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Ch6 The Stability of Linear Feedback SystemsTheconceptofstabilityTheRouth-HurwitzstabilitycriterionTherelativestability1Ch6 The Stability of Linear F6.1 The concept of stabilityAstablesystemisadynamicsystemwithaboundedoutputtoaboundedinput(BIBO).Theissueofensuringthestabilityofaclosed-loopfeedbacksystemiscentraltocontrolsystemdesign.Anunstableclosed-loopsystemisgenerallyofnopracticalvalue.absolutestability,relativestability26.1 The concept of stability Absolutestability:Wecansaythataclosed-loopfeedbacksystemiseitherstableoritisnotstable.Thistypeofstable/notstablecharacterizationisreferredtoasabsolutestability.Relativestability:Giventhataclosed-loopsystemisstable,wecanfurthercharacterizethedegreeofstability.Thisisreferredtoasrelativestability.3 Absolute stability:We ca44556.2TheRouth-Hurwitzstabilitycriterion66.2 The Routh-Hurwitz stabilitwhere 7where 7Anecessaryandsufficientconditionforafeedbacksystemtobestableisthatallthepolesofthesystemtransferfunctionhavenegativerealparts.8A necessary and sufficient conAnecessarycondition:Allthecoefficientsofthepolynomialmusthavethesamesignandbenonzeroifalltherootsareinleft-handplane(LHP).Thecharacteristicequationiswrittenas9A necessary condition:All theHurwitzandRouthpublishedindependentlyamethodofinvestigatingthestabilityofalinearsystem.Thenumberofrootsofq(s)withpositiverealpartsisequaltothenumberofchangesinsignofthefirstcolumnoftheRoutharray.Routh-Hurwitz stability criterion10Hurwitz and Routh published inCASE1Noelementinthefirstcolumniszero.CASE2Zerointhefirstcolumnwhilesomeotherelementsofrowcontainingazerointhefirstcolumnarenonzero.CASE3Zerosinthefirstcolumn,andotherelementsoftherowcontainingthezeroarealsozero.11CASE1 No element in the firConsiderthecharacteristicpolynomialTheRoutharrayis 12Consider the characteristic poCase3ConsiderthecharacteristicpolynomialTheRoutharrayisTheauxiliarypolynomial13Case 3Consider the characteris1414Designexample：weldingcontrol15Design example：welding contro6.3Therelativestability Therelativestabilityofasystemcanbedefinedasthepropertythatismeasuredbytherelativerealpartofeachrootorpairofroots.Axisshiftandexamples166.3 The relative stability The1717ConsidercontrolsystemDeterminetherangeofKsatisfyingthestabilityandallpolesM.Step4Therootlocusontherealaxisalwaysliesinasectionoftherealaxistotheleftofanoddnumberofpolesandzeros.Step5Determinethenumberofseparateloci,SL,thenumberofseparatelociisequaltothenumberofpoles.33Step3 Locate the poles and zerExample7.1Second-ordersystem34Example7.1 Second-order systemStep6 The root loci must be symmetricalwith respect to thehorizontalrealaxiswithangles.Step7 The root loci proceed to the zeros at infinity alongasymptotescenteredatandwithangles.TheselinearasymptotesarecenteredatapointontherealaxisgivenbyTheangleoftheasymptoteswithrespecttotherealaxisis35Step6 The root loci must be sExample7.2Fourth-ordersystem36Example7.2 Fourth-order syste3737Step8 Determine the point at which the locus crosses theimaginaryaxis(ifitdoesso),usingtheRouth-Hurwitzcriterion.TheactualpointatwhichtherootlocuscrossestheimaginaryaxisisreadilyevaluatedbyutilizingtheRouth-HurwitzCriterion.Step9Determinethebreakawaypointontherealaxis(ifany).LetorStep10TheangleoflocusdeparturefromapoleisTheangleoflocusarrivalfromazerois38Step8 Determine the point at w3939404041414242Step11 Determine the root locations that satisfy the phasecriterionatroot.Thephasecriterionisq=1,2.Step12Determinetheparametervalueataspecificrootusingthemagnituderequirement.Themagnituderequirementatis43Step11 Determine the root locaExample7.4Fourth-ordersystem44Example7.4 Fourth-order syste45457.3ParameterDesignbytheRootLocusmethodThismethodofparameterdesignusestherootlocusapproachtoselectthevaluesoftheparametersTheeffectofthecoefficienta1maybeascertainedfromtherootlocusequation467.3 Parameter Design by the Ro474748484949505051517.4SensitivityandtheRootLocusTherootsensitivityofasystemT(s)canbedefinedasthesensitivityofasystemperformancetospecificparameterchangeswehave527.4 