La Place Article

8/6/2019 La Place Article

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Troublesat theOrigin:ConsistentUsageandPropertiesof theUnilateralLaplaceTransform

KentH.Lundberg,HaynesR.Miller,andDavidL.TrumperMassachusettsInstituteofTechnology

The Laplace transform is a standard tool associated with the analysis of signals, models, and controlsystems, and is consequently taught in some form to almost all engineering students. The bilateral andunilateral forms of the Laplace transform are closely related, but have somewhat different domains ofapplication.Thebilateral transform ismostfrequentlyseen in thecontextofsignalprocessing,whereastheunilateraltransformismostoftenassociatedwiththestudyofdynamicsystemresponsewheretheroleofinitialconditionstakesongreatersignificance.Inourteachingwehavefoundsomesignificantpitfallsassociated with teaching our students to understand and apply the Laplace transform. These confusionsextendtothepresentationofthismaterialinmanyoftheavailablemathematicsandengineeringtextbooksas well.

The most significant confusion in much of the textbook literature is how to deal with the origin inthe

application

of

the

unilateral

Laplace

transform.

That

is,

many

texts

present

the

transform

of

a

time

functionf(t) as(1)L{f(t)}=

0

f(t)est dt

without properly specifiying the meaning of the lower limit of integration. Said informally, does theintegralincludetheoriginfully,partially,ornotatall?Thisissuebecomessignificantassoonassingularityfunctionssuchastheunitimpulseareintroduced.Whileitisnotpossibletodevotefullattentiontothisissue within the context of a typical undergraduate course, this skeleton in the closet as Kailath [8]calleditneedstobebroughtoutfullyintothelight.OurpurposeinwritingthisarticleistoputforwardaconsistentsetofLaplacetransformdefinitionsandpropertieswhichallowthecorrectanalysisofdynamicsystems in the presence of arbitrary initial conditions and where the system is driven by functionswhich include singularities. We also present reasonable mathematical support for these properties, aswell as a consistent treatment of singularity functions, without becoming fully enmeshed in the theoryof generalizedfunctions,whichquicklybecomes too fardivorcedfromapplications.

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To properly learn about and apply the unilateral Laplace transform, students need to be taught aconsistent set of properties that correctly handle problems with arbitrary inputs and initial conditions.Theproperformof the unilateralLaplace transformfully includes theorigin

(2)L{f(t)} =

f(t)est dt0

as indicated by the 0 notation. Thus the integral includes interesting events which happen at t = 0,suchasimpulsesorhigher order singularityfunctions,steps,andthebeginningsofothertransients.Sometextsreferto thisformas theL transform.However,sinceweregard thisastheonlycorrectusageoftheunilateraltransforminthecontextofdynamicsystems,wewillomitanyadditionalnotation,andusethesymbolL torepresent the transformasdefined in(2).

Followingfrom thisdefinitionof the transform is the time derivative rule(t)} = sF(s) f(0) (3)L{f

wherebytheinitialconditionsexistingbeforeanyt = 0 transientarebroughtintotheanalysis.Notethatthe0 indicatesthattheresponsewillbecalculatedintermsofwhatwetermthepre initial conditions.

Also associatedwith this definition is the the initial value theoremlim sF(s) = f(0+). (4)

s1

where the notation 1 indicates that the limit is taken along the positive real axis. It is interestingto note here that the value calculated by (4) is associated with thepost initial values at t = 0+. Thisformoftheinitial value theoremisthecorrectresult,anditisalsothedesiredone,sincewearemostlyinterested in the initialvalueafteranydiscontinuitiesatt = 0.

Moregenerally,ifF(s) isabletobewrittenasapolynomialplusafunctionF(s) convergingtozeroass 1 then

lim sF(s) = f(0+) . (5)s1

Theseproperties(3),(4), (5)andsome extensionsare developed morefully in the appendices.With this overview in hand, the remainder of the paper is organized as follows: We first motivate

the discussion with two simple dynamic system examples such as might be presented in a sophomoreengineering course. The responses of these systems are calculated via the Laplace transform definitionand properties presented above. Next, we discuss the application of the Laplace transform to abstractsignals, independent of any dynamic systems context, to clarify the need for the consistent definitionspresentedabove.Finally,thearticleconcludeswithappendiceswhichintroduceanddevelopthetransform

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vI(t) =

vO

+

1, t < 0

0, t > 0

C

R

Fig.1. Schematicofahigh pass electricalfilterdrivenbyanunstep.TheinitialstateofthissystemisthecapacitorvoltagevC(0

) = 1,and thus the initialoutputvoltage isvO(0) = 0.

properties with reasonable mathematical support. In order to do this, we need to think carefully abouthowsingularityfunctionsaredefined,howthesearecombinedwithregularfunctionstoformgeneralizedfunctions,andhowrequiredmathematicaloperationson these generalizedfunctionscanbeconsistentlydefined.

