Laplace Transforms. Class 16. Laplace Transforms. Laplace Transforms of Common Functions. Inverse Laplace Transform, L -1: Chapter 3.

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1 Laplace Tranform Definiion Cla 6 Laplace Tranform Reminder: Dean Lecure omorrow am, JSB Audiorium Dr. L. Dougla Smoo Energy & Climae Change Imporan analyical mehod for olving linear ordinary differenial equaion. - Applicaion o nonlinear ODE? Mu linearize fir. Laplace ranform play a key role in imporan proce conrol concep and echnique. -Example: Tranfer funcion Frequency repone Conrol yem deign Sabiliy analyi The Laplace ranform of a funcion, f(, i defined a F () L [ f () f() e d (-) where F() i he ymbol for he Laplace ranform, L i he Laplace ranform operaor, and f( i ome funcion of ime,. Noe: The L operaor ranform a ime domain funcion f( ino an domain funcion, F(). i a complex variable: a bj, j Invere Laplace Tranform, L -: By definiion, he invere Laplace ranform operaor, L -, conver an -domain funcion back o he correponding ime domain funcion: () L F f Imporan Properie: Boh L and L - are linear operaor. Thu, () () () () ax by (-) L ax by al x bl y where: - x( and y( are arbirary funcion - a and b are conan - Similarly, [ x( and Y L[ y( X L L ax by ax b y Laplace Tranform of Common Funcion. Conan Funcion Le f( a (a conan. Then from he definiion of he Laplace ranform in (-), a a a L ( a) ae d e (-) 6

2 . Sep Funcion The uni ep funcion i widely ued in he analyi of proce conrol problem. I i defined a: for < S() for (-) Becaue he ep funcion i a pecial cae of a conan, i follow from (-) ha L S() (-6) 7. Derivaive Thi i a very imporan ranform becaue derivaive appear in he ODE we wih o olve. In he ex (p.), i i hown ha df L F d f Similarly, for higher order derivaive: ( n ) ( n ) (-9) iniial condiion a Fir derivaive n d f n n n () L F f f ( ) n d... f f (-) 8 where: - n i an arbirary poiive ineger - k k d f f ( ) k d Special Cae: All Iniial Condiion are Zero Suppoe n f f... f. Then n d f n L F n d In proce conrol problem, we uually aume zero iniial condiion. Reaon: Thi correpond o he nominal eady ae when deviaion variable are ued, a hown in Ch.. 9. Exponenial Funcion b Conider f () e where b >. Then, b b ( b ) L e e e d e d ( b ) e b b. Recangular Pule Funcion I i defined by: (-6) f ( h w Time, 6. Impule Funcion (or Dirac Dela Funcion) The impule funcion i obained by aking he limi of he recangular pule a i widh, w, goe o zero bu holding he area under he pule conan a one. (i.e., le h ) Le, impule funcion w Then, δ ( ) δ ( L for < f () h for < w (-) for w The Laplace ranform of he recangular pule i given by h w ( e ) F (-)

3 Table.. Laplace Tranform See page of he ex. Laplace able (con.) Pracice a. S( (Sep funcion wih a magniude of ) b. e -6 in c. d y d where 6 6 d y dy, d d, y() F F Soluion of ODE by Laplace Tranform Procedure:. Take he L of boh ide of he ODE.. Rearrange he reuling algebraic equaion in he domain o olve for he L of he oupu variable, e.g., Y().. Perform a parial fracion expanion.. Ue he L - o find y( from he expreion for Y(). 6 Pracice Solve he following equaion: dy - y e y() d Y () Y ( ) Y y( e e ( ) Y ( ) Y Ue # in Table. ( )( ) ( )( ) Check Anwer: y() / / y( e e y ( e e y( e e y( e e y ( y( e 7 Parial Fracion Expanion Baic idea: Expand a complex expreion for Y() ino impler erm, each of which appear in he Laplace Tranform able. Then you can ake he L - of boh ide of he equaion o obain y(. Example: Y ( )( ) Perform a parial fracion expanion (PFE) α α (-) (-) where coefficien α and α have o be deermined. 8

