chapter laplace transform章拉普拉斯变换

H62SPCChapter3:LaplaceTransform2016-2017BlockDiagramReductionTechniquesIBlocksinCascadeKeyPoint:Thekeythingforallblockdiagrammanipulationandreductionisthatthefunctionforthesystemoutput(orthetotalsystemtransferfunction)shouldneverchangeasaresultofblockdiagrammanipulationG1xy

G2u

yG1G2xBlockDiagramReductionTechniquesIIMovingatakeoffpointaheadofablockMovingatakeoffpointbehindablockYXZGYXZGGy=Gxz=Gxy=Gxz=GxYXZGYXG

Zy=Gxz=xy=Gx

BlockDiagramReductionTechniquesIIIMovingsummingjunctionszxy++

Gzxy++

GG

zxy++

Gzx++G

y

BlockDiagramReductionTechniquesIVReductionoffeed-forwardpaths(BlocksinParallel)YXG++Hxy

BlockDiagramReductionTechniquesVReductionoffeedbackloopsYXG++HxyeHy

BlockDiagramReductionTechniquesVIReductionoffeedbackloopsYXG+-HxyeHy

Thisoneisverycommonlyusedinclosedloopcontrolsystemanalysis!BlockDiagramReductionTechniquesVIISystemswithMultipleInputsThereisoftenmorethanoneinputintoasystem…G1xzG2u++yXandYarebothinputsintothesystem,zistheoutput

Note-thiscouldalsobesolvedusingthesuperpositiontheorem--Assumey=0,calculateZ-Assumex=0,calculateZ-FullzisthesumofthesetworesultsFirstOrderSystemsIfanelementofenergystorageisassociatedwithanelementofenergydissipationthenthenatureoftheoutputisgivenby:

x=inputvariabley=outputvariableT=Timeconstantk=gainExample:vvRvLiRL

Comparetostandardform:

ResponseofafirstOrderSystem:UnitStepWeusea“StepInput”totesttheresponseofasystemtoinstantaneouschangesininput:x(t)=u(t):Itispossibletomathematicallyprovethatthesolutiontothedifferentialequationis:y0k

tTransientStateandSteadyState5TTransientStateSteadyStateResponseofafirstOrderSystem:UnitCosinevRvLiR

TheDOperator

DisamathematicaloperatorwhichrepresentstheprocessofdifferentiationwithrespecttotimeExample:

KeyPointsBlockDiagramReductionDeterminingsystemresponseWehavealreadydeducedthattheresponseofsystemstostimuliisusuallydeterminedbyadifferentialequationThismeansthatforagiveninput(astepinputforexample),inordertodeterminehowsystemresponds,wemustsolvethedifferentialequation.Thiscanbecarriedoutusingtheusualtechniques,butthereisabetterway,whichlendsitselfverywelltocontroldesignasitgivesusatransferfunction.ThemethodusesLAPLACETRANSFORMSDifferentialEquationInputConvertusingtheLaplaceTransformSolvesysteminLaplacedomainConvertbackintothetimedomainSolutionPierre-SimonLaplace:TheFrenchNewtonDevelopedmathematicsinastronomy,physics,andstatisticsBeganworkincalculuswhichledtotheLaplaceTransformFocusedlateroncelestialmechanicsOneofthefirstscientiststosuggesttheexistenceofblackholesLaplaceTransform:IdeasTheLaplaceTransformconvertsintegralanddifferentialequationsintoalgebraicequationsThisislikephasors,but:Appliestogeneralsignals,notjustsinusoidsHandlesno-steady-stateconditionsAllowsustoanalyzeComplicatedcircuitswithsources,Ls,Rs,andCsComplicatedsystemswithintegrators,differentiators,gainsHistoryoftheTransform

Eulerbeganlookingatintegralsassolutionstodifferentialequationsinthemid1700’s:Lagrangetookthisastepfurtherwhileworkingonprobabilitydensityfunctionsandlookedatformsofthefollowingequation:Finally,in1785,LaplacebeganusingatransformationtosolveequationsoffinitedifferenceswhicheventuallyleadtothecurrenttransformTheLaplaceTransform

Notes:sisusuallycomplex(notreal)sisaconstantforthepurposeofintegrationTransformationisonlyvalidfort0NotationforLaplaceTransformsTimeDomains-Domain

transformsLowercaseUppercaseWewillbeinterestedinthesignaldefinedfort>=0TheLaplaceTransformofasignal(function)f(t)isthefunctiondefinedby:s

RestrictionsTherearetwogoverningfactorsthatdeterminewhetherLaplacetransformscanbeused:f(t)mustbeatleastpiecewisecontinuousfort≥0|f(t)|≤MeγtwhereMandγareconstantsSincethegeneralformoftheLaplacetransformis:itmakessensethatf(t)mustbeatleastpiecewisecontinuousfort≥0.Iff(t)wereverynasty,theintegralwouldnotbecomputable.ContinuityBoundednessThiscriterionalsofollowsdirectlyfromthegeneraldefinition:Iff(t)isnotboundedbyMeγtthentheintegralwillnotconverge.LaplaceTransformTheoryGeneralTheoryExampleConvergenceLaplaceTransformsSomeLaplaceTransformsWidevarietyoffunctioncanbetransformedInverseTransformOftenrequirespartialfractionsorothermanipulationtofindaformthatiseasytoapplytheinverseLaplaceTransformsofCommonFunctions:UnitRampfunction

