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La Transformada Inversa de Laplace

Module by: Steven J. Cox. E-mail the authorTranslated By: Fara Meza, Erika Jackson

Based on: The Inverse Laplace Transform by Steven J. Cox

Summary: This module introduces the Inverse Laplace Transform. Building on the groundwork done in the Laplace Transform module, this module gives some background into the Inverse Laplace method, and uses MATLAB's ilaplace command to find the actual solutions to a problem.

To Come

En la Función de Transferencia estableceremos que la función para la transformada de Laplace es hh es

ℒ-1⁢h⁢t=12⁢π⁢∫−∞∞e(c+y⁢j)⁢t⁢h⁢(c+y⁢j)⁢tdy ℒ h t 1 2 y c y t h c y t
(1)
donde j≡2-1 2 -1 y el numero real ccson escogidos para que todas las singularidades de hh se encuentren en el lado izquierdo del integral.

Continuando con la Transformada Inversa de Laplace

Con la transformada inversa de Laplace podemos expresar la solución de x′=B⁢x+g x B x g , como

x⁢t=ℒ-1⁢s⁢I−B-1⁢(ℒ⁢g+x⁢0) x t ℒ s I B ℒ g x 0
(2)
Por ejemplo, tomaremos el primer componente de ℒ⁢x ℒ x ,nombrado ℒ x 1 ⁢s=0.19⁢(s2+1.5⁢s+0.27)s+164⁢(s3+1.655⁢s2+0.4078⁢s+0.0039) . ℒ x 1 s 0.19 s 2 1.5 s 0.27 s 1 6 4 s 3 1.655 s 2 0.4078 s 0.0039 . Definimos:
Definition 1: polos
También llamadas como singularidades estos son los punto ss en los cuales ℒ x 1 ⁢s ℒ x 1 s explota.
Son las raíces del denominador,
-1 /100 ,    -329 /400±27316 ,    and    -1/6 . -1 100,   ± -329 400 2 73 16 ,  and  -16.
(3)
Las cuatro son negativas, es suficiente tomar c=0 c 0 así la integración de ecuación 1 continua en el eje imaginario. No suponemos que el lector aya encontrado integraciones en el plano complejo pero esperamos que este ejemplo provea la motivación necesaria para examinarla. Sin embargo antes de esto hay que notar que MATLAB tiene el cálculo necesario para desarrollar este punto. Volviendo a fib3.m observamos que el comando ilaplace produce x 1 ⁢t=211.35⁢e−t100−(0.0554⁢t3+4.5464⁢t2+1.085⁢t+474.19)⁢e−t6+e−(329⁢t)400⁢(262.842⁢cosh273⁢t16)+262.836⁢sinh273⁢t16 x 1 t 211.35 t 100 0.0554 t 3 4.5464 t 2 1.085 t 474.19 t 6 329 t 400 262.842 2 73 t 16 262.836 2 73 t 16

Figura 1: Los 3 potenciales asociados con el modelo del circuito RC .
Figura 1 (fib3_fig1.png)

Los otros potenciales, vistos en esta figura posen una expresión similar. Por favor note que cada uno de los polos de ℒ⁢ x 1 ℒ x 1 se muestra como exponencial x 1 x 1 y que los coeficientes del exponencial son polinomios con grados que son determinados por el orden de su respectiva polos.

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