Chapter8

FrequencyResponseMethodCurriculumSystemSystemmodelingPerformanceissuesanalysiscorrectionTimedomainComplexdomainFrequencydomain8.1IntroductionLinearconstantcoefficientsystemsInputsignal:sinusoidal(frequency,magnitude,andphase)Keyconcepts:(8.1)FrequencyresponseFrequencycharacteristicRepresentationoffrequencycharacteristic(howtogetthem?)Polarplot(8.2)Bodeplot(8.2,8.3)Logmagnitudeandphasediagram(8.6)ApplicationsDeterminethesteady-stateresponsetoasinusoidalinput(8.1)Determinethetransferfunctionbyexperiment(8.4)Analyzeperformancespecificationinthefrequencydomain(8.5)Stabilityanalysis(chapter9)andDesign(chapter10)

Featuresoffrequencydomainanalysismethod⑴Thismethodstudieshowthemagnitudeandphaseofsinusoidalsteady-statevarywiththefrequency⑵Studiesthe

stabilityandperformanceoftheclosed-loopsystembytheopen-loopfrequencycharacteristics⑶Agraphicanalysismethod

⑷Anapproximatemethod8.1Introduction48.1IntroductionThefrequencyresponseofasystemisdefinedasthesteady-stateresponseofthesystemtoasinusoidalinputsignal.

Theresultingoutputsignalforalinearsystem,aswellassignalsthroughoutthesystem,issinusoidalwiththesamefrequencyasinputsinusoidalinthesteadystate;

Itdiffersfromtheinputwaveformonlyinamplitudeandphaseangle,andtheamountofdifferenceisafunctionoftheinputfrequency.Thesteady-stateoutputsignaldependsonlyonthemagnitudeandphaseofT(jω)ataspecificfrequency

ωs.

Y(s)=T(s)R(s)withr(t)=Asinωt.whereareassumedtobedistinctpoles.Theninpartialfractionformwehavewhereaandβ

areconstantswhichareproblemdependent.TakingtheinverseLaplacetransformyieldsDemonstrationexampleStaticoutputofthesystem

Parametersband

canbeobtainedby7

T(jω)canberepresentedby

8Oneadvantageofthefrequencyresponsemethodisthatexperimentaldeterminationofthefrequencyresponseofasystemiseasilyaccomplishedandisthemostreliableanduncomplicatedmethodfortheexperimentalanalysisofasystem.Furthermore,thedesignofasysteminthefrequencydomainprovidesthedesignerwithcontrolofthebandwidthofasystem,aswellassomemeasureoftheresponseofthesystemtoundesirednoiseanddisturbances.Asecondadvantageisthatthetransferfunctiondescribingthesinusoidalsteady-statebehaviorofasystemcanbeobtainedbyreplacingswithjωinthesystemtransferfunctionT(s)(calledasfrequencycharacteristic).Thebasicdisadvantageofthefrequencyresponsemethodforanalysisanddesignistheindirectlinkbetweenthefrequencyandthetimedomain.

ConceptofFrequency-ResponseCharacteristics

ConsidertheRCcircuitshowninthefigure(ur(t)=Asinwt).Obtainuc(t).Modeling10

DefinitionofFrequency-ResponseCharacteristicDefinition1Definition3Definition2Magnitude-FrequencyCharacteristic

Phase-Frequency

Characteristic

ConceptofFrequency-ResponseCharacteristics

11

Considerthesystemshowninthefigure(r(t)=3sin(2t+30o)).Obtaincs(t),es(t).Solution.ConceptofFrequency-ResponseCharacteristics

12

Frequency-ResponseCharacteristicsinGraphicalFormsⅠ.Frequencycharacteristic

Ⅱ.Magnitudeandphasecharacteristic

(Nyquist)Ⅲ.Log-magnitude-frequencycharacteristic

(Bode)Ⅳ.Log-phase-frequencycharacteristic

(Nichols)Magnitude-FrequencyPhase-FrequencyLog-magnitude-frequencyLog-phase-frequencyForConceptofFrequency-ResponseCharacteristics

13

CorrelationbetweenMathematicalmodelsConceptofFrequency-ResponseCharacteristics

14

Themagnitude-phasecurveforFirst-OrderfactorsMagnitude-PhaseFrequencyCharacteristics15

Magnitude-phasecharacteristicsoftypicalfactorsProve：Theamplitude-phasecharacteristicsoffirst-orderfactorsareasemicircle.

