Transformation de Laplace

In mathematica e, in particular, in analyse functional, le transformation de Laplace de un function $f(t)$, definite pro tote numero real $t\geq 0$, es le function $F(s)$, definite per:

Transformation de Laplace
instantia de: integral transform[*]
subclasse de: integral transform[*]
Commons: Laplace transformation

$F(s)=\left\{{\mathcal {L}}f\right\}(s)=\int _{0^{-}}^{\infty }e^{-st}f(t)\,dt.$

Le limite inferior de $0^{-}$ es un notation parve pro significar $\lim _{\epsilon \rightarrow +0}-\epsilon \$ e garantir le inclusion del function delta de Dirac $\delta (t)\$ al 0 se ha tal impulsion in $f(t)$ al 0.

Iste integral transformation ha un numero de proprietates que face lo utilisabile pro analisar linear systemas dynamic. Le major significative avantage es que differentiationes e integrationes cambia se in multiplicationes e divisiones, respectivemente, con $s\$. (Iste es similar al modo con que logarithmos cambia operationes de multiplication de numeros pro le summa de su logarithmos.) Iste cambia equationes integral e equationes differential al equationes polynomial, que es multe plus facile de solver. Le transformation inverse de Laplace es le integral de Bromwich, que es un integral complexe date per:

$f(t)={\frac {1}{2\pi i}}\int _{\gamma -i\infty }^{\gamma +i\infty }e^{st}F(s)\,ds.$

ubi $\gamma \$ es un numero real selectionate de modo que le cammino de integration es in le region de convergentia de $F(s)\$, normalmente requirente $\gamma >\operatorname {Re} (s_{p})\$ pro tote singularitate $s_{p}\$ de $F(s)\$. Se tote singularitates es in le semi-plano sinistre, que es $\operatorname {Re} (s_{p})<0\$ pro tote $s_{p}\$, le $\gamma \$ pote esser fixate al zero e le formula integral inverse supra cambia se identic al le inverse transformation de Fourier.

Le transformation de Laplace pote esser extendite al transformation de Laplace duo-lateral o transformation de Laplace bilateral per ponente le intervallo de integration sur tote le axe real; se iste es facite le ordinari o unilateral transformation cambia se simplemente in un caso special consistente del transformationes facente uso de function degrau unitari de Heaviside in le definition del function essente transformate.

De plus, le transformation, ambe uni o bi-lateral es alicun vices definite un paucmente differentemente, per

$F(s)=\left\{{\mathcal {L}}f\right\}(s)=s\int _{0^{-}}^{\infty }e^{-st}f(t)\,dt.$

Le transformation de Laplace es multe usate in ingenieria. Le egresso de un systema dynamic linear pote esser calculate per convolutante su responsa impulsive unitari con le signal de ingresso. Performante iste calculo in le spatio de Laplace cambia le convolution in un multiplication, que multe vices pote facer le calculo plus facile.

Le transformation de Laplace es nominate in honor de Pierre-Simon Laplace.

Notation de ingenieros e physicos

Un multe vices conveniente abuso de notation, prevalente specialmente inter ingenieros e physicos, consiste in scriber isto in le sequente forma:

$F(s)=\left\{{\mathcal {L}}f\right\}(s)=\int _{0^{-}}^{\infty }e^{-st}f(t)\,dt.$

Quando on dice "le transformation de Laplace" sin qualification, le unilateral transformation es normalmente intendite. Le bilateral transformation es definite como a sequer:

$F_{B}(s)=\left\{{\mathcal {B}}f\right\}(s)=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.$

Le transformation de Laplace F(s) typicamente existe pro tote numero real s > a, ubi a es un constante que depende del comportamento de crescimento de f(t), enquanto le transformation bilateral es definite in le intervallo a < s < b.

Le transformation de Laplace pote etiam esser usate pro solver equationes differential e es usate extensivemente in ingenieria electric.

