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

......Iftherearenoinitialconditionsthenthesee????(??),??2????and??3????respectivelyExample: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.,????=0??????????Then????=1????(??)i.e.,time-domainintegralesdivisionbyfrequencyvariablesExample:????=??(??),so????=1;gisaunitstepfunction????=1??fisaunitstepfunction,then????=1??;gisaunitrampfunction(g(t)=tfort>=0), ????=1??2IntegralintheLaplaceDomainII

Multiplicationbyt

Multiplicationbyt:Example

ConvolutionTheconvolutionofsignalsfandg,denoted?=?????,isthesignal???=0?????????????????Sameas???=0?????????????????;inotherwords?????=?????(verygreat)importancewillsooneclearIntermsofLaplaceTransform:????=??????(??)LaplaceTransformturnsconvolutionintomultiplication.Convolution:ProveLet’sshowthat??????=????????????=??=0∞(??=0?????????????????)???????????=??=0∞??=0????????????????????????????Whereweintegrateoverthetriangle0≤??≤??Changeorderofintegration:????=??=0∞??=??∞??????????????????????????Changeviabletto??=?????;????=????;regionofintegrationes ??≥0,??≥0Convolution:Example

FindingtheLaplaceTransform

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