Sensitivity and the Root L53535454555556567.5Three-term(PID)ControllersThecontrollerprovidesaproportionalterm,anintegrationterm,andaderivativeterm577.5 Three-term(PID)Controller58585959606061616262SummaryInthischapter,wehaveinvestigatedthemovementofthecharacteristicrootsonthes-planeasthesystemparametersarevariedbyutilizingtherootlocusmethod.Therootlocusmethod,agraphicaltechnique,canbeusedtoobtainanapproximatesketchinordertoanalyzetheinitialdesignofasystemanddeterminesuitablealterationsofthesystemstructureandtheparametervalues.Furthermore,weextendedtherootlocusmethodforthedesignofseveralparametersforaclosed-loopcontrolsystem.Thenthesensitivityofthecharacteristicrootswasinvestigatedforundesiredparametervariationsbydefiningarootsensitivitymeasure.63Summary In this chapter,AssignmentE7.4E7.864AssignmentE7.464Ch8 Frequency Response MethodsBasicconceptoffrequencyresponseFrequencyresponseplotsDrawingtheBodediagramPerformancespecificationinthefrequencydomain65Ch8 Frequency Response Method8.1 Basic concept of frequency responseThefrequencyresponseofasystemisdefinedasthesteady-stateresponseofthesystemtoasinusoidalinputsignal.Theresultingoutputsignalforalinearsystem,isalsoasinusoidalinthesteadystate;itdiffersfromtheinputwaveformonlyinamplitudeandphaseangle.668.1 Basic concept of frequencLetinputTheLaplacetransformationTheoutput undeterminedcoefficient67Let inputThe Laplace transfor6868iscomplexvector69is complex vector69FrequencyCharacteristics TransferfunctionandLaplacetransformFrequencycharacteristicsandFouriertransform70Frequency Characteristics TranFrequencycharacteristic,Transferfunctionanddifferentialequationareequivalentinrepresentationofsystem.71Frequency characteristic,TranFrequencycharacteristicandTransferfunction72Frequency characteristic and TComputationoffrequencyresponse73Computation of frequency respo8.2FrequencyresponseplotsPolarplotBodediagramNicholschartFrequencyresponseplotsoftypicalelements748.2 Frequency response plotsPo7575frequency response of an RC filter76frequency response of an RC fi77777878 Theprimaryadvantageofthelogarithmicplotistheconversionofmultiplicativefactorintoadditivebyvirtueofthedefinitionoflogarithmicgain79 The primary advantage of tBode diagram of an RC filter80Bode diagram of an RC filter808181Nichols chart0o180o-180ow0-20dB20dB82Nichols chart0o180o-180ow0-20Frequencyresponseplotsoftypicalelements GainPoleatoriginZeroatorigin 83Frequency response plots of tyPoleontherealaxis(jwT+1)Zeroontherealaxis(jwT+1)TwocomplexpolesTwocomplexzeros84Pole on the real axis(jwT+1)88585868687878888Bodediagramofatwin-Tnetwork89Bode diagram of a twin-T netwo90908.3 Drawing the Bode diagram918.3 Drawing the Bode diagram9192929393949495959696Drawing 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.97Drawing Bode diagram:97(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/dec120lgKw98(1)La(w)=20lgK 20lgw-20 d8.4Performancespecificationinthefrequencydomain Attheresonantfrequency,amaximumvalueofthefrequencyresponse,isattained.Thebandwidthisthefrequency,atwhichthefrequencyresponsehasdeclined3dBfromitslow-frequencyvalue.998.4 Performance specification 100100101101102102Ingeneral,themagnitudeindicatestherelativestabilityofasystems.Thedesirablefrequency-domainspecificationsareasfollows:1.Relativitysmallresonantmagnitude:,forexample.2.Relativitylargebandwidthssothatthesystemtimeconstantissufficientlysmall.103In general,the magnitude -20-40or-60-20-40wL(w)-60frequency characteristicslowmidhigh104-20-40or-60-20-40wL(w)-60freq8.5Logmagnitudeandphasediagrams1058.5 Log magnitude and phase diDesignexample:Engravingmachinecontrolsystem106Design example:Engraving mach107107108108109109SummaryBasicconceptoffrequencyresponseFrequencyresponseplotsDrawingtheBodediagramPerformancespecificationinthefrequencydomain110SummaryBasic concept of frequeAssignmentE8.1E8.5E8.6111AssignmentE8.1111
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