Laplace Transform. National Chiao Tung University Chun-Jen Tsai 10/19/2011

Transcription

1 plc Trnorm Nionl Chio Tung Univriy Chun-Jn Ti /9/

2 Trnorm o Funcion Som opror rnorm uncion ino nohr uncion: d Dirniion: x x, or Dx x dx x Indini Ingrion: x dx c Dini Ingrion: x dx 9 A uncion my hv nicr propry in h rnormd domin! /7

3 Ingrl Trnorm I x, y i uncion o wo vribl, hn dini ingrl o w.r.. on o h vribl ld o uncion o h ohr vribl. Exmpl: xy dx y Impropr ingrl o uncion din how ingrion cn b clculd ovr n inini inrvl: K, d lim b b K, d /7

4 plc Trnorm Diniion: b uncion dind or, hn h ingrl { } d i id o b h plc rnorm o, providd h ingrl convrg. Th rul o h plc rnorm i uncion o, uully rrrd o F. 4/7

5 Exmpl: {} By diniion: d lim b b d lim b b lim b b, providd >. Th ingrl divrg or <. 5/7

6 Exmpl: {} By diniion: d Uing ingrion by pr nd lim, >, w hv: { } {} d 6/7

7 7/7 Exmpl: { } By diniion: d d >, } {

8 Gmm Funcion Γx Th gmm uncion i dind or x > Sinc Γ x x d. Γ x x d x x x d. For x >, w hv Γx x Γx, hror, gmm uncion i clld h gnrlizd coril uncion. No: Γ nd Γx x! i x i poiiv ingr. 8/7

9 Exmpl: { } I w l u, u/, nd d du/, hn by diniion, or >, { } Γ u d u du, or ll > o h >. I n i nonngiv ingr, Γn n!, nd w hv n n! { }, > n. 9/7

10 Exmpl: {in } By diniion: { in } in d in co d co d, > co 4 in d 4 { in } { in }, > /7

11 /7 inriy o {} For um o uncion, w cn wri whnvr boh ingrl convrg or > c. Hnc, ] [ d g d d g β α β α } { } { } { G F g g β α β α β α

12 Exmpl: { in } { in } { } {in } / { co 6} / [/ / 6], >. /7

13 Uni Sp Funcion Th uni p uncion u i dind o b, < u., u i on dnod u. No h u i only dind on h non-ngiv xi inc h plc rnorm i only dind on hi domin. 8 /7

14 Rwri o Picwi Funcion A picwi dind uncion cn b rwrin in compc orm uing u. For xmpl, g, h, < i h m g gu hu. y y 4/7

15 plc Trnorm o u, > By diniion, Thror, { u } u d lim b b. { u } >, >. d 5/7

16 Trnorm o Bic Funcion { } { } n n! { }, n,,, n {co k} k {in k} k k {coh k} k {inh k} k k { u } 6/7

17 Exinc o {} Thorm: I i picwi coninuou on [,, nd h i o xponnil ordr or > T, whr T i conn, hn {} convrg. b Diniion: A uncion i id o b o xponnil ordr c i hr xi conn c, M >, nd T > uch h M c or ll > T. 7/7

18 Exmpl: Exponnil Ordr Th uncion,, nd co r ll o xponnil ordr c or >, inc w hv,, co. M c, c > T 8/7

19 9/7 Proo o Exinc o {} By h ddiiv inrvl propry o dini ingrl, Th ingrl I xi ini inrvl, picwi coninuou. Now, I xi wll {} convrg. } { I I d d T T., c c M c M d M d M d I T c T c T c T T c >

20 /7 Exmpl: Trnorm o Picwi Evlu {} or Soluion: { }., > d d d d <.,, y

21 Invr plc Trnorm I F i h plc rnorm o uncion, nmly, {} F, hn w y h i h invr plc rnorm o F, h i, Exmpl: {F}. {/}, {/ }, nd {/ }. /7

22 /7 Exmpl: Invr Trnorm Evlu {/ 5 } Soluion: Evlu {/ 7} Soluion: ! 4! in

23 /7 inriy o {} Th invr plc rnorm i lo linr rnorm; h i, or conn α nd β, Exmpl: Evlu { 6 / 4} } { } { } { G F G F β α β α in co

24 Exmpl: Pril Frcion / Evlu Soluion: Thr xi uniqu conn A, B, C uch h: 6 9 A 4 A 4 B 4 C 4 By compring rm, w hv B C 4 4/7

25 5/7 Exmpl: Pril Frcion / Pril rcion: Thror

26 Trnorming Driviv Wh i h plc rnorm o '? { } d Thror { } { } F d No h hi drivion only work i i coninuou uncion 6/7

27 Driviv Trnorm Thorm Thorm: I h uncion i coninuou nd picwi mooh or nd i o xponnil ordr, o h hr xi nonngiv conn M, c, nd T uch h M c or T. Thn { } xi or > c, nd { } F. Proo: Prorm ini pic-by-pic ingrion o d 7/7

28 Gnrl Driviv Trnorm Thorm: I, ',, n r coninuou on [, nd r o xponnil ordr nd i n i picwi coninuou on [,, hn { } F n n n n n, whr F {}. 8/7

29 Solving inr IVP / Th plc rnorm o linr DE wih conn coicin bcom n lgbric quion in X. Th i, bcom or n d x d x n n d d n n x n n d x d x n n d d { } n n n { x} { }, n [ n X n x n x x n ] n [ n X n x x n ] X F 9/7

30 Solving inr IVP / Givn iniil condiion, x x, x' x,, x n x n, w hv ZX I F, or dy bhvior F X Z I, Z rnin bhvior whr Z n n n n nd I n n n n x n n n n x' n x n. /7

31 dy d y in, y Exmpl: / 6 Sinc dy { y} {in }, d {dy/dx} Y - y Y-6, nd {in} / 4, w hv 6 Y 6 Y, 4 or 6 Y 6, Y 4. 4 /7

32 /7 Exmpl: / Aum h w hv A 8, B, C 6. Thror, C B A y y in co 8 6, in y y d dy

33 /7 Exmpl: Soluion: 5,, 4 y y y y y } { } { 4 y d dy d y d 4 ] [ Y y Y y y Y Y Y y } {

34 Trnorm o Ingrl Thorm: Th plc rnorm o h ingrl o picwi coninuou uncion o xponnil ordr i F τ dτ { }. Th invr orm i: - F τ dτ. Rcll h: {'} F. 4/7

35 Proo o Trnorm o Ingrl Sinc i picwi coninuou, by undmnl horm o clculu, i g τdτ, g i coninuou nd g' whr i coninuou. Bcu i o xponnil ordr, hr xi conn M nd c uch h c c g d M τ τ τ dτ < c g i o xponnil ordr. Thu, {} {g'} {g} g. Bu g, hror, M { } F τ dτ { g }. M c c. 5/7

36 6/7 Exmpl: Invr by Ingrion Sring wih in, F / w hv:. co in, in co, co in d d d τ τ τ τ τ τ τ

37 Bhvior o F I i picwi coninuou on [, nd o xponnil ordr or > T, hn lim {}. Proo: Sinc i picwi coninuou on T, i i ncrily boundd on h inrvl. Th i M. Alo, M γ or > T. I M dno h mximum o {M, M } nd c dno h mximum o {, γ}, hn or > c: { } M c c d M M c c d. 7/7

38 8/7 Pril Frcion Dcompoiion Finding invr plc rnorm uully involv pril rcion dcompoiion, l P b polynomil uncion wih dgr l hn n: inr cor dcompoiion: whr A, A,, A n r conn. Qudric cor dcompoiion whr A,, A n nd B,, B n r conn.,... n n n A A A P, ] [... ] [ ] [ n n n n b B A b B A b B A b P

39 -xi Trnlion Thorm Thorm: I {} F nd i ny rl numbr, hn { } F. Proo: { } d d F. 9/7

40 4/7 Exmpl: { 5 } nd { co 4} Soluion: ! } { } { 6 6 } 4 {co } 4 co {

41 4/7 Invr o -xi Trnlion Th invr plc rnorm o F, cn b compud by muliply {F} by, ollow. Exmpl: Compu {5/ }. Sinc } { } { F F, 5.

42 4/7 Exmpl: y" 6y' 9y Solv h DE wih iniil condiion y, y' Y. 4! 4! 4 5 y

43 Convoluion o Two Funcion I nd g r picwi coninuou on [,, hn pcil produc, dnod by g, i dind by h ingrl g τ g τ dτ nd i clld h convoluion o nd g. Th convoluion i uncion o. No h g g. Exmpl: in τ in τ dτ in co. 4/7

44 44/7 Convoluion Thorm Thorm: I nd g r picwi coninuou on [, nd o xponnil ordr, hn Proo: τ β, d dβ, o h { } { } { }. G F g g. τ β β τ β β τ τ β τ β τ d d g d g d G F { }. g d d g G F τ τ τ

45 45/7 Exmpl: { }. in d τ τ τ { } { } { } in in d τ τ τ

46 46/7 Invr Form o Convoluion Thorm: Exmpl: F G / k, { }. g G F - k - [ ]. co in co co, in in k k k k d k k k d k k k k - τ τ τ τ τ

47 Solving Ingrl Equion W cn u convoluion horm o olv dirnil quion wll ingrl quion. For xmpl, h Volrr ingrl quion: g τ h τ dτ, whr g nd h r known. 47/7

48 48/7 Exmpl: Soluion: noic h h. Tk h plc rnorm o ch rm: Th invr rnorm hn giv: F F. d τ τ τ

49 Dirniion o Trnorm Thorm Thorm: I i picwi coninuou nd i o xponnil ordr, hn d { } { } F. Proo: d d d F d d d d d d { } { } { F } - [ ] { } d 49/7

