2. The Laplace Transform
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1 . The Lplce Trnform. Review of Lplce Trnform Theory Pierre Simon Mrqui de Lplce ( French tronomer, mthemticin nd politicin, Miniter of Interior for 6 wee under Npoleon, Preident of Acdemie Frncie under Loui XVI. Complex Vrible vrible coniting of rel nd imginry quntitie σ + ω Grphicl Repreenttion of Complex Number in complex plne (Re rel, Im imginry or (σ, ω Exmple, σ + ω Im o o4 o - 3 Re -
2 Complex Function Function of complex number re clled complex function. A n exmple: G ( G + x G y Both G x nd G y re rel quntitie. The mgnitude of G( i G( ( G x + G y The ngle θ of G( i θ tn - (G y / G x The ngle te poitive vlue when meured counterclocwie from the poitive rel xi. The complex conugte of the function G( i G( G x G y Pole nd Zero pole -vlue for which the function G( tend towrd infinity zero -vlue for which the function G( equl zero -
3 Exmple The complex function K( + ( + 5 G( ( + ( + 5( + 5 h - zero t -, -5 - ingle pole t, -, -5, - double pole (multiple pole of order t -5. Note tht G(, i.e i n infinite zero while -, - re finite zero. Exmple The bove complex function i equivlent to ( K ( + / ( + 5 / K ( + / ( + 5 / ( + 5 / G 5 3 ( + / ( + 5 / ( + / ( + 5 / ( + 5 / For lrge vlue of lim K G( 3 i.e. G( h triple zero t. If infinite zero re included, G( h the me number of pole nd zero, 5 pole nd 5 zero. -3
4 Euler Theorem co θ + inθ e θ Im in θ co θ+ in θe θ coθ + in θ θ co θ Re From thi reult co( θ + in( θ coθ inθ e θ nd coθ inθ (e (e θ θ + e e θ θ -4
5 . Review of Lplce Trnform The Lplce trnform i ued to - trnform ytem of differentil eqution (of rel independent vrible, time into et of lgebric eqution (of complex vrible,. - obtin eily olution of the et of lgebric eqution. - obtin olution of the originl problem function of time, pplying the Invere Lplce trnform, Definition: f (t function of independent vrible time t, defined non-zero for t i.e. ƒ(t for t< F( Lplce trnform of ƒ(t; Lplce trnform of trnform f(t from the t-pce into F( in the - pce complex vrible L n opertor indicting tht the quntity tht it prefixe i to be trnformed by the Lplce integrl t f (te dt, i.e F( L{ƒ(t} The Lplce trnform of function ƒ(t i given by F( L{ƒ(t} f (te t dt L[ƒ(t] i n opertor pplied to f(t tht doe the following: - multiplie ƒ(t with e -t -integrte with regrd to time t the product between nd : nd return complex function F(. t f ( t e dt -5
6 Exmple: Unit tep function u(t u(t t u(t for t < for t > undefined for t, i.e. cn te ny vlue between nd. U( L{u(t} e t dt U( h no finite zero nd one zero vlue pole. -plne repreenttion of zero nd pole reult in one pole mred x in the origin Im x Re -6
7 Exmple: Unit rmp function u(t t f(t for t t for t > t F( L{f(t} t e dt U( h no finite zero nd two zero vlue pole xx. -plne repreenttion of zero nd pole reult in two pole mred xx in the origin Im xx Re -7
8 Exmple: Exponentil function f(t t f(t for t e -t for t > F( L{f(t} e t e t dt e (+ t dt + F( h no finite zero nd one non-zero pole p -. -plne repreenttion of zero nd pole reult in pole - mred x Im x Re p - -8
9 Exmple: Sinuoidl function f(t for t in (ωt for t > where, Euler identity give in ωt (e ωt F( L{f(t} e ωt in( ωt e t dt (e ( ω t + e ( ω t e t dt ω + ω The pole of F( re given by + ω or - ω tht give tow imginry pole p ω nd p - ω nd one non-zero pole p -. -plne repreenttion i Im p ω x Re p - ω x -9
10 Lplce Trnform Tble See Tble - Lplce Trnform Pir in the textboo. Propertie of Lplce Trnform Linerity For ƒ(t nd g(twith Lplce trnform F( L{f(t} G(L{g(t} their liner combintion f(t + b g(t h the Lplce trnform F( + b G( -
11 Time trnlted function A f(t time trnlted by time durtion i f(t- i.e. f(t for t h the me vlue the trnlted function f(t- t t f(t for t<, cn be written f(t-(t- where unit tep function trnlted by i given by (t- for t> for t< Given F( L{f(t} L{f(t - } e - F( for Exmple: Lplce Trnform of Pule Function f(t of mplitude A nd durtion i f(t(a/ (t - (A/ (t- L{f(t} (A/(- exp(- -
12 Rel Differentition Given F( L{f(t} nd initil condition of f(t, Lplce trnform of derivtive of f(t re obtined follow -firt derivtive of f(t d L [ f (t] F( f ( dt where ƒ( i the initil vlue of ƒ(t evluted t t. -econd derivtive of f(t d L [ dt f (t] F( f ( f &( -n-th derivtive of f(t d L [ dt n n n n n f (t] F( f ( f &(... f (n ( f (n ( For zero initil vlue Lplce trnform of the nth derivtive of ƒ(t i given by x n F(. A time derivtive in the time domin become multipliction by in the Lplce domin. Exmple Given tht co (ωt (/ω d/dt [in (ωt] nd ω L{ in (ωt } + ω ω L{ co (ωt } L{ d/dt [in (ωt] } ω + ω + ω -
