Correct Answer - Option 4 : 3

__Concept__:

The general expression for a second order transfer function is given as:

\(\frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{\omega _n^2}}{{{s^2} + 2ζ {\omega _n}s + \omega _n^2}}\)

ωn = Natural undamped frequency

ζ = damping ratio

Closed loop transfer function of negative feedback system is given by,

\(\frac{C(s)}{R(s)}=G(s)H(s)=\frac{G(s)}{1+G(s)H(s)}\)

For unity feedback system, H(s) = 1

So, \(G(s)H(s)=\frac{G(s)}{1+G(s)}\)

__Calculation__:

Given, \(G(s)=\frac{9}{s(s+3)}\)

∴ \(G(s)H(s)=\large{\frac{\frac {9}{s(s+3)}}{1+ \frac {9}{s(s+3)}}=\frac{9}{s^2+3s+9}}\)

Comparing this with the general expression of the transfer function of the second-order control system, we can write:

ωn^{2} = 9

**ωn = 3 rad/s**

Also, 2ζωn = 3

Putting ωn = 3 in the above equation, we get:

2 × ξ × 3 = 3

ζ = 0.5

Since ζ < 1 , the nature of the time response of the system will be under damped.