I. APPLICATION EXAMPLESInthissectionwepresentapairofexampleproblemsdrawnfromelectricalandmechanicalengineering,

respectively. To correctly calculate the transient response of these systems requires care in applying thetransform(2); thesetof propertiespresented above(3),(4), and (5)yield the correctanswer.A. First Order High Pass FilterDrivenby an Unstep

First,consider thehigh pass filtershown inFigure1,which isdrivenbyan unstep functionvI(t) =

1 t < 0.

0 t > 0

TheLaplace transformof this input isVi(s) = 0,L{vI(t)}=

which certainly seems uninteresting. We also specify the initial condition vO(0) = 0, and thus thecapacitor is initiallycharged tovC(0) = 1V.

Tofindthetotalsystemresponsetotheinitialstateandthisinput,westartwiththedifferentialequationd

(vIvO)vOC = 0dt R

ordvO vO dvI

+ = .dt RC dt

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1

0.5 0 0.5 1 1.5 2 2.5 3

0.2

0

0.2

0.4

0.6

0.8

1

Input Voltage vI(t)

1 0.5 0 0.5 1 1.5 2 2.5 3

1

0.8

0.6

0.4

0.2

0

0.2

Output Voltage vO

(t)

Fig.2. Responseof thehigh pass filter drivenbyan unstep.

TheLaplace transformof this differential equation,using thecorrectformof the derivative rule, issVo(s)vO(0) + Vo(s) = sVi(s)vI(0),

RC

which reduces toVo(s) =

sVi(s)vI(0) + vO(0)s + 1/RC

.

Theassociatedpre initial valuesarevI(0) = 1 andvO(0) = 0,andwehavealreadycalculatedVi(s) =0. Thus theexpressionsimplifies to

Vo(s) =1

s + 1/RC. (6)

Inverting this transformgives the output timewaveformaset/RC; t > 0,vO(t) =

whichisshowninFigure2alongwiththeinputwaveform.Wecanfurtherapplytheinitialvaluetheoremto(6) as:

vO(0+) = lim sVo(s) = 1.

s

Of course, simple time domain arguments lead to these same results with less effort, but our intentionhere is to demonstrate the proper application ofLaplace techniques insolvingsuchproblems.

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m

k b

y

x

Fig. 3. An idealized automobile suspension system approaching a rectangular speed bump. In the analysis, the center of thewheel isassumed tofollow thebump exactly.

B. Second Order Car SuspensionAnother

example

which

illuminates

the

proper

use

of

the

Laplace

transform

technique

is

the

idealized

second order automobile suspension systemshowninFigure3.Inthissimplifiedmodel,weassumethatthecenterofthewheelfollowsthestepcontourexactly,suchthattheinputmotionx(t) takestheformofaunitstep.Wearetheninterestedincalculatingtheresultingcarbodyoutputmotiony(t).Weexaminethisresponseunderthreesetsofinitialconditionsinordertogivesomeinsightintothetypesofsolutionsthatmightbe calculatedwith theLaplace technique.

The differential equationdescribing thesystem ismy = b(xy) + k(xy).

TheLaplace transformsof thederivativesarex}= sX(s)x(0),L{

sY(s)y(0),L{y}=and

y}= s2Y(s)sy(0)y(0).L{Therefore, thedifferentialequation transforms to

ms2Y(s)msy(0)my(0) =b sX(s)x(0)sY(s) + y(0) +k (X(s)Y(s))

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5 0 5 10 15

0

0.5

1

1.5

(a)

5 0 5 10 150

0.5

1

1.5

2

(b)

5 0 5 10 150

0.5

1

1.5

2

(c)

Fig.4. Responseof the idealizedsuspension system to an inputunit step with threesets of initialconditions:(a) initialrest;(b)initialpositiony(0) = 1,zeroinitialvelocity y(0) = 0;and(c)initialposition y(0) = 1 andinitialvelocity y(0) =b/m.The lastsetof initialconditionsresults in immediateconvergence tofinalvalue.