Relation al altere transformationes

Transformation de Fourier

Le continue transformation de Fourier es equivalente al calculo del transformation de Laplace con argumento complexo.

${\mathcal {F}}f(\omega )={\mathcal {L}}f(i\omega )=\int _{0^{-}}^{\infty }e^{-i\omega t}f(t)\,\mathrm {d} t.$

Iste equivalentia es usualmente usate pro determinar le spectro de frequentia de un signal o systema dynamic. Nota que le ${\frac {1}{\sqrt {2\pi }}}$ constante non es non includite.

Transformation de Mellin

Le transformation de Mellin e su inverse es relationate al transformation de Laplace bilateral per un simple cambio de variabiles. Se in le transformation de Mellin

$\left\{{\mathcal {M}}g\right\}(s)=\int _{0}^{\infty }\theta ^{s}g(\theta ){\frac {d\theta }{\theta }}$

nos pone $\theta =\exp(-t)$, nos obtene un transformation de Laplace bilateral. Desde que un transformation de Laplace ordinari pote esser scripte como um caso special de um transformation bilateral, e desde que le transformation bilateral pote esser scripte como le summa de duo transformationes unilateral, le theoria del transformationes de Laplace, Fourier e Mellin sta al base del mesme subjecto. Nonobstante, un puncto de vista differente e differente problemas characteristic es associate a cata un le iste tres major transformationes integral.

Le transformation-Z

Le equivalentia con le transformation-Z non es tanto directe como pro le transformationes de Fourier o Mellin. Toma un signal continuo, su transformation de Laplace e su transformation-Z e nomina los:

• Signal continuo: $f(t)$
• transformation de Laplace: $F(s)$
• transformation-Z: $F(z)$

Multiplica $f(t)$ per un Dirac comb e nomina le resultato $f^{*}(t)$

$f^{*}(t)=f(t)\delta _{T}(t)=\sum _{n=0}^{\infty }f(nT)\delta (t-nT)$

e tomante le transformation de Laplace resulta in

$F^{*}(s)=\int _{0^{-}}^{\infty }f^{*}(t)e^{-st}\,dt.=\int _{0^{-}}^{\infty }f(t)\delta (t-nT)e^{-st}\,dt.=\sum _{n=0}^{\infty }f(nT)e^{-nTs}$

Ergo le equivalentia pote esser establite:

$F^{*}(s)=\left.F(z)\right|_{z=e^{sT}}$

Iste equation refere le valores mensurate del signal continue al signal sequential discrete resultante del transformation-Z.

Proprietates e theoremas

• Linearitate

${\mathcal {L}}\left\{af(t)+bg(t)\right\}=a{\mathcal {L}}\left\{f(t)\right\}+b{\mathcal {L}}\left\{g(t)\right\}$

• Differentiation

${\mathcal {L}}\{f'\}=s{\mathcal {L}}\{f\}-f(0)$

${\mathcal {L}}\{f''\}=s^{2}{\mathcal {L}}\{f\}-sf(0)-f'(0)$

${\mathcal {L}}\left\{f^{(n)}\right\}=s^{n}{\mathcal {L}}\{f\}-s^{n-1}f(0)-\cdots -f^{(n-1)}(0)$

• Division del frequencia

${\mathcal {L}}\{tf(t)\}=-F'(s)$

• Integration del frequencia

${\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\int _{s}^{\infty }F(\sigma )\,d\sigma$

• Integration

${\mathcal {L}}\left\{\int _{0}^{t}f(\tau )\,d\tau \right\}={\mathcal {L}}\left\{1*f(t)\right\}={1 \over s}{\mathcal {L}}\{f\}$

• Theorema del valor initial

$f(0^{+})=\lim _{s\to \infty }{sF(s)}$

• Theorema del valor final

$f(\infty )=\lim _{s\to 0}{sF(s)}$, tote le polos in le semi-plano a sinistra.