(Lowerhalfofthecircle)Magnitude-PhaseFrequencyCharacteristics16

Magnitude-phasecharacteristicsFromtheshapeofthecurve,weknowthatFromthestartingpoint：Fromj0From

j1：Obtainthetransferfunctionfromthemagnitude-phasecharacteristicsshowninthefigure.Magnitude-PhaseFrequencyCharacteristics17

UnstableFirst-OrderFactors⑸ReciprocalFirst-OrderFactorsMagnitude-PhaseFrequencyCharacteristics18Example8.2PolarplotoftransferfunctionThefrequencycharacteristicisThemagnitudeandphaseangleare8.2FrequencyResponsePlotsUsetherealandimaginarypartsofG(jω)as8.2FrequencyResponsePlots

§Magnitude-PhaseFrequencyCharacteristics(Nyquist)§Magnitude-PhaseFrequencyCharacteristicsofTypicalFactors⑴Thegain⑵Derivativefactor⑶Integralfactor⑷First-orderfactor21AmplitudeandPhaseFrequency⑹OscillationlinkAmplitudeandPhaseFrequencyCharacteristicsofTypicalLink22AmplitudeandPhaseFrequencyCharacteristicsResonance

frequency

wrandresonantpeaking

Mr

Example4:When23AmplitudeandPhaseFrequencyCharacteristicsResonancefrequencyResonantpeakingwr,Mr

不存在24AmplitudeandPhaseFrequencyCharacteristics

AmplitudeandphasecharacteristicsTheamplitudeandphasecharacteristicsisshowninfigure.Determinethetransferfunction.Fromtheshapeofcurve,wehaveFromstartingpoint：Fromj(w0)：From|G(w0)|：25AmplitudeandPhaseFrequencyCharacteristics

UnstableOscillationlink26

⑺ReciprocalQuadraticFactor

Magnitude-PhaseFrequencyCharacteristics27

⑻DelayFactorMagnitude-PhaseFrequencyCharacteristics28

NyquistPlotsofTypicalFactors⑴⑵⑶⑻⑸⑷⑹⑺Magnitude-PhaseFrequencyCharacteristics29

NyquistPlotofOpen-LoopTransferFunctionsNyquistPlotofOpen-loopTransferFunctionsStartingpoint

Endingpoint

30

NyquistPlotofOpen-LoopTransferFunctions31

A:

B:NyquistPlotofOpen-LoopTransferFunctions

32

SketchtheNyquistplotforSolution.Asymptotes:Intersectionpointwithrealaxis:

NyquistPlotofOpen-LoopTransferFunctions

33Example8.4Bodediagramofatwin-TnetworkThedeterminationofthefrequencyresponseusingthepole-zerodiagramandvectorstojωThetransferfunctionofthenetworkis8.2FrequencyResponsePlots34Ifthezerosareat±j1,andthepolesareatAtω=0Atω=1/τ

When8.2FrequencyResponsePlots358.2FrequencyResponsePlotsThelimitationsofpolarplotsTheadditionofpolesorzerostoanexistingsystemrequirestherecalculationofthefrequencyresponseFurthermore,calculatingthefrequencyresponseinthismanneristediousanddoesnotindicatetheeffectoftheindividualpolesorzeros.BodeDiagramsSemilogCoordinate37

BodeDiagrams

⑴Magitudemultiplication=Logarithmaddition

Convenientforsegmentaddition；LongitudinalaxisAbscissaAxisFeaturesofthecoordinateFeaturesScaledby

lgw，dec“Decade”按lgw

刻度，dec“十倍頻程”Markedbyw.Distancereflectingratio按w標定，等距等比“Decibel”

