Y () s. Lecture 8. Laplace Transform. () t. Introduction u S. Laplace Transform: definition

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1 Iroducio u S y cur 8 u U alac Traorm Dirial quaio Tim domai alac domai Algbraic quaio y Y - uca rrarii - Baic Auomaic Corol uca rrarii - Baic Auomaic Corol alac Traorm: diiio Obrvaio :R C C C : whr σ j C [ ] [ ] I i a comlx ucio o comlx variabl I i idd o valu o or < [ ] Δ d hi i cary o imo om codiio ovr or h igral xic covrgc axi. uca rrarii - Baic Auomaic Corol uca rrarii - Baic Auomaic Corol

2 Examl S raorm Th uiary amliud i h ollowig im ucio: [ ] Th alac raorm o h i: < d i R > w will o coidr h covrgc axi ay mor, rom ow o uca rrarii - Baic Auomaic Corol 5 Examl Imul raorm Th ucio i did by h ollowig rorii: im, im d Wha do h rlaio ma? I i oibl o hi o a ucio did a ollow: <, > ad ay ha: im lim uca rrarii - Baic Auomaic Corol Thror, o calcula h alac Traorm o h imul: im lim [ im ] lim [ ] [ ] d d [ ] [ im ] lim lim uca rrarii - Baic Auomaic Corol 7 Haviid horm Rlaio bw ad imul im im τ d τ im τ dτ d d uca rrarii - Baic Auomaic Corol 8

3 alac Traorm: rorii. iariy [ ] [ ]. Tim domai hi [ ], < τ, τ > [ τ ] [ β ] β, β C uca rrarii - Baic Auomaic Corol 9 uca rrarii - Baic Auomaic Corol Examl Tim hi. alac domai hi [ ] a a uca rrarii - Baic Auomaic Corol uca rrarii - Baic Auomaic Corol

4 Examl alac domai hi Examl alac domai hi a ma, a > a a [ ] a [ i] j j j j j j j ha i [ i ] j j j j j j uca rrarii - Baic Auomaic Corol uca rrarii - Baic Auomaic Corol. alac domai diriaio [ ] [ ] d d Examl alac domai diriaio [ a ] d d a a whr [ ] a a uca rrarii - Baic Auomaic Corol 5 uca rrarii - Baic Auomaic Corol

5 5. Tim domai diriaio [ ] d d [& ] Examl Tim domai diriaio [ co] d i d d i d I i dicoiuou a h lim uca rrarii - Baic Auomaic Corol 7 uca rrarii - Baic Auomaic Corol 8 Exio o highr ordr drivaiv cod ordr [&& ] & & [ ] & &. Tim domai igraio [ ] [ ] τ dτ -ordr d d d d uca rrarii - Baic Auomaic Corol 9 uca rrarii - Baic Auomaic Corol

6 Examl Tim domai igraio ram τ dτ [ ram ] ram ram 5 uca rrarii - Baic Auomaic Corol a im ram a i co a a co a a a a uca rrarii - Baic Auomaic Corol a i Rmarabl raorm 7. Traorm o vcor x x M X [ x ] x [ x ] M [ x ] X M X Pol ad zro o a alac raorm Pol: valu o whr Zro: valu o whr Prorii.. [ Ax ] A [ x ] AX [ x& ] X x [ ] x& M M [ x& ] X x X x Rmarabl ca: raioal N D N ad D ar olyomial i Pol: roo o Zro: roo o D N uca rrarii - Baic Auomaic Corol uca rrarii - Baic Auomaic Corol

7 coidr agai h iiial chm i ordr o iroduc h maig o h ivr alac raorm: u Dirial quaio u S Tim domai y Y - y alac domai Algbraic quaio uca rrarii - Baic Auomaic Corol 5 U Ivr alac raorm Thr i a bijcio rlaio i h ucio ar idical: or xc or o ull dimio igl oi - uca rrarii - Baic Auomaic Corol Evaluaio o h ivr alac raorm [ ] Exlici ormula -> a comlx igral ha w will o udy Iiial valu horm ial valu horm Haviid xaio oly i S i raioal Iiial valu horm [ ] lim i ii I i dicoiuou a h horm i modiid a, ha i lim uca rrarii - Baic Auomaic Corol 7 uca rrarii - Baic Auomaic Corol 8

