Lead Compensators Design Using Frequency...

Click here to load reader

  • date post

    12-Mar-2018
  • Category

    Documents

  • view

    226
  • download

    4

Embed Size (px)

Transcript of Lead Compensators Design Using Frequency...

  • 5/4/2012

    1

    LeadCompensatorsDesignUsingFrequencyResponseTechniques

    AhmedAbuHajar,Ph.D.

    FrequencyResponseofFeedbackSystemTransferFunctionofFeedbackSystem: Let180 bethefrequencyatwhichthe

    phaseofL(j)is1800

    FrequencyResponseofFeedbackSystem

    bili id i

    ImportantNotes:1 PositiveFeedback,systemisnotstable(oscillateat180)

    2 Infinite([email protected]

    StabilityConsiderations

    Letassumethatthereisafrequencyx suchthat 3 StableandTolerableGainat180

  • 5/4/2012

    2

    PhaseMargin&GainMarginPhaseMargin(PM)

    Let0 bethefrequencyatwhichtheOpenLoopGainis0dB(unitygain).

    PMmustbepositive

    PM > 30o 60oPM>30o 60o.

    GainMargin(GM)

    @180,|L(j180)|6dB.

    -180o

    0

    PM

    180

    LeadCompensatorDesignusingFrequencyresponseTheTransferFunction:

    Km

    z p

    z p

    m

    mTheMaximumPhasem

    m:[email protected]

    ThegainisincreasedbecauseofKc andbecauseof thezeroswhichgives+20dB/Dec

  • 5/4/2012

    3

    DesigningLeadCompensatorsDesignObjective1 Increasethephaseatthefrequency0dBcrossing,henceincreasesPM2 ImprovesthevelocityerrorconstantKv byselectingtheproperKc.

    Steps:1 FindtheLoopGainL(s)=G(s)H(s).2 DetermineK=Kc thatwouldgivethedesiredKv.

    3 DrawBodePlotofL1(s)=KG(s)H(s).MustcompensateforK changesthePM requirement. EvaluatethePM.

    5 Determine fromtherequiredPM.Thenewzerocrossingoccursatfrequencyiswherem islocated.It i h th i f L ( ) i i

    4 Add5o12o becauseoftheincreaseinthegainduetothezero

    6 Determinethelocationofthezeroandthepoleas:

    7 ChoosethevalueofKc fromK=KC.8 ChecktheGMandPMrequirements.Ifnotmet,reiteratetheprocess.(Usually,youneedtogotostep4andaddmorephase).

    ItiswherethegainofL1(s)isgivenas:

    LeadCompensatorExample

    DesignaleadcompensatorusingBodePlotMethodsuchthat

    G(s)C(s)R(s)

    1 ThevelocityerrorconstantKv =20sec12 PhaseMarginPM=50o3 GainMarginGM>10.

    C(s)R(s)G(s)Gc(s)

    Solution

  • 5/4/2012

    4

    LeadCompensatorExampleDrawBodePlotofL1(s)

    UsingapproximatedbodeplotPMisfoundtobe17o.UsingMatlab,exactPMwasfoundtobe17.9o.

    Weneedtoevaluatem ofthecompensatortoget50o +(5o 12o)

    Themaximumphaseofthecompensator

    LeadCompensatorExampleSolvefor

    Thegain(Km)causedbytheearlyzero

    Thenewzerocrossing(m)iswheretheL1(j)= 6.2dB

    FromtheGraph,thatislocatedatm =9rad/sec

  • 5/4/2012

    5

    LeadCompensatorExample

    ThezeroandthepoleandKC ofthecompensatorsarefoundasfollows

    Thecompensatorisgivenas

    LeadCompensatorExample

  • 5/4/2012

    6

    LeadCompensatorExample

    LeadCompensatorExample

  • 5/4/2012

    7

    LagCompensatorsDesignUsingFrequencyResponseTechniques

    AhmedAbuHajar,Ph.D.

    Lag CompensatorTheTransferFunction

    p z

    Km

    p zTheObjective:

    Toprovideattenuation(reductioninmagnitude)afterz.Byreducingthemagnitudebeforethe0dBcrossing(0),thephasemarginimproves.

    TheattenuationisdeterminedbytheratioThelocationofzandpmustbebefore0.

    DesignObjectives:ImprovingthesteadystateerrorwhileimprovingthePM.

  • 5/4/2012

    8

    Lag CompensatorExample

    G(s)C(s)R(s)

    Design a Lag Compensator using Frequency Response method such that:DesignaLagCompensatorusingFrequencyResponsemethodsuchthat:1 ThevelocityerrorconstantKv =5sec12 PhaseMarginPM=40o C(s)R(s)

    G(s)Gc(s)

    Solution

    Thevelocityerrorconstantisgiveas:

    Lag CompensatorExample

    Kchangesthe0dBcrossingfrequency0.InvestigatetheeffectofKon0.

    DrawBodePlotofL1(s)

    UncompensatedthePMis30o

    DuetoK=5,thePMis12o

    Notstablesystem.

  • 5/4/2012

    9

    Lag CompensatorExampleNow,weneedtolocatethefrequencyatwhichwegetaphasemarginFromthegraph,@=0.7,PM=40o.Add12o correctionfactor,(acquiredbytrialanderror)

    @0 =0.5=PM=52.

    [email protected] =0.5

    Choosez= 0.01 before0 p= 0.01

    ThecompensatedPM= 40.7o

    Lag CompensatorExample

  • 5/4/2012

    10

    Lag CompensatorExample

    Lag CompensatorExample

    /ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

    /CreateJDFFile false /Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice