Laplaceova transformacija

Laplasova transformacija (nazvana po Pjer-Simon Laplasu) je integralna transformacija, koja datu kauzalnu funkciju f(t) (original) preslikava iz vremenskog domena (t = vreme) u funkciju F(s) u kompleksnom spektralnom domenu. Laplasova transformacija, iako je dobila ime u njegovu čast, jer je ovu transformaciju koristio u svom radu o teoriji verovatnoće, transformaciju je zapravo otkrio Leonard Ojler, švajcarski matematičar iz osamnaestog veka.

Pojam originala

Funkcija t->f(t) naziva se originalom ako ispunjava sledeće uslove:

1. f je integrabilna na svakom konačnom intervalu t ose

2. za svako t<0, f(t)=0

3. postoje M i s₀, tako da je ${\displaystyle |f(t)|\leq Me^{s_{0}t}}$

Definicija Laplasove transformacije

${\displaystyle F(s)={\mathcal {L}}\left\{f(t)\right\}=\int _{0}^{\infty }e^{-st}f(t)\,dt.\qquad (s=\sigma +\mathrm {i} \omega$ ;\quad \sigma >0;\quad t\geq 0)}

Funkcija F(s) je »slika« ili laplasova transformacija »originala« f(t).

Za slučaj da je ${\displaystyle s=i\omega }$ dobija se jednostrana Furijeova transformacija:

${\displaystyle F(\omega )={\mathcal {F}}\left\{f(t)\right\}}$

${\displaystyle ={\mathcal {L}}\left\{f(t)\right\}|_{s=i\omega }=F(s)|_{s=i\omega }}$

${\displaystyle =\int _{0}^{\infty }e^{-\imath \omega t}f(t)\,\mathrm {d} t.}$

Osobine

Linearnost

${\displaystyle {\mathcal {L}}\{\sum _{k=1}^{n}\alpha _{k}f_{k}(t)\}=\sum _{k=1}^{n}\alpha _{k}F_{k}(s)}$

Teorema sličnosti

Ako je ${\displaystyle a>0}$, tada je ${\displaystyle {\mathcal {L}}\{f(at)\}={1 \over a}F({s \over a})}$, pri čemu je ${\displaystyle {\mathcal {L}}\{f(t)\}=F(s)}$

Diferenciranje originala

Ako je ${\displaystyle a>0}$ i ${\displaystyle {\mathcal {L}}\{f(t)\}=F(s)}$, tada je ${\displaystyle {\mathcal {L}}\{f^{\prime }(t)\}=sF(s)-f(0)}$

Diferenciranje slike

Ako je ${\displaystyle {\mathcal {L}}\{f(t)\}=F(s)}$, tada je ${\displaystyle {\mathcal {L}}\{tf(t)\}=-F^{\prime }(s)}$, odnosno indukcijom se potvrđuje da važi ${\displaystyle {\mathcal {L}}\{t^{n}f(t)\}=(-1)^{n}F^{(n)}(s)}$

Integracija originala

Ako je ${\displaystyle a>0}$ i ${\displaystyle {\mathcal {L}}\{f(t)\}=F(s)}$, tada je ${\displaystyle {\mathcal {L}}\{\int _{0}^{t}t(\tau )d\tau \}={1 \over s}F(s)}$

Integracija slike

Ako postoji integral ${\displaystyle \int _{0}^{\infty }F(s)ds}$, tada je ${\displaystyle {\mathcal {L}}\{{f(t) \over t}\}=\int _{s}^{\infty }F(\sigma )d\sigma }$

Teorema pomeranja

${\displaystyle {\mathcal {L}}\{{e^{s_{0}t}f(t)}\}=F(s-s_{0})}$

Teorema kašnjenja

${\displaystyle {\mathcal {L}}\{{f(t-\tau )}\}=-e^{-s\tau }F(s),\tau \geq 0}$

Laplasova transformacija konvolucije funkcija

${\displaystyle {\mathcal {L}}\{f*g\}={\mathcal {L}}\{f\}{\mathcal {L}}\{g\}}$

Ova osobina je poznata kao Borelova teorema. Napomena: definicija konvolucije je: ${\displaystyle (f\ast g)(x)=\int _{-\infty }^{+\infty }f(x-t)\cdot g(t)\cdot dt=\int _{-\infty }^{+\infty }f(t)\cdot g(x-t)\cdot dt}$

Laplasova transformacija periodičnih funkcija

Ako ${\displaystyle f(t)}$ ima osobinu ${\displaystyle f(t+T)=f(t)}$, tada važi ${\displaystyle {\mathcal {L}}\{f(t)\}=\int _{0}^{T}{e^{-su} \over {1-e^{-sT}}}f(u)du}$

Dokaz

${\displaystyle {\mathcal {\,}}\int _{0}^{\infty }e^{-st}f(t)dt=\int _{0}^{T}e^{-st}f(t)dt{\mid }_{t=u}+\int _{T}^{2T}e^{-st}f(t)dt{\mid }_{t=u+T}\int _{2T}^{3T}e^{-st}f(t)dt{\mid }_{t=u+2T}+...\,}$

${\displaystyle {\mathcal {\,}}\int _{0}^{\infty }e^{-st}f(t)dt=\int _{0}^{T}e^{-st}f(u)du+\int _{T}^{2T}e^{-s(u+T)}f(u+T)du+\int _{2T}^{3T}e^{-s(u+2T)}f(u+2T)du+...\,}$

${\displaystyle {\mathcal {\,}}\int _{0}^{\infty }e^{-st}f(t)dt=\int _{0}^{T}e^{-su}f(u)du+e^{-sT}\int _{0}^{T}e^{-su}f(u)du+e^{-2sT}\int _{0}^{T}e^{-su}f(u)dT+...\,}$

${\displaystyle {\mathcal {\,}}\int _{0}^{\infty }e^{-st}f(t)dt=(1+e^{-sT}+e^{-2sT}+...)\int _{0}^{T}e^{-sT}f(T)du{\mid }_{u=t}\,}$

Odakle sledi: ${\displaystyle {\mathcal {v}}\{f\}={1 \over 1-e^{-Ts}}\int _{0}^{T}e^{-st}f(t)\,dt}$

Tabela najčešće korišćenih Laplasovih transformacija

Jednostrana Laplasova transformacija ima smisla samo za ne-negativne vrednosti , stoga su sve vremenske funkcije u tabeli pomožene sa Hevisajdovom funkcijom.

ID	Funkcija	Vremenski domen ${\displaystyle x(t)={\mathcal {L}}^{-1}\left\{X(s)\right\}}$	Laplasov -domen (frekventni domen) ${\displaystyle X(s)={\mathcal {L}}\left\{x(t)\right\}}$	Oblast konvergencije za kauzalne sisteme
1	idealno kašnjenje	${\displaystyle \delta (t-\tau )\ }$	${\displaystyle e^{-\tau s}\ }$
1a	jedinični impuls	${\displaystyle \delta (t)\ }$	${\displaystyle 1}$	${\displaystyle \mathrm {} \ s\,}$
2	zakašnjeni -ti stepen sa frekvencijskim pomeranjem	${\displaystyle {\frac {(t-\tau )^{n}}{n!}}e^{-\alpha (t-\tau )}\cdot u(t-\tau )}$	${\displaystyle {\frac {e^{-\tau s}}{(s+\alpha )^{n+1}}}}$	${\displaystyle {\textrm {Re}}\{s\}>0\,}$
2a	-ti stepen (za ceo broj )	${\displaystyle {t^{n} \over n!}\cdot u(t)}$	${\displaystyle {1 \over s^{n+1}}}$	${\displaystyle {\textrm {Re}}\{s\}>0\,}$
2a.1	-ti stepen (za realno )	${\displaystyle {t^{q} \over \Gamma (q+1)}\cdot u(t)}$	${\displaystyle {1 \over s^{q+1}}}$	${\displaystyle {\textrm {Re}}\{s\}>0\,}$
2a.2	Hevisajdova funkcija	${\displaystyle u(t)\ }$	${\displaystyle {1 \over s}}$	${\displaystyle {\textrm {Re}}\{s\}>0\,}$
2b	zakašnjena Hevisajdova funkcija	${\displaystyle u(t-\tau )\ }$	${\displaystyle {e^{-\tau s} \over s}}$	${\displaystyle {\textrm {Re}}\{s\}>0\,}$
2c	rampa funkcija	${\displaystyle t\cdot u(t)\ }$	${\displaystyle {\frac {1}{s^{2}}}}$	${\displaystyle {\textrm {Re}}\{s\}>0\,}$
2d	frekvencijsko pomeranje -tog reda	${\displaystyle {\frac {t^{n}}{n!}}e^{-\alpha t}\cdot u(t)}$	${\displaystyle {\frac {1}{(s+\alpha )^{n+1}}}}$	${\displaystyle {\textrm {Re}}\{s\}>-\alpha \,}$
2d.1	eksponencijalno opadanje	${\displaystyle e^{-\alpha t}\cdot u(t)\ }$	${\displaystyle {1 \over s+\alpha }}$	${\displaystyle {\textrm {Re}}\{s\}>-\alpha \ }$
3	eksponencijalno približavanje	${\displaystyle (1-e^{-\alpha t})\cdot u(t)\ }$	${\displaystyle {\frac {\alpha }{s(s+\alpha )}}}$	${\displaystyle {\textrm {Re}}\{s\}>0\ }$
4	sinus	${\displaystyle \sin(\omega t)\cdot u(t)\ }$	${\displaystyle {\omega \over s^{2}+\omega ^{2}}}$	${\displaystyle {\textrm {Re}}\{s\}>0\ }$
5	kosinus	${\displaystyle \cos(\omega t)\cdot u(t)\ }$	${\displaystyle {s \over s^{2}+\omega ^{2}}}$	${\displaystyle {\textrm {Re}}\{s\}>0\ }$
6	sinus hiperbolikus	${\displaystyle \sinh(\alpha t)\cdot u(t)\ }$	${\displaystyle {\alpha \over s^{2}-\alpha ^{2}}}$	${\displaystyle {\textrm {Re}}\{s\}>|\alpha |\ }$
7	kosinus hiperbolikus	${\displaystyle \cosh(\alpha t)\cdot u(t)\ }$	${\displaystyle {s \over s^{2}-\alpha ^{2}}}$	${\displaystyle {\textrm {Re}}\{s\}>|\alpha |\ }$
8	eksponencijalno opadajući sinus	${\displaystyle e^{\alpha t}\sin(\omega t)\cdot u(t)\ }$	${\displaystyle {\omega \over (s-\alpha )^{2}+\omega ^{2}}}$	${\displaystyle {\textrm {Re}}\{s\}>\alpha \ }$
9	eksponencijalno opadajući kosinus	${\displaystyle e^{\alpha t}\cos(\omega t)\cdot u(t)\ }$	${\displaystyle {s-\alpha \over (s-\alpha )^{2}+\omega ^{2}}}$	${\displaystyle {\textrm {Re}}\{s\}>\alpha \ }$
10	-ti koren	${\displaystyle {\sqrt[{n}]{t}}\cdot u(t)}$	${\displaystyle s^{-(n+1)/n}\cdot \Gamma \left(1+{\frac {1}{n}}\right)}$	${\displaystyle {\textrm {Re}}\{s\}>0\,}$
11	prirodni logaritam	${\displaystyle \ln(t)\cdot u(t)}$	${\displaystyle 1 \over s}\,\left[\ln(s)+\gamma \right]}$	${\displaystyle {\textrm {Re}}\{s\}>0\,}$
12	Beselova funkcija prve vrste, reda	${\displaystyle J_{n}(\omega t)\cdot u(t)}$	${\displaystyle {\frac {\omega ^{n}\left(s+{\sqrt {s^{2}+\omega ^{2}}}\right)^{-n}}{\sqrt {s^{2}+\omega ^{2}}}}}$	${\displaystyle {\textrm {Re}}\{s\}>0\,}$ ${\displaystyle (n>-1)\,}$
13	modifikovana Beselova funkcija prve vrste, reda	${\displaystyle I_{n}(\omega t)\cdot u(t)}$	${\displaystyle {\frac {\omega ^{n}\left(s+{\sqrt {s^{2}-\omega ^{2}}}\right)^{-n}}{\sqrt {s^{2}-\omega ^{2}}}}}$	${\displaystyle {\textrm {Re}}\{s\}>|\omega |\,}$
14	Beselova funkcija druge vrste, nultog reda	${\displaystyle Y_{0}(\alpha t)\cdot u(t)}$	${\displaystyle -{2\sinh ^{-1}(s/\alpha ) \over \pi {\sqrt {s^{2}+\alpha ^{2}}}}}$	${\displaystyle {\textrm {Re}}\{s\}>0\,}$
15	modifikovana Beselova funkcija druge vrste, nultog reda	${\displaystyle K_{0}(\alpha t)\cdot u(t)}$
16	funkcija greške	${\displaystyle \mathrm {erf} (t)\cdot u(t)}$	${\displaystyle {e^{s^{2}/4}\left(1-\operatorname {erf} \left(s/2\right)\right) \over s}}$	${\displaystyle {\textrm {Re}}\{s\}>0\,}$
Objašnjenja: ${\displaystyle u(t)\,}$ predstavlja Hevisajdovu funkciju. ${\displaystyle \delta (t)\,}$ predstavlja Dirakovu delta funkciju. ${\displaystyle \Gamma (z)\,}$ predstavlja Gama funkciju. ${\displaystyle \gamma \,}$ je Ojler-Maskeronijeva konstanta. ${\displaystyle t\,}$, je realan broj koji obično predstavlja vreme, iako može da se odnosi na bilo koju dimenziju. ${\displaystyle s\,}$ je kompleksna ugaona frekvencija, a ${\displaystyle {\textrm {Re}}\{s\}}$ je njen realni deo. ${\displaystyle \alpha \,}$, ${\displaystyle \beta \,}$, ${\displaystyle \tau \,}$, and ${\displaystyle \omega \,}$ su realni brojevi. ${\displaystyle n\,}$, je ceo broj. Kauzalni sistem je sistem u kome je impulsni odziv nula za svako vreme . Vidi još: kauzalnost.