8 Examl Iiial valu horm & [ ] [ co ] lim? [& ] i ac co & lim i ac & i ial valu horm [ ] Hyoh ha oly: ol wih gaiv ral ar, or ull ol, ha i lim lim i ii uca rrarii - Baic Auomaic Corol 9 uca rrarii - Baic Auomaic Corol Examl ial valu horm [ co ] Pol i ± j Th ial valu horm cao b alid! [ ] lim uca rrarii - Baic Auomaic Corol Ivr raorm uig Haviid xaio I ca b alid oly i i raioal N D b a b a m m b a m m Th ida i o dcomo i a um o lm who ivr raorm ar ow. uca rrarii - Baic Auomaic Corol

9 uca rrarii - Baic Auomaic Corol Oly h ollowig ca will b coidrd: wih diic ral ol wih mulil ral ol wih comlx cojuga ol wih h domiaor ordr qual o h umraor m uca rrarii - Baic Auomaic Corol K i i i a D K i ol i co j i j i, Ral diic ol uca rrarii - Baic Auomaic Corol 5 Examl Calcula h ivr raorm o 7 mu b idical 7 uca rrarii - Baic Auomaic Corol I i cary o olv a ym o hr quaio ad hr uow

10 Th i i oibl o calcula h ivr raorm Ral mulil ol > D I ha mulil ol β β β K Or quivally: 5 5, β! β β uca rrarii - Baic Auomaic Corol 7 uca rrarii - Baic Auomaic Corol 8 Examl Calcula h ivr raorm o uca rrarii - Baic Auomaic Corol 9 mu b idical Th i i oibl o calcula h ivr raorm ram Or quivally:, uca rrarii - Baic Auomaic Corol

11 Comlx cojuga ol σ j σ j D β γ σ β γ βσ βσ β σ ol i σ ± j σ γ βσ σ σ σ σ co σ i β γ σ γ βσ σ σ... β co i... uca rrarii - Baic Auomaic Corol uca rrarii - Baic Auomaic Corol Examl Calcula h ivr raorm o 5 β γ 5 mu b 5 idical 5 β γ 5 β β γ 5 γ 5 β γ β β γ 5 γ uca rrarii - Baic Auomaic Corol Th i i oibl o ma h ollowig xaio:...? 5 Thr i o a immdia ivr raorm I i oibl o rwri h domiaor o h cod rm i a dir way: 5 Wha i h ivr raorm? uca rrarii - Baic Auomaic Corol

12 uca rrarii - Baic Auomaic Corol 5 Rmmbr ha: σ σ σ, co, i σ σ Th: uca rrarii - Baic Auomaic Corol Th i i oibl o calcula h ivr raorm, i co uca rrarii - Baic Auomaic Corol 7 I h umraor ad h domiaor hav h am ordr, i h xaio o w mu iroduc a coa rm. K im... N D Zro rlaiv ordr m uca rrarii - Baic Auomaic Corol 8 Examl Calcula h ivr raorm o mu b idical

13 Th i i oibl o calcula h ivr raorm y u y & Examl alac Traorm ad dirial quaio Y U [ y ] [ u ] Y y Y U im Or quivally: im, Wih u y y Y U Y Y Y uca rrarii - Baic Auomaic Corol 9 uca rrarii - Baic Auomaic Corol 5 Y y Cocluio Simly alyig h vry baic rul o alac Traorm w ar abl o olv dirial quaio, wihou comuig comlx igral oluio agrag ormula ad comuig h xoial o marix A uca rrarii - Baic Auomaic Corol 5 uca rrarii - Baic Auomaic Corol 5