⑵Representsfrequencycharacteristicinlargescale;⑶L(w)canbedeterminedbyexperiment,socanG(s).AnintroductionforBodediagrams(LogarithmicPlots)38Theprimaryadvantageofthelogarithmicplotistheconversionofmultiplicativefactors,suchas(jωτ+1),intoadditivefactors,20log(jωτ+1)byvirtueofthedefinitionoflogarithmicgain.ThegeneraltransferfunctionisGeneralcase39ThelogarithmicmagnitudeofG(jω)isThephaseangleplotis401.ConstantgainKb2.Poles(orzeros)attheorigin(jω)3.Poles(orzeros)attherealaxis(jωτ+1)4.Complexconjugatepoles(orzeros)Notes:WecandeterminethelogarithmicmagnitudeplotandphaseangleforthesefourfactorsandthenutilizethemtoobtainaBodediagramforanygeneralformofatransferfunctionTypicalfactors418.2FrequencyResponsePlotsThecurvesforeachfactorareobtainedandthenaddedtogethergraphicallytoobtainthecurvesforthecompletetransferfunction.Furthermorethisprocedurecanbesimplifiedbyusingtheasymptoticapproximationstothesecurvesandobtainingtheactualcurvesonlyatspecificimportantfrequencies.42ConstantGainKbThelogarithmicgainfortheconstantKbis8.2TheBodediagramoftypicalfactorsThegaincurveisahorizontallineontheBodediagram.Ifthegainisanegativevalue,-Kb,thelogarithmicgainremains20logKb.Thenegativesignisaccountedforbythephaseangle,-180°.Poles(orzeros)attheorigin(jω)LogarithmicmagnitudePhaseangledB8.2TheBodediagramoftypicalfactorsPolesorZerosontheRealAxisLogarithmicmagnitudeTheasymptoticcurveforω<<1/τis20log1=0dB,andtheasymptoticcurveforω>>1/τis–20logωτ,whichhasaslopeof–20dB/decade.Theintersectionofthetwoasymptotesoccurswhenω=1/τ,thebreakfrequency.Theactuallogarithmicgainis–3dBwhenω=1/τThephaseangleis8.2TheBodediagramoftypicalfactorsThephaseanglecurve8.2TheBodediagramoftypicalfactors46

TheLogarithmicplotoffirst-orderfactorsissymmetricaboutthe

(w=1/T,j=-45

)point.Prove：Suppose47ComplexConjugatePolesorZerosThelogarithmicmagnitudeforapairofcomplexconjugatepolesisThephaseangleisWhenu<<1,themagnitudeis

andphaseangleis00

Whenu>>1,themagnitudeis

andphaseangleis-1800

8.2TheBodediagramoftypicalfactors8.2FrequencyResponsePlots8.2FrequencyResponsePlotsThemaximumvalueofthefrequencyresponse,MPω,occursattheresonantfrequencyωr

Theresonantfrequencyisandthemaximumvalueofthemagnitude|G(ω)|is8.2FrequencyResponsePlots

BodeDiagramsReview

TheBodediagramoftypicalfactors⑴TheGain⑵DerivertiveFactor⑶IntegralFactor⑷First-OrderFactor52

⑸ReciprocalFirst-OderFactorBodeDiagramsReview53

⑹QuadraticFactorsBodeDiagramsReview54

BodeDiagramsReview⑺ReceprocalQuadraticFactors55

⑻DelayLinkBodeDiagramsReview568.3AnexampleofdrawingtheBodediagramTheBodediagramofatransferfunctionG(s)ThefactorshaveAconstantgainK=5ApoleattheoriginApoleatω=2Azeroatω=10Apairofcomplexpolesatω=ωn=5057588.3AnexampleofdrawingtheBodediagram5960618.3AnexampleofdrawingtheBodediagram62Insummary,onemayobtainapproximatecurvesforthemagnitudeandphaseshiftofatransferfunctionG(jω)inordertodeterminetheimportantfrequencyranges.Withintherelativelysmallimportantfrequencyranges,theexactmagnitudeandphaseshiftcanbereadilyevaluatebyusingtheexactequations.TheexactG(jω)canbeplottedbyMatlab

63BodeDiagramForOpen-loopSystemsThestepstosketchBodediagramforopen-loopsystem⑴Changingopen-loop

transferfunction

G(jw)

intotheendofastandardform⑵Listingtheturningfrequencyinturn.⑶確定基準線0.2Inertiallink0.5First-ordercompositedifferential

1OscillationLink基準點斜率⑷DrawingthediagramFirst-orderInertiallink-20dB/decCompositedifferential+20dB/decSecond-orderOscillationLink-40dB/decCompositedifferential-40dB/decw=0.2

