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Ch6 The Stability of Linear Feedback SystemsnTheconceptofstabilitynTheRouth-HurwitzstabilitycriterionnTherelativestability2020/11/2416.1 The concept of stabilityAstablesystemisadynamicsystemwithaboundedoutputtoaboundedinput(BIBO).Theissueofensuringthestabilityofaclosed-loopfeedbacksystemiscentraltocontrolsystemdesign.Anunstableclosed-loopsystemisgenerallyofnopracticalvalue.absolutestability,relativestability2020/11/242精品资料3你怎么称呼老师？如果老师最后没有总结一节课的重点的难点，你是否会认为老师的教学方法需要改进？你所经历的课堂，是讲座式还是讨论式？教师的教鞭“不怕太阳晒，也不怕那风雨狂，只怕先生骂我笨，没有学问无颜见爹娘”“太阳当空照，花儿对我笑，小鸟说早早早”4Absolutestability:Wecansaythataclosed-loopfeedbacksystemiseitherstableoritisnotstable.Thistypeofstable/notstablecharacterizationisreferredtoasabsolutestability.Relativestability:Giventhataclosed-loopsystemisstable,wecanfurthercharacterizethedegreeofstability.Thisisreferredtoasrelativestability.2020/11/2452020/11/2462020/11/2476.2TheRouth-Hurwitzstabilitycriterion2020/11/248where 2020/11/249Anecessaryandsufficientconditionforafeedbacksystemtobestableisthatallthepolesofthesystemtransferfunctionhavenegativerealparts.2020/11/2410Anecessarycondition:Allthecoefficientsofthepolynomialmusthavethesamesignandbenonzeroifalltherootsareinleft-handplane(LHP).Thecharacteristicequationiswrittenas2020/11/2411HurwitzandRouthpublishedindependentlyamethodofinvestigatingthestabilityofalinearsystem.Thenumberofrootsofq(s)withpositiverealpartsisequaltothenumberofchangesinsignofthefirstcolumnoftheRoutharray.Routh-Hurwitz stability criterion2020/11/2412CASE1Noelementinthefirstcolumniszero.CASE2Zerointhefirstcolumnwhilesomeotherelementsofrowcontainingazerointhefirstcolumnarenonzero.CASE3Zerosinthefirstcolumn,andotherelementsoftherowcontainingthezeroarealsozero.2020/11/2413ConsiderthecharacteristicpolynomialTheRoutharrayis 2020/11/2414Case3ConsiderthecharacteristicpolynomialTheRoutharrayisTheauxiliarypolynomial2020/11/24152020/11/2416Designexample：weldingcontrol2020/11/24176.3Therelativestability nTherelativestabilityofasystemcanbedefinedasthepropertythatismeasuredbytherelativerealpartofeachrootorpairofroots.nAxisshiftandexamples2020/11/24182020/11/2419ConsidercontrolsystemDeterminetherangeofKsatisfyingthestabilityandallpolesM.Step4Therootlocusontherealaxisalwaysliesinasectionoftherealaxistotheleftofanoddnumberofpolesandzeros.Step5Determinethenumberofseparateloci,SL,thenumberofseparatelociisequaltothenumberofpoles.2020/11/2435Example7.1Second-ordersystem2020/11/2436Step6 The root loci must be symmetricalwith respect to thehorizontalrealaxiswithangles.Step7 The root loci proceed to the zeros at infinity alongasymptotescenteredatandwithangles.TheselinearasymptotesarecenteredatapointontherealaxisgivenbyTheangleoftheasymptoteswithrespecttotherealaxisis2020/11/2437Example7.2Fourth-ordersystem2020/11/24382020/11/2439Step8 Determine the point at which the locus crosses theimaginaryaxis(ifitdoesso),usingtheRouth-Hurwitzcriterion.TheactualpointatwhichtherootlocuscrossestheimaginaryaxisisreadilyevaluatedbyutilizingtheRouth-HurwitzCriterion.Step9Determinethebreakawaypointontherealaxis(ifany).LetorStep10TheangleoflocusdeparturefromapoleisTheangleoflocusarrivalfromazerois2020/11/24402020/11/24412020/11/24422020/11/24432020/11/2444Step11 Determine the root locations that satisfy the