Solving for theoutputY(s) givesY(s) =

(bs + k)X(s) +

bx(0) + (ms + b)y(0) + my(0)

ms2 + bs + k ms2 + bs + k

Using thissolution,wecanproperly find the systemresponse toarbitrary inputsand initial conditions.We now assume that the input takes the form of a unit step x(t) = u(t), which has a pre initial

valuex(0) = 0. The transform isX(s) = 1/s. The solution under this unit step is then calculated forthreesetsofinitialconditions.Thefirstsetofinitialconditions(a)arezero(initialrest):y(0) = 0 andy(0) = 0.Theresultingoutput transform is

bs + kY(s) = .

s(ms2 + bs + k)

Invertingthisresultyieldsthezero state stepresponseshowninFigure4(a).Notethatinthissystem,forastepinput,thepositioniscontinuousacrosst = 0,andthevelocitytakesastepwithvalueb/m m/sec.This is true independentof initialconditionsand thusapplies to the three casesstudiedhere.

Thesecondsetofinitialconditions(b)areinitialpositiony(0) = 1,zeroinitialvelocity y(0) = 0.That is, the system position starts at the level which it will settle to in steady state. In this case, theoutput transformexpression is

ms2 + 2bs + kY(s) =

s(ms2 + bs + k),

or1 b

Y(s) = + .s ms2 + bs + k

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Inverting this expression gives the time response shown in Figure 4(b). The observed motion resultsfromtheforceimpulseb(t) whichthedamperappliestothemassatthemomentofthestep.Applyingthe initial value theorem to Y(s) and sY(s), respectively yields the post initial values y(0+) = 1,and

y(0

+

) = b/m.The

post initial

velocity

is

simply

an

expression

of

the

change

in

momentum

induced

inthemassbytheforceimpulseappliedfromthedamper;thischangeinmomentumoccursinallthreecases studiedhere.

The third set of initialconditions is selected toyield immediateconvergence to thefinalvalue of theresponse. That is, we choose the initial position asy(0) = 1, and set the initial velocity to y(0) =b/m,such thattheinitialvelocitywillbeexactlycanceledbytheforce impulsefromthedamper.Forthese initialconditions, theoutput transform is

ms2 + bs + k 1Y(s) = = .

s(ms2 + bs + k) s

Werecognize thisas the transformof theunitstep,which givesy(t) = 1 ; t > 0.

ThiswaveformisshowninFigure4(c),alongwithapre initial velocityofb/m,shownbythestraightline segment of position fort < 0. Of course, in theanalysiswe do not concern ourselves with exactlyhowthispre initial velocityisestablished,noranydetailsofthetimevariationpriortothetransient.Theonlyrequiredprior knowledge is thevelocity andposition immediatelybefore the transient(t = 0).

II . SIGNAL EXAMPLESWhile the Laplace transform is frequently associated with the solution of differential equations, the

need to clearly distinguish0+ and0 is independentofanydynamic systemscontext. In the followingexample, adapted from Problem 11.17 in Siebert [4], we apply the unilateral transform to three signalsand their derivatives. This clarifies that the need for using theL form (2), (3), (4), and (5) is reallya matter of properly defining signals and their transforms, and is not fundamentally connected to thesolution ofdifferentialequations.

Consider the threesignalsf(t),g(t),andh(t) asshown inFigure5,f(t) = eat

g(t) = eatu(t)

h(t) = eatu(t)u(t),7


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which are plotted for the valuea = 1. All three functions are nonsingular and agree for positive time,therefore they all have the same Laplace transform. However, their derivatives differ for t 0. Inparticular,thederivativesincludedifferingamountsofanimpulseatt = 0,andthustheLaplacetransformsof

their

derivatives

must

differ.

Our

choice

of

Laplace

transform

properties

should

give

consistent

and

correct results when operating on these signals and their derivatives. The associated transforms arecalculated below to show that this is the case. We also demonstrate the consistent use of the initialvalue theorem in thecontextof thesesignals.

eatA. Functionf(t) =eatConsiderthefunctionf(t) = withassociatedpre initial valuef(0) = 1.TheLaplacetransform

off(t) is1

L{eat}= .s + a

The timederivative off(t) isf(t) = aeat,

and theLaplace transformof the timederivative isa . (7)L{aeat}= s + a

Using thederivativerules

sF(s)f(0) =a1 = a

s + s + a

produces thesameresult.Theresultsfrom the initial value theorem ares

f(0+) = lim = 1,s + as1

andf(0+) = lim

sa= a.

s + as1

There is consistency between the time domain and Laplace domain calculation of the signals and theirinitialvalue.