Inertiallink-20w=0.5

First-ordercompositedifferential+20w=1

OscillationLink-40第一轉(zhuǎn)折頻率之左的特性及其延長線64BodeDiagramForOpen-loopSystems⑸Correction⑹Check①Whentheturningfrequencyoftwoinertiallinksareclosetoeachother②Whenoscillation

x(0.38,0.8)

①TherightmostslopeofL(w)isequalto

-20(n-m)dB/dec

②Thenumberofturningpoint=(Inertial)+(First-ordercompositedifferential)+(Oscillation)+(Second-ordercompositedifferential)③j(w)

-90°(n-m)基準點斜率w=0.2

Inertiallink-20w=0.5

First-ordercompositedifferential+20w=1

OscillationLink-4065

BodeDiagramForOpen-loopSystemsBasepoint

.SketchBodediagramSolution.①Standardform②Turningfrequencies③Baseline④Plotting

Slope

⑤

CheckTherightmostslopeofL(w)is-20(n-m)=0Thenumberofturningpoints=3j(w)tendsto

-90o(n-m)=0o

66

BodeDiagramForOpen-loopSystemsSketchtheBodeDiagramandtheNyquistPlot.

Solution.①BaselinePointSlope②③④Check

TherightmostslopeofL(w)is-20(n-m)=-80dB/decThenumberofturningpoints=3j(w)

-90o(n-m)=-360o67

BodeDiagramForOpen-loopSystemsSketchtheBodeDiagramandthe

NyquistPlot.

68

ObtainthetransferfunctionfromtheBodediagram.Solution.Fromtheplot

CorrespondingrelationbetweentheBodediagramandNyquistPlot:Turningfrequency

CutoffFrequency

wc：69

Solution.Fromthediagram:ObtainthetransferfunctionfromtheBodediagram.70

CorrespondingrelationbetweentheBodediagramandNyquistPlot：

CutoffFrequency

wc：71

BodeDiagramForOpen-loopSystemsObtainG(s)fromtheBodediagram.Solution.SolutionⅡSolutionⅠSolutionⅢProof：72

BodeDiagramForOpen-loopSystemsObtainG(s)andsketch

j(w)andtheNyquistplotforgivenL(w)ofaminimumphasesystem.Solution⑴III⑵Sketching

j(w)⑶73

BodeDiagramForOpen-loopSystems⑴⑵⑶⑷748.5PerformanceSpecificationintheFrequencyDomainDiscusstherelationshipbetweentheexpectedtransientresponseandthefrequencyresponseofthesystemForasimplesecond-ordersystem,wehavealreadyresulttothisproblem.Thetransferfunctionofsecond-orderclosed-loopsystemisThefrequencyresponseofthesystemisThesecond-ordersystem,thedampingrationofthesystemisrelatedtothemaximummagnitudeMpω

,thefrequencyωr-resonantfrequencyThebandwidth,ωB,isameasureofasystem’sabilitytofaithfullyreproduceaninputsignal8.5PerformanceSpecificationintheFrequencyDomain8.6LogMagnitudeandPhaseDiagramThereareseveralalternativemethodsofpresentingthefrequencyresponseofafunctionGH(jω).(1).Thepolarplot(2).TheBodediagramNowweintroduceanalternativeapproachtoportrayingthefrequencyresponsegraphically(3).Log-magnitude-phasediagram:thelogarithmicmagnitudeindBversusthephaseangleforarangeoffrequencies.Forexample,atransferfunctionisForexample,atransferfunctionis8.7DesignExample:EngravingMachineControlSystemTheengravingmachineTherearetwomotorsinthex-axisAseparatemotorisusedforbothy-axisandz-axisTheblockdiagrammodelforthex-axispositioncontrolsystemisThegoalistoselectanappropriategainK,utilizingfrequencyresponsemethods,sothatthetimeresponsetostepcommandsisacceptable8.7DesignExample:EngravingMachineControlSystemFirst,obtaintheopen-loopandclosed-loopBodediagramTheopen-looptransferfunction(frequencydomain)Wearbitrarilyselectk=2Thenopen-looptransferfunction8.7DesignExample:EngravingMachineControlSystemTheclosed-looptransferfunctionWeassumethatthesystemhasdominantsecond-orderroots,thesystemmaybesecond-orderformFromtheBodediagram,weget8.7DesignExample:EngravingMachineControlSystemWeareapproximatingT(s)asasecond-ordersystem,thenwehaveForseco