Consider the second$-$order system defined by:

$\frac{Y(s)}{U(s)}=\frac{1}{s^2+2s+1}$

If the input $u(t)$ is a unit impulse $(t)$ :

For the undamped system where $=0,$ solve for and sketch the response $y(t).$ Find the natural frequency ${}_n$ and the period $T$ of the oscillation and show them clearly on the sketch.

For the under$-$damped system where $=0\mathrm{.}2,$ solve for and sketch the response $y(t).$ Find the damped natural frequency ${}_d,$ the period $T$ of the oscillation, the maximum peak $M_p($overshoot$),$ and the settling time $t_s$ and show them clearly on the sketch.

For the critically$-$damped system where $=1,$ solve for and sketch the response $y(t).$