1f(t)tLaplaceTransformsofCommonFunctions:Sinusoid

f(t)t1f(t)tExponentialDecayfunction

f(t)t

Sinusoidalfunction

LaplaceTransformsofCommonFunctionsIIf(t)tDampedSinusoidfunction

LaplaceTransformsofCommonFunctionsIIIf(t)tTheunitimpulse(deltadirac)function

Unitarea

....Workingforthisistedious…

Properties:LinearityTheLaplaceTransformislinear:iffandgareanysignals,andaisanyscalar,wehave:i.e.homogeneity&superpositionhold.Example:Properties:One-to-one

What“almost”means?Iffandgdifferonlyatafinitenumberofpoints(wheretherearen’timpulses),thenF=GTimeScalingdefinesignalgbyg(t)=f(at),wherea>0;then G(s)=(1/a)F(s/a)makessense:timesarescaledbya,frequenciesby1/a.Let’scheck:Whereτ=atExponentialScaling

TimeDelay

Example:Timedelay

DerivativesintheLaplaceDomainI

sF(s)

Wheref(0)istheinitialcondition(i.e.it’svalueatt=0)ofthefunction.Ifthereisn’tonethenf(0)=0Example:Derivation

DerivativesintheLaplaceDomainII

Similarexpressionscanbederivedforhigherorderdifferentials

......Iftherearenoinitialconditionsthenthesee𝑠𝐹(𝑠),𝑠2𝐹𝑠and𝑠3𝐹𝑠respectivelyExample:RLCircuitTransferfunctionvvRvLiRL

Withnoinitialconditions:

iI(s)di/dtsI(s)vV(s)Assumingthevoltage,V(s),istheinput,andthecurrentwe’reconsidering,I(s)istheoutput,wecanconvertthisintoatransferfunction:

Example:RLCCircuitTransferfunction

vvRvLivC

Thistime,let’sassumethatthecapacitorvoltageistheoutputthatwewanttoderiveatransferfunctionforWithzeroinitialconditions:vc

VC(s)dvc/dtsVC(s)vV(s)

Rearrangingasatransferfunction:

IntegralintheLaplaceDomainIILetgbetherunningintegralofasignalf,i.e.,𝑔𝑡=0𝑡𝑓𝜏𝑑𝜏Then𝐺𝑠=1𝑠𝐹(𝑠)i.e.,time-domainintegralesdivisionbyfrequencyvariablesExample:𝑓𝑡=𝛿(𝑡),so𝐹𝑠=1;gisaunitstepfunction𝐺𝑠=1𝑠fisaunitstepfunction,then𝐹𝑠=1𝑠;gisaunitrampfunction(g(t)=tfort>=0), 𝐺𝑠=1𝑠2IntegralintheLaplaceDomainII

Multiplicationbyt

Multiplicationbyt:Example

ConvolutionTheconvolutionofsignalsfandg,denotedℎ=𝑓∗𝑔,isthesignalℎ𝑡=0𝑡𝑓𝜏𝑔𝑡−𝜏𝑑𝜏Sameasℎ𝑡=0𝑡𝑓𝑡−𝜏𝑔𝜏𝑑𝜏;inotherwords𝑓∗𝑔=𝑔∗𝑓(verygreat)importancewillsooneclearIntermsofLaplaceTransform:𝑯𝒔=𝑭𝒔𝑮(𝒔)LaplaceTransformturnsconvolutionintomultiplication.Convolution:ProveLet’sshowthatℒ𝑓∗𝑔=𝐹𝑠𝐺𝑠𝐻𝑠=𝑡=0∞(𝜏=0𝑡𝑓𝜏𝑔𝑡−𝜏𝑑𝜏)𝑒−𝑠𝑡𝑑𝑡=𝑡=0∞𝜏=0𝑡𝑒−𝑠𝑡𝑓𝜏𝑔𝑡−𝜏𝑑𝜏𝑑𝑡Whereweintegrateoverthetriangle0≤𝜏≤𝑡Changeorderofintegration:𝐻𝑠=𝜏=0∞𝑡=𝜏∞𝑒−𝑠𝑡𝑓𝜏𝑔𝑡−𝜏𝑑𝑡𝑑𝜏Changeviabletto𝑡=𝑡−𝜏;𝑑𝑡=𝑑𝑡;regionofintegrationes 𝜏≥0,𝑡≥0Convolution:Example

FindingtheLaplaceTransform

LaplaceTransformtablesLaplaceTransformforODEsEquationwithinitialconditionsLaplacetransformislinearApplyderivativeformulaRearrangeTaketheinverseLaplaceTransforminPDEsLaplacetransformintwovariables(alwaystakenwithrespecttotimevariable,t):Inverselaplaceofa2dimensionalPDE:CanbeusedforanydimensionPDE:ODEsreducetoalgebraicequationsPDEsreducetoeitheranODE(iforiginalequationdimension2)oranotherPDE(iforiginalequationdimension>2)TheTransformreduc