Consider the closed$-$loop system given by:

$\frac{C(s)}{R(s)}=\frac{{}_n^2}{s^2+2{}_{n}s+{}_n^2}$

a$)$ Determine the values of $$ and ${}_n$ so that the system responds to a step input with approximately $5\%$ overshoot and a peak time of $1$ sec $.$

b$)$ Calculate the rise time and settling time.

c$)$ Calculate the response of the system to a step input.

d$)$ Calculate the response of the system to an input $r(t)=10(t)$ where $(t)$ is the unit impulse function.