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eatB. Functiong(t) = u(t)eatThefunctiong(t) = u(t) hasanassociatedpre initial valueg(0) = 0.TheLaplacetransformof

g(t) is thesameasforf(t)1

L{eat

u(t)}= .s + aHowever, the timederivativenow includes an impulse

g(t) = (t)aeatu(t).TheLaplace transformof this timederivative is

a s(t)}= 1 =L{g

s + a s + a

which isdifferentfrom theresultabove(7).Using the correctderivativerules ssG(s)g(0) = =a 0s + s + a

we geta consistentresult.The initial value theoremgivess

g(0+) = lim = 1s + as1

producingthecorrectvalueatt = 0+.Wecanalsoapplythemoregeneralinitialvaluetheorem(5)tothetransform of the derivative as follows (see the discussion in the appendix for more details). Expandingout the nonsingularpartof the transform gives

a

G(s) = 1

a 1 + F

(s).s +

And thus,g(0+) = lim sF(s) = a

s1

which is thecorrectvalue.eatu(t)u(t)C. Functionh(t) =

Finallyconsider thefunctionh(t) =

1 t < 0 = eateat t > 0 u(t)u(t),

which hasanassociatedpre initial valueh(0) = 1.TheLaplace transformof thissignal is thesameasfortheother two(H(s) = G(s) = F(s)),andsowedontlistithere.Computingthetimederivativegives

h(t) = 2(t)aeatu(t).9


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TheLaplace transformof this timederivative isa 2s + a

(t)}= 2 = .L{hs + a s + a

Usingthe

correct

derivative

rule

s 2s + a

sH(s)h(0) = + 1 =s + a s + a

gives a consistent result. Finally, the initial value theorem gives a correct result for both h and itsderivative,h(0+) = 1 andh(0+) = a, althoughwedontshow thedetailshere.

Inconclusion,theformulas(2),(3),(4),and(5)givecorrectresults.Wehopethatthesignalexamplespresentedabove help toclarify theapplicationof theseformulas.

III. THE HORROR OF0+Several versions of the Laplace transform are often found in the literature, which differ in the lower

limitof integration. Somebooksquote theLaplace integralasL0{f(t)}= f(t)est dt (8)

0

which createsan ambiguity withrespect tosignularities at theorigin.Otherbookssolve the ambiguity at theoriginbyspecifying the0+ versionof the transform

L+{f(t)}= f(t)est dt. (9)0+

To differentiate between thisformand thepreferedformL{f(t)}= f(t)est dt

0

theseformsof the transform are oftenreferred to asL andL+ [5].UsingtheL+ formoftheLaplacetransformrendersthewaveformsinSectionIIandtheirderivatives

indistinguishable. AconsistentsetofLaplace transform propertiescanbeconstructedusingL+,namely,L+{f(t)}= f(t)est dt

0+

L+{f(t)}= sF(s)f(0+)

andlim sF(s) = f(0+).s

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Note that 0 appears as 0+ in all of the above equations. There is a symmetric, perhaps attractive,consistency in this appearance. However, this form of the Laplace transform produces unuseful, trivialanswers to manyengineering problems. Mostembarrassingly, note that

L+{(t)}=

(t)est dt = 00+

which shouldstrike thereaderasutternonsense, at leastforpracticalapplications.IV. CONCLUSION

ForaconsistentsetofpropertiesfortheunilateralLaplacetransformthatareuseful toengineers,usethedefinition

dt,L{f(t)}= f(t)est0

thederivativerule(t)}= sF(s)f(0),L{f

and the initial value theoremlim sF(s) = f(0+).s

These properties give students general tools that can be used to solve differential equations with bothnon zero initial conditions and inputs. These also work properly in the presence of input singularities.Otherpresentations of theunilateral transform are notacceptable.

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2

1 0 1 2

0

1

2

3

4f(t)

2

1 0 1 2

4

3

2

1

0f(t)

2 1 0 1 22

1

0

1

2g(t)

2 1 0 1 22

1

0

1

2

1

g(t)

2 1 0 1 22

1

0

1

2h(t)

2 1 0 1 22

1

0

1

2

2

h(t)

Fig. 5. Three functionsf(t) = eat,g(t) = eatu(t),h(t) = eatu(t) u(t), plotted fora = 1, and their derivatives,definedforall time.Thesewaveformsareusedfor illuminating transformproperties.