Inverzna Laplasova transformacija

U opšti slučaj, original f(t) date slike F(s) dobija se rešavanjem Bromvičovog integrala:

${\displaystyle {\mathcal {L}}^{-1}\left\{F(s)\right\}={\frac {1}{2\pi \imath }}\int _{\gamma -\imath \infty }^{\gamma +\imath \infty }e^{st}F(s)\,ds\qquad \gamma >s_{0},}$

gde je ${\displaystyle s_{0}}$ realni deo bilo kog singulariteta funkcije ${\displaystyle F(s)}$.

S obzirom da se ovde integrali kompleksna promenljiva, potrebno je koristiti metode kompleksne matematičke analize. Mnogi primeri inverzne Laplasove transformacije navedeni su u tabeli iznad. U praksi, funkcije se transformišu u primere iz tablice, na primer razlaganjem na proste faktore.

Diskretna Laplasova transformacija

Za funkciju celobrojne promenljive ${\displaystyle f(n)}$ njena diskretna Laplasova transformacija se definiše kao:

${\displaystyle F^{\ast }(s)=\sum _{n=0}^{\infty }e^{-sn}f(n)}$

Konvergencija ovog reda zavisi od ${\displaystyle s}$.

Sve osobine i teoreme regularne Laplasove transformacije imaju svoje ekvivalente u diskretnoj Laplasovoj transformaciji.

Primena

U matematici Laplasova transformacija se koristi za analiziranje linearnih, vremenski nepromenljivih sistema, kao: električnih kola, harmonijskih oscilatora, optičkih uređaja i mehaničkih sistema. Ima primene u rešavanju diferencijalnih jednačina i teoriji verovatnoće.

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Vanjske veze

• http://mathworld.wolfram.com/LaplaceTransform.html
• http://www.mathe.braunling.de/Laplace.htm Arhivirano 2006-07-21 na Wayback Machine-u
• http://mo.mathematik.uni-stuttgart.de/aufgaben/L/laplace_transformation.html
• http://www3.htl-hl.ac.at/homepage/bok/dt/mathe/mindex.html Arhivirano 2016-03-04 na Wayback Machine-u
• http://www.seeit.de/xedu/formeln/Lars%20Weiser/laplace.pdf Arhivirano 2005-05-20 na Wayback Machine-u
• http://www-hm.ma.tum.de/archiv/mw4/ss05/folien/Laplace.pdf Arhivirano 2016-03-06 na Wayback Machine-u
• http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&Page=1020