控制工程-Ch6The Stability of Linear Feedback Systems

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,Attheresonantfr