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APPENDICES

Many treatments of the unilateral Laplace transform in the textbook literature are self contradictory. These inconsistencies arise from a reluctance to use the generalized derivative; a consistent and easilyunderstood treatment can be based on a uniform use of the generalized derivative. In the followingappendices, we provide a reasonable mathematical foundation for the use of generalized functions inassociation with dynamic systems and, in particular, in the context of the Laplace transform. The bookbyHoskins[1] givesfurtherperspectiveon the issuespresentedbelow.

APPENDIX IGENERALIZED FUNCTIONS

Discontinuoussignalsareapartoflife.Generalizedfunctionsariseinanyconsistenttreatmentoftheirderivatives. Onerequirement isauniform descriptionof theLaplace transformof thederivative.

Inmanytextbooksdealingwiththeone sided Laplacetransform,restinitialconditionsaretakenasa standard case. At the least these conditions include f(0) = 0, so any signal f(t) with f(0+) = 0presents us with a discontinuity at t = 0. This assumption is overly restrictive, as we should expect aconsistent set of Laplace definitions to admit non zero pre initial conditions on states and signals. Therest initial condition isjustaspecialcase.

Mostelementarytextbooks,bothinmathematicsandinengineering,attempttominimizethediscussionofsingularityfunctionsbygivinganyt = 0 discontinuityspecialtreatment.Alongtheway,theydisguisethe fact that there is a discontinuity at zero at all, and this exacerbates the confusion of functions suchas1 and the unit step function (which have the same Laplace transform and the same valuef(0+) butwhich have derivativeswithdifferentLaplace transforms).

Thetreatmentofgeneralizedfunctionspresentedbelowisjustasmallpartofthestandardmathematicaltheory of distributions, but what we present is both sufficiently concrete to be quickly understood, andsufficientlygeneraltobewidelyuseful.Wewilldescribethebehaviorofsingularityfunctionsinawaywhichwehopesupportsthecommonintuitionaboutthem.Becauseofourfocusontheone sided Laplacetransform, we will only consider generalized functions on the interval [0,), and we incorporate intotheinformationcarriedbyageneralizedfunctionthevaluesat0 ofthefunctionandallitsderivatives.

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APPENDIX IIDEFINITION OF GENERALIZED FUNCTIONS

A functionf(t) on [0,) ispiecewise continuous if there is a sparse setS[0,) such thatf(t)is continuous on the complement ofS, and for everya [0,) the limitsf(a+) andf(a) exist. (AsetSofnumbers issparse ifno intervaloffinitewidthcontains infinitely manyof itselements.)

This last clause requires special interpretation in casea = 0. In this case we are still demanding thevalueoff(0),but this isnowadditional information,notdeterminedby thevaluesof thefunctionon[0,).Thismightbe called apre initial condition, incontrastwithf(0+) which ispost initial.

Afunctionf(t) onaninterval[0,) ispiecewisesmoothifthereisasparsesetS[0,) suchthatallderivativesoff(t) existonthecomplementofS,andarethemselvespiecewisecontinuousfunctionson [0,).

Thus part of the data of a piecewise smooth function on [0,) is the sequence of valuesf(n)(0),n = 0, 1, 2 . . ..Thisdata isnot implied by valuesoff(t) fort 0.

Inthissetting,theHeavisideunitstepfunctionu(t) isdefinedtotakeonthevalue1forallt > 0 andhave zero pre initial conditions,u(n)(0) = 0 for alln 0. We leave the reader to choose a value foru(0); the value ofageneralizedfunctionatany single point isnot significant, asexplainedbelow.

Asingularityfunction isaformal sumof theformfs(t) = ck,l

(l)(tak)k,l

where(l) isthelth derivativeoftheunitimpulse.Hereak isanincreasingsequencein[0,).Foreachvalue of k, only finitely many of the ck,l are allowed to be nonzero. The sum can be over a finite orinfiniterangeofvaluesofk.If it is infinite,werequire that lim ak = .

kTheak for which someck,l is nonzero are the singular points offs(t). The singularity at t = a

offs(t) is the part of this sum havingak = a. If this sum is empty,fs(t) is said to be nonsingular att = a.

