chapter finaland initial value theroem frequency response performance specification system error章節(jié)終值初值定理頻率響應(yīng)性能規(guī)格系統(tǒng)誤差

FinalValueTheoremFinalValueTheoremTheFINALVALUETHEOREMcanbeusedtodeducethevalueatwhichasystemwillsettle,onceallofthetransientshavediedawayThisisveryusefulforlookingatSTEADYSTATEERRORwhichisthedifferencebetweenthedemand,R(s),andtheoutputofasystem,C(s)afterallofthetransientshaveendedThefinalvaluetheoremisdefinedas:So,inordertofindtheresponseofatransferfunctiontoaninputafterallofthetransientshavediedawaywemultiplyitbys,thensets=0intheresultingequation,andsolvewhat’slefttoworkoutthefinalvalue.TherearetwoimportantcircumstancestowhichFVTcannotbeapplied

UNSTABLESYSTEM-thereisnotfinalvalue!2.PURELYOSCILLATORYSystem-i.e.rootsoncomplexaxis…becausethiswillcontinuetooscillate(nofinalvalue)Findthefinalvalue(i.e.thesteadystatevalue)ofthefollowingsystem,assumingastepchangeinputof10

6FinalValueTheorem:ExampleFinalValueTheorem:ProcedureProcedureforapplyingthefinalvaluetheorem1.Derivetheoutputtransferfunction,C(s)2.MultiplyC(s)bys3.CalculatevalueofsC(s)whenstendstozero4.Theresultisthevalueofc(t)whenttendstoinfinityC.InitialvaluetheorembyinitialvaluetheorembyfinalvaluetheoremChapter3FrequencyResponseofSystemsFrequencyResponseThereareothermethodswhichcanbeusedtodeterminetheoperationalbehavioursofasystem-anotheronewhichwewillconsideristheFREQUENCYRESPONSEConsidertheFollowingSystemIfr(t)=Acos(t)DifferentMagnitudeDifferentPhaseSameFrequency(LinearSystem)TherelativemagnitudeandphasedependsonG(s)andonthefrequencyoftheinputsignal-thiswilllinktotheworkthatyoudoonfiltersontheSignalProcessingpartofthecourse.IMPORTANTNOTE:Thesearesteadystaterelationships-sowedonotgaindirectinformationaboutthetransientresponsefromthis-butwecanusethefrequencyresponseinformationtogetanideaofthetransientresponseofthesystem

R(s)C(s)c(t)=Bcos(t+)Let’sconsidertheprobleminaslightlydifferentformatConsiderthefollowingsystem:BothBandwillvarywithfrequency,.ThereforewehaveC(j)expressedasacomplexfunctionof

OscillatorVariesandA(Magnitude)SYSTEMUNDERTESTCouldbeoneofthefiltercircuitsyou’velookedatDisplayAnalyserorPhase&GainMeteretc.FrequencyResponse:DerivationIExperimentalDerivation:

Bycontrollingtheoscillatoratdifferentfrequenciesandlookingatoutputofthesystem,itispossibletoderivethegainandphaseforarangeoffrequenciesandtabulatingthemLikethis…Whydowetakethelogofthegain?Becauseitmakesiteasiertolookatsystemswhichhaveverysignificantchangesingainovertheconsideredfrequencyrange-thiswilleclearwhenyoulookatBodePlots(seelater)Furthertothis-usinglogscanmakeBodePlotsveryeasytoderivefromatransferfunction(seelater)

B/A

20log(B/A)FrequencysetonoscillatorGainofSYSTEMPhaseofSYSTEMLogofGAINFrequencyResponse:DerivationIIGraphicalRepresentationoftheFrequencyResponseINyquistPlotsIfwetakeeachoftheseinturnanddrawthemonseparatephasordiagramsforeachfrequency:

B/A

.........FirstconsiderthephaseandgainatarangeoffrequenciesforthesystemundertestM1M2

2M3

3

3

2

1

1NyquistPlotscontinuedThisisusuallyplottedwithoutthelinesofMforclarity.NyquistplotsandinverseNyquistplotsareusedforstabilityanalysisandarerelatedtotheNyquistStabilityTheorem(notcoveredinthismodule)AnotherfrequencybasedresponseistheuseofBodeDiagrams-thesedemonstratethemagnitudeandphaseofasystemasafunctionoffrequency.ThesewillbecoveredbyDr.WoolfsonastheyarealsousedextensivelyforfilteringapplicationsPlottingallthree(andmore!)onasinglediagramgives:M2M1M3NyquistPlotLocusofphasorsforM(j)GraphicalRepresentationoftheFrequencyResponseIINyquistplotexampleOpenloopsystemhaspoleat2Closed-loopsystemhaspoleat1Ifwemultiplytheopen-loopwithagain,K,thenwecanmovetheclosed-looppole’spositiontotheleft-halfplanePerformanceSpecificationandSystemErrorPerformanceSpecificationLet’sconsiderasecondorderresponsetressMpMp-PeakOvershootMaximumamountbywhichtheresponsepassesthereferenceinthefirstoscillationtr-RiseTimeTimetoreachthereferencepointthefirsttime(otherdefinitionsexist)ess-SteadyStateErrorDifferencebetweenreferenceandactualoutputwhenalloscillationshavediedawaytimeWhyisPerformanceSpecificationimportant?Example1:1.4VdcsourcewhichfeedsaCPUonaPCess=CouldaffectoperationoftheCPUifthesteadystateerrorissignificantMp=CouldaffectoperationandlifetimeoftheCPUorresultinimmediatedamageiftoohightr=IftooslowtheCPUmaynotoperatecorrectlyorthevoltagesourcemaynotrespondwelltochangesinloadExample2:Computercontrolledcuttingmachineess=PooraccuracyincuttingasdemandedpositionisnotachievedMp=Toolgoespastreferenceandruinsthepiecebeingprintedtr=Cuttingofpiecetakesalongtime