50 nh-ordr Trnorm Dirniion Thorm: I F {} nd n,,, hn n n n d { } F n d Proo: Th proo cn b don by mhmicl inducion. Hr, w only chck h nd -ordr c. d d d d { } { } { } { } Exmpl: Compu { in k}. d d { in k} {in k} d d k k k k 5/7

51 5/7 Exmpl: x" 6x co4, x, x' Soluion: {x"} {6x} {co 4}, Sinc, rom prviou xmpl,. in 4 8 in in x k k k X

52 5/7 Ingrion o Trnorm Thorm Thorm: I i picwi coninuou wih xponnil ordr, nd h lim / xi nd i ini. Thn, In ddiion, w hv Proo:. d F σ σ { } { }. - - d F F σ σ [ ] { }. / / d d d d d d F σ σ σ σ σ σ σ σ

53 5/7 Exmpl: {inh /} W ir vriy h h uncion i boundd whn : Now, w hv. lim lim inh lim { }. ln ln inh inh d d d σ σ σ σ σ σ σ σ

54 54/7 -xi Trnlion Thorm Thorm: I F {} nd >, hn { u } F. Proo: v, dv d, d d u d u d u { } { } dv v u v

55 55/7 Invr o -xi Trnlion I {F} nd >, h invr orm o h - xi rnlion horm i: Exmpl:. } { u F. i, i, < u

56 56/7 Alrniv Form o -xi Trnlion For g h lck h prci hid orm g, w cn driv n lrniv orm: Exmpl: Sinc gπ co π co, }. { } { } { g u g dv v g d g u g v. } {co } {co u π π π

57 57/7 Exmpl: No h co u π, w hv {y'} {y} {co u π}, < π π y y y, co, 5,, Y π π π 5. co in 5 co in 5 π π π π π π u u y

58 Sri Circui Th currn in circui i govrnd by h ingrodirnil quion di d Ri C i τ dτ E. E R C 58/7

59 Exmpl: Singl-loop RC Circui Givn.h, R Ω, C., i, nd E -U-, ind i. Soluion: Sinc. di d i { } i τ dτ U nd i τ dτ I /, w hv I.I I. I,. 59/7

60 Exmpl: coninud, 8, < i /7

61 Trnorm o Priodic Funcion I priodic uncion h priod T, T >, hn T. Th plc rnorm o priodic uncion cn b obind by ingrion ovr on priod. Thorm: I i picwi coninuou on [,, o xponnil ordr, nd priodic wih priod T, hn { } T T d. 6/7

62 Proo o Priodic Trnorm Thorm Proo: { } T d T l u T, hn h nd rm bcom T Thror d T u T u u d, T du u du T T { } d { } T { } d. T T { } 6/7

63 6/7 Exmpl: Squr-Wv Trnorm Find h rnorm o qur-wv. Soluion: On priod o E cn b dind : < <,, E { }. d d d E E E 4

64 Exmpl: Priodic Inpu Volg / Th DE or i in ingl-loop R ri circui i di Ri E. d Drmin i whn i nd E i h qurwv in h prviou xmpl. Soluion: / I RI I R / Sinc x x x x K K 64/7

65 65/7 Exmpl: Priodic Inpu Volg / Sinc w hv By pplying h -xi rnlion horm:, / / / / R R R R. / R R I / / / u u R u u u R i R R R

66 Exmpl: Priodic Inpu Volg / Thror i R R / R n n R n/ u n. For xmpl, i R,, nd < 4, w hv i /7

67 Uni Impul Qui on, h inpu o phyicl ym i hor priod, lrg mgniud uncion. Thi yp o uncion cn b dcribd by δ,,, < <. Th uncion δ i clld uni impul bcu / y δ d. 67/7

68 Dirc Dl Funcion Din δ limδ. Th uncion δ i clld Dirc dl uncion. δ i chrcrizd by:, i δ,, ii δ d. 68/7

69 69/7 Trnorm o δ Thorm: For >, {δ }. Proo: [ ] { } { } { }. lim lim,, u u δ δ δ δ No h {δ}. δ i no norml uncion inc {δ}.

70 7/7 Exmpl: Two IVP / Solv y" y 4δ π, wih iniil condiion y, y', nd b y, y'. Soluion : Th plc rnorm i: Y Y 4 π,. 4in co. 4 π π π u y Y., 4 in co, co < π π y y - 4π π

71 Exmpl: Two IVP / Soluion b Th plc rnorm i π 4 Y. Thror, y 4in π u π y, 4in, < π π. - π 4π 7/7

72 7/7 Impul Rpon Conidr nd -ordr linr ym wih uni impul inpu : x" x' x δ, x, x'. Applying plc rnorm o h ym: w i h zro- rpon o h ym o uni impul, hror, w i clld h impul rpon o h ym.. w Z x W Z X

73 7/7 inr Dynmic Sym Rcll h or gnrl linr dynmic ym, w hv W /Z i clld h rnr uncion o h ym. No h. Z I Z F X { } { }. I W F W x zro- rpon zro-inpu rpon