We also comment here that the unit step function u(t) = (1)(t) is taken as unitless. Succeedingderivativesintroduceincreasingnegativepowersoftime.Theunitsofthesingularityfunctions(l)(t) arethussec(l+1).Ifwewishtousethesefunctionsinassociationwithphysicalquantities,thenthefunctionsmust be multiplied by weighting terms with appropriate units. For example, in a voltage waveform,the functions (l)(t) must be multiplied by weighting terms ck,l which have units of Volt sec(l+1). Asanotherexample,aforceimpulseF0(t) hasaweightingtermF0 withunitsofN sec. ThetermF0 then

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corresponds to the area under a finite duration force event, such as a hammer blow, which we mightmodelwithan impulse function.

Ageneralizedfunction is a formalexpressionf(t) = fr(t) + fs(t)

whoseregularpartfr(t) isdeterminedbyapiecewisesmoothfunctionandwhosesingularpartfs(t)isasingularityfunction. The regularpartcontributes the dataf(n)(0) = fr(n)(0).

There isasubletyhere.Twopiecewisesmoothfunctionsdetermine thesamegeneralizedfunctions if(andonlyif)theydifferfromeachotheronasparseset.ThisiswhythevalueoftheHeavisidefunctionatt = 0 isirrelevant.Thiscaveatisnecessaryinorderfortheusualuniquenesstheoremforsolutionsofdifferentialequations toholdforgeneralizedfunctions.

Generalized functions can be differentiated to give other generalized functions. The generalizedderivativeofapiecewisesmoothfunctionfr(t) isgivenbytheordinaryderivativeoffr(t) atitssmoothpoints, along with the specification that (f)(n)(0) = fr(n+1)(0), plus the sum of delta functions oftheform

(f(a+)f(a))(ta)for each pointa at whichf(t) isdiscontinuous. Thederivative of a singularity function simply follows

(l) = (l+1).thenotationGeneralizedfunctionscannotgenerallybe multiplied together,andsuchgeneralproductsneveroccur

inapplicationsoftheLaplacetransform.However,iff(t) isasmoothfunctionthenitsproductwithanygeneralized functiong(t) isdefined,and theproductrule

(fg)(t) = f(t)g(t) + f(t)g(t)holdsinthisgenerality.Indeed,thisruleisusedtoderivethedefinitionoftheproduct.Westartwiththerelation

f(t)(ta) = f(a)(ta). (10)We can use this and the product rule to express all other products of the formf(t)(n)(t a), wheref(t) is a smooth function, as linear combinations of (nk)(t a)s with constant coefficents. To seethis,differentiate(10)andapply this same identitywithf(t) replacingf(t) tofind

f(t)(ta) = f(a)(ta)f(a)(ta).15


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Continuing in the sameway leadsby induction to the identitynn

f(t)(n)(ta) = (1)k

f(k)(a) (nk)(ta).kk=0

Thisformulaspecifieshowageneralizedfunction istobemultipliedbyasmoothfunctionf(t).Italsoexplains exactly how smoothf(t) has to be before the productf(t)(n)(t a) makes sense: it and itsderivativesuptof(n)(t) shouldexistatt = a andbecontinuousthere.ThisresultalsoappearsinHoskins[1] asequation3.26.

Theproductrulecanbeintegratedtogettheusualintegrationbypartsformula,validforanysmoothfunctionf(t) andanygeneralizedfunctiong(t): b+ b+ b+

f(t)g(t) dt = f(t)g(t) f(t)g(t) dta aa

Ifwe apply this to thecasea = 0,we will use the valuesf(0) andg(0) which comeas part of thedefinitionofthesefunctions.Partofthedefinitionofbeingsmooth(asopposed topiecewisesmooth)is

f(t)g(t) dt = (f(b)g(b+)f(0)g(0)) f(t)g(t) dt.0 0

APPENDIX II IEQUATION (3),TH E DERIVATIVE RUL E

An important feature of generalized functions is that (assuming appropriate growth conditions) one

(0)f ,so:b+

candefine theirLaplace transforms.ThebasicsingularityfunctionssatisfynsL{(n)(t)}=

Its easy tocheck that thedifferentiation rule is compatiblewith this.The derivative rule (3)

(t)}= sF(s)f(0).L{fcanbedirectlyderivedfrom thedefinition(2) through integrationbyparts:

thatf(0) = b+

e

stfL{f

(t)}= (t) dt .0Integrating by parts

u dv = uv v duu = est dv = f(t) dt

du = sest dt v = f(t)

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results in

estf(t) dt0

estf(t)

estf(t) dt= + s0 0

= f(0) + sL{f(t)}Provided one systematically uses the generalized derivative, this formula is valid for any generalizedfunctionf(t) provideds hasreal part largeenoughfor the improper integrals toconverge.