TheeffectofdampingonMpandtr

SteadyStateErrorAnalysis:ErrorfunctionConsideringourclosedloopsystem-let’sderiveatransferfunctionwhichtellsushowtheError,E(s),respondstoaninputR(s)

G(s)E(s)R(s)C(s)+-

WecouldapplytheFinalValueTheoremtothisinordertoderivewhatthefinalerrorvaluewillbe-thisisoursteadystateerror,ess

SteadyStateErrorAnalysis:StepInput

Thiscanalsobeconsideredintheform:

WhereK0iscalledthePositionErrorConstantandiscalculatedas

SteadyStateErrorAnalysis:RampInput

Thiscanalsobeconsideredintheform:

WhereK1iscalledtheVelocityErrorConstantandiscalculatedas

SteadyStateErrorAnalysis:ParabolicInput

Thiscanalsobeconsideredintheform:

WhereK2iscalledtheAccelerationErrorConstantandiscalculatedas

SystemTypesandInputTypesInputType(m):

UnityStepInput:UnityRampInput:

UnityParabolicInput:Type0InputType1InputType2InputTypemInput

SystemType(n):Determinedbythenumberoffreeintegratorsinthesystemtransferfunctioni.e.

AreType1Systems

AreType2Transferfunctions

AreTypenTransferfunctionsSteadyStateErrorAnalysis:GeneralIngeneraltermswecanlinkthesteadystateerrorconstantstothesystemnumber,n,andinputnumber,m,asfollows

InputType(m)SystemType(n)012010200n=SystemNumberm=InputNumberNote:Ifm>nthesteadystateerrorisinfinite;ifm=nithasafinitevalue,if(m<n)theerroriszeroHowdoweimprovesystemresponseI(s)I(s)*+-

Consideraunityfeedbacksystem-herewefeederrorintoanamplifierwhichappliesvoltagetoacircuit-forsimplicitywewillassumethatthefeedbackloophasunitygain

A

Itisclearthatbyimprovingthegainoftheamplifier,wecanreducethiserrorbutpracticallyspeakingtherearelimitstohowhighthiscanbeset-sohowdoweimprovethesituation?ControlDesignNote:Type0input,Type0systemControlDesignControldesigndescribestheapproachofaddingpolesandzerostothesystembeingcontrolledinanefforttoimproveit’sresponsetochangesininputorchangesintheenvironment.Weknowthatincreasingthesystemnumberwilldecreasetheerrorinthestepresponse-let’sstartwiththis.Wewilladdanintegratortotheforwardpathtransferfunction-withagainK.I(s)I(s)*+-

A

Sowehaveimprovedtheerror-whatabouttheshapeoftheresponse?ControlDesignIII(s)I(s)*+-

A

NotethatifweassumethatAisfixedbythehardwarethatweareusing-Kisdeterminedeithersettingthedampingfactortoacertainvalueorthenaturalfrequency-theyarenotindependent…h(huán)owcanweimprovethis?Let’saddanintegratorintheforwardpath:ControlDesignIIII(s)I(s)*+-

A

Nowwehavetwovariablestosetthedampingfactorandthenaturalfrequencyoftheresponse-wecanthereforegettheresponsethatwewantfromthesystemInthiscasewehaveusedaProportional+Integral(P+I)controllerMethodsofControlDesignThemethodusedabove,whereweessentiallycomparethecoefficientsofthecharacteristicequationtothestandardformofthesameorderworksforsimplecasesonly-thissitheonlymethodwewilluseinthismoduleFormorecomplexorhigherordersystemswecan:Reducesystemorderbyneglectingnon-dominantpoles-onlyworksinsomecasesUseControlDesignMethodsRootLocusDesignMethodBodePlotDesignMethodBothofthelattermethodsarestudiedinH63CSD-CONTROLSYSTEMSDESIGNHowdoweimplementthecontroltransferfunctionthatwehavedeveloped?UsingOperationalAmplifiersUsingamicroprocessorormicrocontroller(coveredinH63CSD)ImplementingTransferFunctionsusingOp-AmpsTransferfunctioncanbeimplementedinelectroniccircuitformusingOp-AmpsThesearecommonlyusedinthecontrols