Copyrigh F.L. Lewi 999 All righ reerved EE 434 - Conrol Syem LECTURE REVIEW OF LAPLACE TRANSFORM LAPLACE TRANSFORM The Laplace ranform i very ueful in analyi and deign for yem ha are linear and ime-invarian (LTI. Beginning in abou 9, ranform echnique were applied o ignal proceing a Bell Lab for ignal filering and elephone long-line communicaion by H. Bode and oher. Tranform heory ubequenly provided he backbone of Claical Conrol Theory a praciced during he World War and up o abou 96, when Sae Variable echnique began o be ued for conrol deign. Pierre Simon Laplace wa a French mahemaician who lived 749-87, during he age of enlighenmen characerized by he French Revoluion, Roueau, Volaire, and Napoleon Bonapare. where Given a ime funcion f(, i unilaeral Laplace ranform i given by F ( = f ( e d, = + i a complex variable. Differen auhor may ake he lower limi a + (i.e. no including effec occurring exacly a ime = inead of Laplace ranform ha a lower limi of heory a he ULT. One may wrie F ( = f ( e e d. The bilaeral and i no a ueful for feedback conrol and, comparing hi o he Fourier ranform, one ee ha f( may have a Laplace ranform hough i Fourier ranform doe no exi. Thi i due o he fac ha he weighed funcion f ( e decay faer han f ( o ha i Fourier ranform may exi. The invere Laplace ranform i a complex inegral given by + f ( = π F( e d, j
where he inegraion i performed along a conour in he complex plane. Since hi i ediou o deal wih, one uually ue he Cauchy heorem o evaluae he invere ranform uing f ( = Σ encloed reidue of F( e. In hi coure we hall ue lookup able o evaluae he invere Laplace ranform. In hi coure we hall ue following noaion for he uni e ignal: u ( u - ( u - ( Uni impule Uni ep Noe ha each funcion i he inegral of he previou funcion. Uni ramp An abbreviaed able of Laplace ranform i given here. The ex ha a more deailed able. Noe ha we are dealing wih he -ided ranform o ha all ime funcion hould be conidered o be muliplied by he uni ep. I i imporan o repreen a complex pair of pole in a good way. We hall no pli a complex pole pair ino wo ingle roo uing 'j'. Inead, we hall wrie ( +α + β. Noe ha hi ha roo where: ( + α ( + α + β = β + α = ± jβ =. = α ± jβ We hall dicu he complex pole pair in deph when we dicu yem performance and econd-order yem.
Table of Laplace Tranform Time Funcion Tranform Pole/Zero Plo u ( u ( j u ( ( n u e α ( n! n+ + α -α in β β + β jβ -jβ coβ + β jβ -jβ α e b aα ( a coβ + in β β a + b ( +α + β jβ -b/a -α -jβ 3
PROPERTIES OF THE LAPLACE TRANSFORM Several properie of he Laplace ranform are imporan for yem heory. Thu, uppoe he ranform of x (, y( are repecively X (, Y (. Then one ha he following properie. Laplace Tranform Properie Time-Domain Operaion Lineariy propery ax ( + by( Time caling propery x(a Time hifing propery x( Frequency hifing propery a x( e Convoluion propery x( * y( Differeniaion Propery x' ( Differeniaion Propery ( x n ( Inegraion propery x(τ dτ Frequency Domain Operaion ax ( + by ( a X X ( e a X ( + a X ( Y ( X ( x( n X ( X ( + ( x( L x ( n n x( τ dτ In hi able, a upercrip encloed in parenhei denoe differeniaion of ha order. An aerik denoe convoluion, x( * y( = x( τ y( τ dτ = x( τ y( τ dτ. The upper and lower convoluion limi are a hown ince all of our ime funcion are caual (i.e. muliplied by he uni ep in hi coure. Noe ha, in he inegraion propery here i a correcion erm in he ranform if he ime funcion x( i no zero o he lef of ime =. Since all our funcion are caual, hi erm i uually equal o zero in hi coure. The ime caling propery ae he fac ha if ime i caled one way, hen frequency cale in an oppoie manner. For example, if a > o ha he ime axi i compreed, hen he frequency cale i expanded. 4
Erwin Schrodinger developed an elegan heory of quanum mechanic uing he premie ha poiion and momenum are Laplace ranform pair. Thi allowed him o how he imporance of hi famou wave equaion. In hi conex, he ime caling propery i known a he Heienberg Uncerainy Principle. In fac, i ae ha if poiion i known more accuraely (i.e. i probabiliy deniy funcion i on a compreed cale, hen momenum i known le accuraely (i.e. i PDF i on an expanded cale. A an inereing final commen, i urn ou ha in quanum mechanic here are oher ranform pair beide poiion/momenum. One of hee i ime/energy. INITIAL AND FINAL VALUE THEOREMS Two heorem are indipenable in feedback conrol deign. Iniial Value Theorem (IVT. lim X ( = x(. The IVT ay ha he (fa ranien behavior of x( i deermined by he high-frequency conen of i pecrum X(. Final Value Theorem (FVT. lim lim X ( = x(. The FVT ay ha he eady-ae, or DC behavior, of x( i deermined by he lowfrequency conen of i pecrum X(. Example + 3 Le a ime funcion x( have he ranform X ( =. Uing he IVT 3 + 4 + 3 and he FVT one can deermine he iniial and final value of x( wihou ever having o find he funcion x( ielf. Thi i very imporan for quick deign inigh laer in he coure. Boh he IVT and he FVT rely on he quaniy 3 + 3 X ( =. 3 + 4 + 3 5
According o he IVT, aking he limi of X( a goe o infiniy yield he iniial value of x ( =. To ake hi limi noe ha he highe-order erm in he numeraor and denominaor dominae. According o he FVT, aking he limi of X( a goe o zero yield he final value for x( of. To ake hi limi noe ha he lowe-order erm in he numeraor and denominaor dominae. Taking he invere ranform (which we hall cover oon of X( yield 3 x = ( e e u (. A kech of hi funcion i given below. ( x( - 6