[40 points] A linear time-invariant continuous-time system has a transfer function:
$H(s) = \frac{s + 2}{(s + 1)^2 + 4}$
The input $x(t) = C \cos(\omega_0 t + \theta)u(t)$ is applied to the system for $t \ge 0$ with zero initial
conditions. The resulting steady-state response $y_{ss}(t)$ is $y_{ss}(t) = 6\cos(t + 45^\circ)$, $t \ge 0$.
1) Find C, $\omega_0$, and $\theta$. [Hint] Know the difference between $H(\omega)$ and $H(s)$: $H(\omega)$ captures
steady-state response and $H(s)$ captures both transient and steady-state response.
2) Knowing that when a system is at rest (initial conditions are zero) initially, the zero-state
response can be expressed as $y_{zs}(t) = y_{tr}(t) + y_{ss}(t)$ where $y_{zs}(t)$ is the zeros-state
response, $y_{tr}(t)$ is the transient response, and $y_{ss}(t)$ is the steady-state response of the
system to the input $x(t)$; compute the transient response $y_{tr}(t)$ and the Laplace
transform $Y_{tr}(s)$ where $Y_{tr}(s) = \mathcal{L}T[y_{tr}(t)]$. For full credit, you have to show the
details on how you got the solution; presenting a solution just by using MATLAB will
not give you much credit.