APPENDIX IVEQUATION (4),TH E INITIALV ALUE THEOREM

The initial value theorem islim sF(s) = f(0+).

s1

Actually, it is more accurate to call this the post initial value theorem, since it yields the result at0+,butwewillstaywithstandardterminologyandrefertoitastheinitialvaluetheorem.Thisresultcanbederivedviaseveralapproacheswhichprovidealternateinsights.Weshowthreeapproachesbelow,whichare based upon a) a formal application of the derivative rule, b) a qualitative argument from the siftingpropertyofsest in the limit, and c) the initial singularity theorem.

A. From theDerivativeRuleThe initial value theorem(4) canbederived from thederivativerule (3).If

(t)}= sF(s)f(0)L{fthen

sF(s) =

f(t)est dt + f(0).0

Taking the limit ass goes to infinityalong the realaxisgiveslim sF(s) = lim

f(t)est dt + f(0)

0s1 0+s1 = lim f(t)e0 dt +

f(t)est dt + f(0)

0+0s1 0+ = lim f(t) + 0 + f(0) = f(0+)

s1 0

which is theexpectedresult.17


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polynomialwhichcarriespreciselytheinformationaboutthesingularityoff(t) att = 0.Toexpressthiswe willuse thenotationF(s) G(s) tomean

lim (F(s)G(s)) = 0 .s1

The initial singularityformulaasserts that ifcl

(l)(t)l

is thesingularityoff(t) att = 0, andF(s) = L{f(t)}, thenlF(s) cls .

l

This comesoutof thevalueL{(n)(t)}= sn togetherwith the two factslim 0 ,

s1

L{fr(t)}=and

lim 0 for a > 0 .s1

L{(n)(ta)}=TheInitialValueTheoremarisesbyapplyingtheInitialSingularityTheoremtof(t).Thesingularity

off(t) att = 0 iscl

(l+1)(t) ,(f(0+)f(0))(t) +l

withLaplace transforml+1

(f(0+)f(0)) + cls ,lso thetderivative rule(3) implies

L{f l+1sF(s)f(0) = (t)}(f(0+)f(0)) + cls .l

Canceling thef(0)s,l+1sF(s) f(0+) + cls .

l

In particular, iff(t) isnonsingular att = 0, thenlim sF(s) = f(0+) .s1

More generally, with the above analysis, we can see that the value f(0+) exists even if there is asingularityatt = 0,asalso noted byKailath[8].That is, ifF(s) is a polynomialplus afunctionF(s)converging tozeroass 1 then

lim sF(s) = f(0+) .s1

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Forexample,consider thefirst order transforms + a

F(s) = = 1 +ab

.s + b s + b

This has a post initial valuef(0+) = a b, even thoughf(t) includes an impulse at t = 0, as can beseenusingourresultabove, orviadirect inversion of the transform.

APPENDIX VINVERSE

Manybooks quote the inverseLaplace transform1 +j

L1{F(s)}=2j j e

stF(s) ds.

However,thisresultisprimarilyoftheoreticalinterest,andwehavenotfounduseforitinundergraduateteaching. The limited order systems typically studied with Laplace techniques are readily solved viapartial fraction expansion and inversion via tables. As well, this formula requires the use of complexintegration,whichisnotarequiredcomponentofmostundergraduateengineeringcurricula.Thuswedonot recommendmore than abrief mention of thisresult inundergraduate engineeringcourses.

APPENDIX VICONFUSION IN TH E TEXTBOOK LITERATURE

ThetextbookliteratureissurprisinglyrandominitstreatmentoftheunilateralLaplacetransform;manyotherwise excellent texts fumble this issue. As is clear from the above discussions, we regard theLfromastheonlysuitableapproachforthestudyofdynamicsystems.Thisformisproperlyadoptedinanumberoftexts,forexample,thoseofFranklin,Powell,andEmami Naeini [12],DeCarloandLin[13],Kailath[8],andSiebert[4].WehavefoundthediscussioninSiebert[4],Chapters2and11,particularlyhelpful.Ourexamplewaveforms insectionIIarebased on Problem 11.17 in Sieberts text.

