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Estabilidad BIBO

Module by: Richard Baraniuk Translated By Fara Meza, Erika JacksonBased on: BIBO Stability by Richard Baraniuk

Summary: Explica las estabilidad de entrada acotada, salida acotada.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

BIBO es para entrada acotada, salida acotada (bounded input, bounded output). BIBO estable es una condición, tal que cualquier entrada acotada nos da una salida acotada. Esto es que conforme nosotros pongamos una entrada estable, nos garantiza obtener una salida estable.

Para entender este concepto, primero debemos ver que significa exactamente acotado. Una señal acotada es cualquier señal tal que existe un valor donde el valor absoluto de la señal nunca es mayor para algún valor. Ya que el valor es arbitrario, lo que queremos decir es que en ningún punto la señal puede tender a infinito.

Figura 1: Una señal acotada es una señal para la cual existe un constante A A tal que ∀t:|ft|<A t f t A
Figura 1 (bibo.png)

Una vez que ya tenemos identificado lo que significa una señal acotada, debemos mover nuestra atención a la condición que un sistema debe poseer para garantizar que si una señal pasa a través del sistema, una señal acotada se presentara en la salida. Resulta que un sistema LTI continuo en el tiempo con una respuesta al impulso ht h t es estabilidad BIBO si y solo si

Condición de Tiempo-Continuo para Estabilidad BIBO

∫-∞∞|ht|dt<∞ t h t (1)
Esto es decir que la función de transferencia es absolutamente integrable.

Extendiendo este concepto a tiempo- discreto tomamos la transición estándar de integración a sumatoria y obtenemos la función de transferencia hn h n , que debe ser absolutamente sumable. Esto es

Condición de Tiempo-Discreto para Estabilidad BIBO

∑n=-∞∞|hn|<∞ n h n (2)

Estabilidad y Laplace

La estabilidad es muy fácil de deducir desde la gráfica de polos y ceros de una función de transferencia. La única condición necesaria para demostrar la estabilidad es demostrar que el eje- ⅈω ω es en la región de convergencia

Figura 2
(a) Ejemplo de una gráfica de polos y ceros para un sistema estable de tiempo-continuo.(b) Ejemplo de una gráfica de polos y ceros para un sistema inestable de tiempo continuo.
Figura 2(a) (bibo2.png)Figura 2(b) (bibo3.png)

Estabilidad y la Transformada Z

La estabilidad para las señales de tiempo-discreto en el dominio-z es tan fácil de demostrar como lo es para las señales de tiempo-continuo en el domino Laplace. Sin embargo en lugar de la región de convergencia ROC (region of convergence) necesaria para contener el eje- ⅈω ω , la ROC debe contener el círculo unitario.

Figura 3
(a) Un sistema estable de tiempo-discreto.(b) Un sistema inestable de tiempo-discreto.
Figura 3(a) (bibo4.png)Figura 3(b) (bibo5.png)

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