Laplace-Transforms

Engs22IntroductiontoLaplaceTransformsp.1LaplaceTransformsWhat?dt(f(t))=F(s)=-stf(t)e∫∞0L[1]wheresisacomplexnumber(havingbothrealandimaginaryparts,s=σ+jω)thatisconstantwithrespecttotime(s≠s(t),ds/dt=0).Toamathematician,thatiswhatmattersabouttheLaplacetransform.Butengineersalmostneverusethatintegral;infactitispossibletousetheLaplacetransformonadailybasis,andactuallyforgetequation[1].InsteadanengineercareabouthowtheLaplacetransformisuseful.Why?TheLaplacetransformisapowerfultoolforanalyzingsystemmodelsconsistingoflineardifferentialequationswithconstantcoefficients.UsesoftheLaplacetransforminthiscontextinclude:1.Asamethodforsolvingdifferentialequations.Aswewillsee,theLaplacetransformprovidesanalternativetoclassicaltime-domainmethodstofindthetimedomainsolutionofdifferentialequations.SolutionofdifferentialequationsviaLaplacetransformsinvolvesalgebraratherthandealingwithdifferentialequations.2.Asausefulwaytothinkaboutandanalyzesystembehavior.Inmanyinstances,relationshipsbetweensysteminputs,outputsandsystemmodelsareclearlyevidentwhenconsideredinthes-domainthatarelessclearwhenconsideredinthetimedomain.Examplesfromamongmanyincludetheexistenceofatransferfunction,definedinthesdomainbutnotdefinedinthetimedomain,whichmultipliesthesysteminputtoobtainthesystemoutput.NotwithstandingtheimportanceoftheLaplacetransformforsolvingdifferentialequations,itisveryimportanttorealizethatthisisonlyapartofthemotivation–andprobablythesmallerpart-forusingthistechnique.Studentsareencouragedtoinvesttheefforttocomfortablethinkingaboutdynamicsystemsandtheirmathematicalrepresentationinthes-domain.TheLaplacetransformisacentralfeatureofmanycoursesandmethodologiesthatbuildonthefoundationprovidedbyEngs22.Amongtheseisthedesignandanalysisofcontrolsystemsfeaturingfeedbackfromtheoutputtotheinput.AUsefulAnalogy.TounderstandtheLaplacetransform,useoftheLaplacetosolvedifferentialequations,andtherelationshipbetweenthes-domainandthetimedomainitisusefultoconsiderthelogarithmfunction.Engs22IntroductiontoLaplaceTransformsp.2ConsiderusingthelogarithmfunctiontocalculatetheproductYX•.AsindicatedinFigure1,wecancalculateYX•directlyusingmultiplication.AlternativelywecancalculatethisproductbytakingthelogarithmofXandY(transformingtothelogarithmdomain),addinglnXandlnY,andthentakingtheinverselogarithmofthesum(exponentiatingit)thissum,togobacktotheoriginaldomain.Wecanalsousethelogarithmfunctiontovisualizesystembehavior.Whereasitisdifficulttodeterminewhetherf(t)=e-atandwhatthevalueofaisbyvisualinspectionand,thisiseasilydonefromaplotofln(f(t))vst.Figure1.Useofthelogarithmfunctiontocalculateaproduct.OriginaldomainLogarithmdomainX,YlnX,lnYX.Yln(X.Y)=lnX+lnYTakelogarithm(transform)AdditionExponentiate(inversetransform)MultiplicationFigure2.Visualizationofsystembehaviorforf(t)andln(f(t)).OriginalDomainLogarithmdomainf(t)tln(f(t))tSlope=aEngs22IntroductiontoLaplaceTransformsp.3AssuggestedbyFigure1,wecancalculateYX•withoutactuallymultiplying—instead,wetransformtothelogarithmdomain,performasimplermathematicaloperation(addition),andthentransformbacktotheoriginaldomain.Ifmultiplicationwerealotharderthanaddition,thiscouldbeveryvaluable.TheideaofsolvingdifferentialequationsusingtheLaplacetransformisverysimilar.WefirsttransformtothesdomainusingtheLaplacetransform.Thatgetsridofallthederivatives,sosolvingbecomeseasy—itisjustalgebrainthesdomain.Thenwetransformbacktotheoriginaldomain(“timedomain”).AnImportantLimitation.TheLaplacetransformandtechniquesrelatedtoitareonlyapplicabletosystemsdescribedbylinearconstant-coefficientmodels.UsingtheLaplaceTransform.Inordertoapplythetechniquedescribedabove,itisnecessarytobeabletodotheforwardandinverseLaplacetransforms.Althoughinprinciple,youcoulddothenecessaryintegrals,peoplehavebeendoingthoseintegralsforcenturies.Theyhavethemprettywellfiguredoutbynow,andyou’vegotbetterthingstodo.SoabetterplanistouseatableofLaplacetransforms.Thebookhasaprettygoodtable,butwe’llprovidealargertable.Thetableisusedprimarilyfortheinversetransform,andfortransforminginputs.Forotherpartsofyourequation(s),itisonlynecessarytoknowafewpropertiesoftheLaplacetransform.SomeofthesearederivedinAppendixII,butyouonlyneedtobeabletousethem,notderivethem.Linearity:(bothsuperpositionandhomogeneity):(s)(s)+bF(t))=aF(f(t))+b(f(t))=a(t)+bf(af212112LLLSuperpositonisusefulbecauseitallowsyoutoseparatetermsinyourequationintosimplerthingsthatyoucaneasilyfindonatable,andhomogeneityisusefulbecauseitallowsyoutofactoroutmultiplicativeconstants,soyoucaninfactmatchyourtermsupwithtableentries.Differentiation:)00-f()=sF(s)-f()=sL(f)f(•LThisisthekeypropertythatmakessolutionsofdifferentialequationspossible.Thederivativegoesaway,andgetsreplacedbyasimplemultiplicationbys.Therearesimilarpropertiesforsecondandhigherderivatives.SeeAppendixII.Engs22IntroductiontoLaplaceTransformsp.4AppendixI:ComplexNumberReview.Aswehaveseeninthecourseofconsideringanalyticalsolutionsofdifferentialequationsinthetimedomain,takingtherootstothecharacteristicequationsofsystemmodelsofteninvolvestakingthesquarerootofanegativenumber.Invaria