phasecriterionatroot.Thephasecriterionisq=1,2.Step12Determinetheparametervalueataspecificrootusingthemagnituderequirement.Themagnituderequirementatis2020/11/2445Example7.4Fourth-ordersystem2020/11/24462020/11/24477.3ParameterDesignbytheRootLocusmethodThismethodofparameterdesignusestherootlocusapproachtoselectthevaluesoftheparametersTheeffectofthecoefficienta1maybeascertainedfromtherootlocusequation2020/11/24482020/11/24492020/11/24502020/11/24512020/11/24522020/11/24537.4SensitivityandtheRootLocusTherootsensitivityofasystemT(s)canbedefinedasthesensitivityofasystemperformancetospecificparameterchangeswehave2020/11/24542020/11/24552020/11/24562020/11/24572020/11/24587.5Three-term(PID)ControllersThecontrollerprovidesaproportionalterm,anintegrationterm,andaderivativeterm2020/11/24592020/11/24602020/11/24612020/11/24622020/11/24632020/11/2464SummaryInthischapter,wehaveinvestigatedthemovementofthecharacteristicrootsonthes-planeasthesystemparametersarevariedbyutilizingtherootlocusmethod.Therootlocusmethod,agraphicaltechnique,canbeusedtoobtainanapproximatesketchinordertoanalyzetheinitialdesignofasystemanddeterminesuitablealterationsofthesystemstructureandtheparametervalues.Furthermore,weextendedtherootlocusmethodforthedesignofseveralparametersforaclosed-loopcontrolsystem.Thenthesensitivityofthecharacteristicrootswasinvestigatedforundesiredparametervariationsbydefiningarootsensitivitymeasure.2020/11/2465AssignmentnE7.4nE7.82020/11/2466Ch8 Frequency Response MethodsnBasicconceptoffrequencyresponsenFrequencyresponseplotsnDrawingtheBodediagramnPerformancespecificationinthefrequencydomain2020/11/24678.1 Basic concept of frequency responseThefrequencyresponseofasystemisdefinedasthesteady-stateresponseofthesystemtoasinusoidalinputsignal.Theresultingoutputsignalforalinearsystem,isalsoasinusoidalinthesteadystate;itdiffersfromtheinputwaveformonlyinamplitudeandphaseangle.2020/11/2468LetinputTheLaplacetransformationTheoutput undeterminedcoefficient2020/11/24692020/11/2470iscomplexvector2020/11/2471FrequencyCharacteristicsn TransferfunctionandLaplacetransformnFrequencycharacteristicsandFouriertransform2020/11/2472nFrequencycharacteristic,Transferfunctionanddifferentialequationareequivalentinrepresentationofsystem.2020/11/2473FrequencycharacteristicandTransferfunction2020/11/2474Computationoffrequencyresponse2020/11/24758.2FrequencyresponseplotsnPolarplotnBodediagramnNicholschartnFrequencyresponseplotsoftypicalelements2020/11/24762020/11/2477frequency response of an RC filter2020/11/24782020/11/24792020/11/2480 Theprimaryadvantageofthelogarithmicplotistheconversionofmultiplicativefactorintoadditivebyvirtueofthedefinitionoflogarithmicgain2020/11/2481Bode diagram of an RC filter2020/11/24822020/11/2483Nichols chart0o180o-180ow0-20dB20dB2020/11/2484Frequencyresponseplotsoftypicalelements n GainnPoleatoriginnZeroatorigin 2020/11/2485nPoleontherealaxis(jwT+1)nZeroontherealaxis(jwT+1)nTwocomplexpolesnTwocomplexzeros2020/11/24862020/11/24872020/11/24882020/11/24892020/11/2490Bodediagramofatwin-Tnetwork2020/11/24912020/11/24928.3 Drawing the Bode diagram2020/11/24932020/11/24942020/11/24952020/11/24962020/11/24972020/11/2498Drawing 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.2020/11/2499(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/dec120lgKw2020/11/241008.4Performancespecificationinthefrequencydomain Attheresonantfrequency,amaximumvalueofthefrequencyresponse,isattained.Thebandwidthisthefrequency,atwhichthefrequencyresponsehasdeclined3dBfromitslow-frequencyvalue.2020/11/241012020/11/241022020/11/241032020/11/24104Ingeneral,themagnitudeindicatestherelativestabilityofasystems.Thedesirablefrequency-domainspecificationsareasfollows:1.Relativitysmallresonantmagnitude:,forexample.2.Relativitylargebandwidthssothatthesystemtimeconstantissufficientlysmall.2020/11/24105-20-40or-60-20-40wL(w)-60frequency characteristicslowmidhigh2020/11/241068.5Logmagnitudeandphasediagrams2020/11/24107Designexample:Engravingmachinecontrolsystem2020/11/241082020/11/241092020/11/241102020/11/24111SummarynBasicconceptoffrequencyresponsenFrequencyresponseplotsnDrawingtheBodediagramnPerformancespecificationinthefrequencydomain2020/11/24112AssignmentnE8.1nE8.5nE8.62020/11/24113
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