheslopeatthebreakfrequency;(4)Thisapproximationcanbecorrectedtotheactualmagnitude.,.,(1)La(w)=20lgK20lgw(2)w1，La(w)=20lgK(3),-20dB/dec,1,20lgK,w,.,8.4Performancespecificationinthefrequencydomain,Attheresonantfrequen