Many authors use theL+ version of the unilateral Laplace transform. This unfortunate choice canlikelybetracedbacktotheclassic1942textbyGardnerandBarnes[9].Thetitleofthispaper,TroublesattheOrigin,wasinspiredbysection3 6 ofAseltine[10].However,whilethistextcorrectlyidentifiesthe problem, it reaches the wrong conclusion! TheL+ version of the transform also seems preferredin mathematically oriented treatments such as the frequently cited book by Zemanian [2]. The controlsystemsbookbyDAzzoandHoupisadoptstheL+ transforminthedefinitionofthederivativeproperty,but isabit unclear as to the lower limiton theLaplace integral.

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However,theL+ formofthetransformhasthehugedisadvantagethatitleavestransientsatt = 0 asessentially an exercise for the reader, presumably to be solved by some time domain technique such asimpulse matching. TheexampleproblemsinsectionIcouldnotbeefficientlysolvedwiththeL+ form;with

this

form

the

transform

of

the

unit

impulse

is

identically

zero!

This

little

problem

is

frequently

circumvented by defining the delta function as occurring completely to the right of t = 0. What thendoes 0+ mean to these authors, and what time value is associated with the initial value theorem? Wesurely cant teach our engineering students to study transients via theL+ transform. We also owe it tothemnot to leave this issuefuzzy by avoidingdiscussionof0+ and0.

Some books add to the confusion by showing bothL+ andL forms, with little or no commentas to when to chose between them. This includes the texts by Ogata [5] [6]. The book by Close andFrederick[3]statesthatbothformsareacceptableaslongasthecorrespondingpropertiesaredevelopedinaconsistentfashion.However, this leads tosomeconvolutedreasoningassociatedwiththe transformof the unit impulse andwith thederivativeproperty.

Furthermore,textbooksthatsimplyclaimL{f(t)}= sF(s)f(0) orlimssF(s) = f(0) withoutfurtherclarificationmustbeconsideredsuspect(evenotherwisefinemathbookssuchasChurchill[11]).The systems book by Palm [7] states that using the one sided transform is the same as assuming thatsignals are zerofort < 0,which isnot true.

In summary, while the books referenced above have much to recommend them, the issue of Laplacetechniques requires clarification in many of these references, and in our teaching of systems. We hopethat thediscussion in thispaperwillbehelpful toward that end.

REFERENCES[1] RolandF.Hoskins.DeltaFunctions:AnIntroductiontoGeneralisedFunctions. HorwoodPublishingLimited,WestSussex,

England,1999.[2] A.H.Zemanian.DistributionTheory andTransformAnalysis. McGraw Hill, NewYork,NY,1965.[3] CharlesM.CloseandDeanK.Frederick.ModelingandAnalysisofDynamicSystems,2nd Ed. JohnWileyandSons,New

York,NY,1995.[4] William McC.Siebert. Circuits,Signals,andSystems,pages 357358. MITPress,Cambridge,MA,1986.[5] Katsuhiko Ogata.ModernControlEngineering,4th Ed. Prentice Hall, Englewood Cliffs,NJ,2002.[6] Katsuhiko Ogata. SystemDynamics,4th Ed.. Prentice Hall, EnglewoodCliffs,NJ,2004.[7] WilliamJ.PalmIII.Modeling,Analysis,andControlofDynamicSystems,2nd Ed.. JohnWileyandSons,NewYork,NY,

2000.[8] Thomas Kailath.LinearSystems,page12. Prentice Hall, EnglewoodCliffs,NJ,1980.[9] MurrayF.GardnerandJohnL.Barnes. Transients inLinearSystems. Wiley,NewYork,NY,1942.

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[10] JohnA.Aseltine. TransformMethod inLinearSystemsAnalysis,pages2729. McGraw Hill, New York,NY,1958.[11] RuelV.Churchill. Complex VariablesandApplications. McGraw Hill, NewYork,NY,1960.[12] Gene Franklin, David Powell, and Abbas Emami Naeini. Feedback Control ofDynamic Systems,4th Ed. Prentice Hall,

UpperSaddleRiver,NJ,2002.[13] RaymondA.DeCarloandPen Min Lin.LinearCircuitAnalysis,2nd Ed. OxfordUniversityPress,NewYork,NY,2001.

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