Question

Transfer Function (35 points). The transfer function of a causal linear time-invariant (LTI) system is described by \begin{equation} G(s) = \frac{1}{s^2 + 9s + 20} \end{equation} (a) Find the poles of the system (roots of the denominator of G(s)). (b) Using your answer for (a), find constants a and b in order to decompose the transfer function as \begin{equation} G(s) = \frac{a}{s+4} + \frac{b}{s+5} \end{equation} (c) Using your answer for (b) and the Laplace transform tables, find the impulse response of the system. (d) Assuming zero initial conditions, find the unit step response of the system in the Laplace domain, Y(s). 1 (e) Assuming zero initial conditions, find the unit step response of the system by computing the inverse Laplace transform of Y(s) analytically (using the Laplace transform tables), i.e., $y(t) = \mathcal{L}^{-1}\{Y(s)\}$. (f) Assuming zero initial conditions, find the unit step response of the system by computing the inverse Laplace transform of Y(s), using the Matlab function 'ilaplace'. (g) Assuming zero initial conditions, find the unit step response of the system by computing the convolution integral in time domain, i.e., $\int_0^t g(t-\tau)u(\tau)d\tau$.

          Transfer Function (35 points). The transfer function of a causal linear time-invariant (LTI) system is described by
\begin{equation*} G(s) = \frac{1}{s^2 + 9s + 20} \end{equation*}
(a) Find the poles of the system (roots of the denominator of G(s)).
(b) Using your answer for (a), find constants a and b in order to decompose the transfer function as
\begin{equation*} G(s) = \frac{a}{s+4} + \frac{b}{s+5} \end{equation*}
(c) Using your answer for (b) and the Laplace transform tables, find the impulse response of the system.
(d) Assuming zero initial conditions, find the unit step response of the system in the Laplace domain, Y(s).
1
(e) Assuming zero initial conditions, find the unit step response of the system by computing the inverse Laplace transform of Y(s) analytically (using the Laplace transform tables), i.e., $y(t) = \mathcal{L}^{-1}\{Y(s)\}$.
(f) Assuming zero initial conditions, find the unit step response of the system by computing the inverse Laplace transform of Y(s), using the Matlab function 'ilaplace'.
(g) Assuming zero initial conditions, find the unit step response of the system by computing the convolution integral in time domain, i.e., $\int_0^t g(t-\tau)u(\tau)d\tau$.

$Transfer Function (35 points). The transfer function of a causal linear time-invariant (LTI) system is described by G(s) = (1)/(s^2 + 9s + 20) (a) Find the poles of the system (roots of the denominator of G(s)). (b) Using your answer for (a), find constants a and b in order to decompose the transfer function as G(s) = (a)/(s+4) + (b)/(s+5) (c) Using your answer for (b) and the Laplace transform tables, find the impulse response of the system. (d) Assuming zero initial conditions, find the unit step response of the system in the Laplace domain, Y(s). 1 (e) Assuming zero initial conditions, find the unit step response of the system by computing the inverse Laplace transform of Y(s) analytically (using the Laplace transform tables), i.e., y(t) = ℒ^-1{Y(s)}. (f) Assuming zero initial conditions, find the unit step response of the system by computing the inverse Laplace transform of Y(s), using the Matlab function 'ilaplace'. (g) Assuming zero initial conditions, find the unit step response of the system by computing the convolution integral in time domain, i.e., ∫0^t g(t-τ)u(τ)dτ.$

Added by Hector C.

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