2. (45%) Consider the following transfer function:
\begin{equation}
G(s) = \frac{Y'(s)}{U'(s)} = \frac{2}{5s + 1}
\end{equation} (3)
The time base is minutes. The initial conditions are u(0) = 1, y(0) = 2. Assume
u'(t) = u(t) – u(0) and y'(t) = y(t) – y(0).
(a) If a unit step change input is applied at t = 0, that is,
$u(t) = \begin{cases} 1 & \text{when } t < 0\\ 2 & \text{when } t \ge 0 \end{cases}$,
what is the value of the output y(t) at time t = 10 min and t = 25 min
respectively?
(15%)
(b) If a rectangular pulse input with a width of 1 min and magnitude of 2 is
applied at t = 0,
$u(t) = \begin{cases} 1 & t < 0\\ 3 & 0 \le t < 1 \text{ min} \\ 1 & t \ge 1 \text{ min} \end{cases}$,
what is the output y(t) when t? ??
(15%)
(c) If a ramp input is applied
$u(t) = \begin{cases} 1 & \text{when } t < 0\\ 2t + 1 & \text{when } t \ge 0 \end{cases}$,
what is the output y(t) at t = 50 min?
(15%)
Justify your answers with quantitative analysis.