13 Rel Integrtion L f (tdt F( + f ( where F( L{ƒ(t} nd ƒ - ( ƒ(t dt evluted t t. Finl Vlue Theorem If F( h ll pole on the left-hnd ide of the imginry xi nd no more thn ingle pole in the origin, then lim t f (t lim F( Stedy tte repone of ytem in the time domin cn be obtined from the limit goe to zero of the Lplce trnform of the function multiplied by. (See Exmple - Initil Vlue Theorem lim t f (t f ( lim F( -3
14 Convolution Integrl If F( L{f(t} nd G( L{g(t} then the invere Lplce trnform of their product H( F( G( denoted ƒ(t*g(t nd clled the convolution of ƒ(t nd g(t i h(t L - {H(} L - {F( G(} t f (t τ g( τ dτ (See Tble - Propertie of Lplce Trnform -4
15 .3 Invere Lplce Trnform Invere Lplce trnform of the complex function F( reult in the correponding time function ƒ(t L [F(] f (t π c c + F(e t d for t > In prctice, Lplce trnform in not obtined uing the bove complex integrtion but by uing Lplce Trnform Pir tble either directly or by proceing F( until it i trnformed in prt found the tble tble. The method for uing indirectly the Lplce Trnform Pir tble by proceing F( until it i trnformed in prt found the tble i the prtil frction expnion method. Prtil Frction Expnion F( B( A( n B n i A i i In control ytem nlyi, F( i frequently occur in the form of rtio of polynomil, clled lo rtionl function where A( nd B( re polynomil in. In ppliction, the highet power of in A( be greter or equl to the highet power of in B(. If not, the numertor B( i divided by the denomintor A( in order to produce polynomil in plu reminder the numertor of new rtionl function. If F( i trnformed in um of component F( F ( + F ( +...F n ( -5
16 Such tht the invere Lplce trnform of F (, F (,, F n ( re vilble from the Lplce Trnform Pir tble L {F(} L {F (} + L {F (} L {F (} f ( + f n ( +...f Prtil frction expnion method i pplied differently, depending the type of pole: ditinct pole b multiple pole c complex conugte pole Rtionl Function with Ditinct Pole After the clcultion of the root of A( n, The zero z, z,..,z m nd B( n i b i i, the pole p, p,..,p n, where n m, the numertor nd denomintor polynomil cn be fctored follow F( B( A( K( + z ( + ( + p ( + p z..( + z m + p n...( Thi form of the cn be converted into um of imple prtil frction tht cn be found in the Lplce Trnform Pir tble n ( F( B( A( K( + z ( + ( + p ( + p z..( + z m + p n ( + p...( + ( + p n ( + p n -6
17 Wht would be required i to find the contnt, clled reidue, for,,,n, correponding to the n-pole p, uch tht the bove right hnd ide um of prtil function i equl to F(. i obtined follow ( + ( + p p B( A( p ( + p + ( + p ( + p ( + p ( + p + ( + p n ( + p n p i.e i obtined from, B( ( + p A( p b Function with Multiple Pole F( B( A( ( + N p ( m B N + p...( p n N n + h pole order of multiplicity higher or equl to, N for p, N for p,., N n for p n, -7
18 F( ( + B( A( N p ( + p N ( m B + p N N N ( + p ( + p Thi cn be trnformed into um of imple prtil frction, which would require to find the contnt N, correponding to the n-pole p, uch tht the bove right hnd ide um of prtil function i equl to F(. N i obtined hown in the exmple from the textboo, pge in (Prtil-Frction Expnion when F( Involve Multiple Pole c Complex conugte pole N...( + ( + N p... + n N ( + p n If p nd p re complex conugte pole, then the reidue nd re lo complex conugte uch tht only one need to be evluted. From Lplce Trnform Pir tble + p n N n L ( + p p t e p t p t p [ ] n t f (t L F( e + e n e for t (See Exmple -4 to -5, A--4-8
19 Prtil-Frction Expnion with MATLAB Conider the following function B(/A( B( n B F ( A( n A i i i MATLAB progrm num [B n B n- B ] den[a m A m- A ] num den The commnd for clculting [r,p,] reidue(num,den The reidue nd the pole p, give the -th prtil-frction for ll,,...,n ( + p (See Exmple -6 nd -7-9
20 .4 Solving Liner Time Invrint-Ordinry Differentil Eqution (LTI-ODE Lplce Trnform LTI-ODE (See Exmple -8 nd -9 Exmple Thi m-dmper-pring (m-b- ytem i ubect to pplied force f(t nd h zero initil condition. Clculte X(. m f(t b x(t Newton econd lw give m & x + bx& + x f (t Lplce Trnform of thi eqution, for zero initil condition, give m X(+bX(+X(F( or -
21 (m +b+x(f( Solving the bove lgebric eqution in X(F( / (m +b+ Invere Lplce trnform for unit impule input force F( give X( / (m +b+ or X(( / m/( +b/m+/m b Obtin x(t for b X(( /m/( + /m (/(/m/( + /m Lplce Trnform Pir tble give L ω ( + ω inωt Denote ω /m uch tht X( cn be written follow X(( /m/( + /m (/( /m/( + /m (/( /m ( /m/( + /m (/( /m (ω/( + ω Conequently x(t (/( /m in ω t (/ m in ( /mt (See Exmple A-- to A-- 7 -
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