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Solución general de un sistema

De por WikiMatematica.org

Gráfica

y(\tau _{k})\left [ H(t-\tau _{k})-H(t-\tau _{k+1}) \right ]

I/O
X_{k}(t)

X_{k}(t)=y(\tau _{k})\left [ h(t-\tau _{k})-h(t-\tau _{k}-\Delta\tau  ) \right ]

X_{k}(t)\approx \sum_{k=1}^{h}X_{k}(t) = \sum_{k=1}^{h}y(\tau _{k})\left [ h(t-\tau _{k})-h(t-\tau _{k}-\Delta\tau  ) \right ]

X_{k}(t)= \lim_{h\to \infty } \sum_{k=1}^{h}y(\tau _{k})\left [ h(t-\tau _{k})-h(t-\tau _{k}-\Delta\tau  ) \right ]

\lim_{h\to \infty }\Delta \tau = 0

X(t)=\lim_{h\to \infty}\sum_{k=1}^{h}y(\tau _{k})\left [\frac{h(t-\tau _{k})-h(t-\tau _{k}-\Delta\tau)}{\Delta \tau } \right ]\Delta \tau

X(t)=\int_{0}^{t}y(\tau )h{}'(t-\tau )d\tau

\therefore X(t)=\int_{0}^{t}y(\tau )g(t-\tau )d\tau

\int_{0}^{t}y(\tau )g(t-\tau )d\tau = y(t)*g(t)

\mathbf{X(t) = y(t)*g(t)}


Recordar que g(t) = h'(t)


Ejemplo

Calcular Respuesta del Sistema

I/O 0.8 \frac{dx}{dt} + x = y\left ( t \right )

con y\left ( t \right )=\begin{Bmatrix}
0; t < 0\\ 
2t; t\geq 0
\end{Bmatrix}

Solución A:

0.8\frac{dx}{dt} + x = 2t;   t\geq 0

1) EHA   0.8 \frac{dx}{dt} + x = 0

   EL    0.8P + 1 = 0

P = \frac{-1}{0.8}

P = -1.25

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