Le valor final es utile perque illo da le comportamento de longe-termino sin havente de facer Decompositiones in fractiones partial o altere algebra difficile. Se le poles del function son in le semi-plano dextre (e.g. $e^{t}$ or $\sin(t)$) le comportamento de iste formula es indefinite.

• cambio in $s$

${\mathcal {L}}\left\{e^{at}f(t)\right\}=F(s-a)$

${\mathcal {L}}^{-1}\left\{F(s-a)\right\}=e^{at}f(t)$

• cambio in $t$

${\mathcal {L}}\left\{f(t-a)u(t-a)\right\}=e^{-as}F(s)$

${\mathcal {L}}^{-1}\left\{e^{-as}F(s)\right\}=f(t-a)u(t-a)$

Nota: $u(t)$ es le function degrau unitari.

• $n$th-potentia cambio

${\mathcal {L}}\{\,t^{n}f(t)\}=(-1)^{n}D_{s}^{n}[F(s)]$

• Convolution

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

Transformationes Commun

• $n$th potentia

${\mathcal {L}}\{\,t^{n}\}={\frac {n!}{s^{n+1}}}$

• Exponential

${\mathcal {L}}\{\,e^{-at}\}={\frac {1}{s+a}}$

• Seno

${\mathcal {L}}\{\,\sin(\omega t)\}={\frac {\omega }{s^{2}+\omega ^{2}}}$

• Coseno

${\mathcal {L}}\{\,\cos(\omega t)\}={\frac {s}{s^{2}+\omega ^{2}}}$

• Hyperbolic seno

${\mathcal {L}}\{\,\sinh(bt)\}={\frac {b}{s^{2}-b^{2}}}$

• Hyperbolic coseno

${\mathcal {L}}\{\,\cosh(bt)\}={\frac {s}{s^{2}-b^{2}}}$

• Natural logarithmo

${\mathcal {L}}\{\,\ln(t)\}=-{\frac {\ln(s)+\gamma }{s}}$

• nme raiz

${\mathcal {L}}\{\,{\sqrt[{n}]{t}}\}=s^{-{\frac {n+1}{n}}}\cdot \Gamma \left(1+{\frac {1}{n}}\right)$

• Function de Bessel del prime specie

${\mathcal {L}}\{\,J_{n}(t)\}={\frac {\left(s+{\sqrt {1+s^{2}}}\right)^{-n}}{\sqrt {1+s^{2}}}}$

• Modificate Bessel function del prime specie

${\mathcal {L}}\{\,I_{n}(t)\}={\frac {\left(s+{\sqrt {-1+s^{2}}}\right)^{-n}}{\sqrt {-1+s^{2}}}}$

• Function error

${\mathcal {L}}\{\,\operatorname {erf} (t)\}={e^{s^{2}/4}\operatorname {erfc} \left(s/2\right) \over s}$

• Function periodic period $T$

${\mathcal {L}}\{f\}={1 \over 1-e^{-Ts}}\int _{0}^{T}e^{-st}f(t)\,dt$

transformation de Laplace	Function del tempore
$1$	$\delta (t)$, impulse unitari
${\frac {1}{s}}$	$u(t)$, degrau unitari
${\frac {1}{(s+a)^{n}}}$	${\frac {t^{n-1}}{(n-1)!}}e^{-at}$
${\frac {a}{s(s+a)}}$	$1-e^{-at}$
${\frac {1}{(s+a)(s+b)}}$	${\frac {1}{b-a}}\left(e^{-at}-e^{-bt}\right)$
${\frac {s+c}{(s+a)^{2}+b^{2}}}$	$e^{-at}\left(\cos {(bt)}+\left({\frac {c-a}{b}}\right)\sin {(bt)}\right)$
${\frac {s\sin \varphi +a\cos \varphi }{s^{2}+a^{2}}}$	$\sin {(at+\varphi )}$