Question $2(25$ marks$)$

a$)$ Consider the second order system as below, analyze the system by finding the relevant response specifications.

$G(s)=\frac{361}{s^2+16s+361}$

i$.$ Natural frequency, $n$

ii$.$ Damping ratio, $$

iii. Settling time, ts

iv$.$ Peak time, tp

v$.$ Overshoot

vi$.$ State nature of this system

$[3$ Marks$]$

$[3$ Marks$]$

$[3$ Marks$]$

$[3$ Marks$]$

$[3$ Marks$]$

$[3$ Marks$]$

b$)$ The transfer function for a process can be expressed as below. Answer the following questions.

i$.$ Calculate the value of gain, K

$[3$ Marks$]$

ii$.$ Calculate the damping ratio, $$

$[1$ Mark$]$

iii. Calculate the natural frequency, wn

$[1$ Mark$]$

iv$.$ Sketch and comment on the system response

$[2$ Marks$]$

$\frac{Y(s)}{U(s)}=\frac{3}{5s^2+7s+2}$