4 To find α : Muliply boh ide by and le - α To find α : Muliply boh ide by and le - α A General PFE Conider a general expreion, N n π ( b ) N Y (-6a) D i i Here D() i an n-h order polynomial wih he roo ( b i ) all being real number which are diinc o here are no repeaed roo. Special Siuaion: Two oher ype of iuaion commonly occur when D() ha: i) Complex roo: e.g., bi ± j ( j ) ii) Repeaed roo (e.g., b ) b For hee iuaion, he PFE ha a differen form. See SEM ex (pp. 7-8) for deail. The PFE i: Y n π i i N( n ) αi i b b i Noe: D() i called he characeriic polynomial. (-6b) 9 Parial Fracion Example Repeaed Facor Addiional Noe 6 ( ) α α α To ge α, muliply boh ide by and e ( 6) α α α ( )( ) ( 6) α 6 / ( )( ) Now ge α ( ( ) ( ) 6 : ) ( )( ( ) ) ( ( ) ( ) 6) Finally ge α : So now olve for f(: f ( e ( )( ( ) ) e α α F() ( ) α α ( ) How do you ge α and α? Muliply ou denominaor and mach like power of. ( )( ) α( ) α( ) ( ) ( ) ( ) α ( ) α ( α ) ( α α ) Therefore, α, and α α. Thi mean ha α -. So Invering F( ) ( ) ( ) f ( e e. Final value heorem (Eq. -8) ( ) [ Y y lim. Iniial value heorem (Eq. -8) ( ) [ Y y lim. Time delay (Real Tranlaion Theorem, Eq. -96) G L{ f ( ) S( )} e F

5 More Pracice More Pracice More Pracice Pracice: Wrie he Laplace form of a funcion ha doe he double e, (a) changing a o a value of, (b) changing o a value of - a min, and (c) changing o a value of a 6 min. e e 6 Wrie he ime domain form of he following Laplace funcion and kech i: e e e ( ) [ S( ) 6( 6) [ S( 6) ( 9) [ S( 9) 9 Deermine he final value of he following funcion: F () F 6 ( )( ) ( 6) ( )( ) ( 6) ()( ) f( 6 8 ime 6 7 Exra 8 Example. Solve he ODE, dy y d y( ) (-6) Fir, ake L of boh ide of (-6), ( Y ) Y Rearrange, Y (-) Take L -, y () L ( ) From Table., How do you ge (-7)?.8 y ().. e (-7) 9 Parial Fracion Expanion. y() L ( ). ( ) (.8)..8. (.8) (.8) (.8) ( ) (.8) L (.8) (.8) e..8 8 e [( e ).8..e. ( e ) (#9 in able wih b )

6 Example. (coninued) Recall ha he ODE, &&& y 6&& y y& 6y, wih zero iniial condiion reuled in he expreion Y (-) 6 6 The denominaor can be facored a ( ) ( )( )( ) 6 6 (-) Noe: Normally, numerical echnique are required in order o calculae he roo. The PFE for (-) i α α α α Y (-) Solve for coefficien o ge,,, 6 6 α α α α (For example, find α, by muliplying boh ide by and hen eing.) Subiue numerical value ino (-): /6 / / /6 Y() Take L - of boh ide: L /6 / / /6 Y L L L L From Table., y() e e e (-) 6 6 Imporan Properie of Laplace Tranform. Final Value Theorem I can be ued o find he eady-ae value of a cloed loop yem (providing ha a eady-ae value exi. Saemen of FVT: lim Y lim y providing ha he limi exi (i finie) for all Re, where Re () denoe he real par of complex variable,. Example: Suppoe, Y ( ) (-) Then, y( ) lim y( lim.

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be