Correct Answer - Option 1 : Underdamped system

__Concept:__

The type of system is determined by the value of the damping factor (ζ), i.e.

1) If 0 < ζ < 1, the system is underdamped.

2) If ζ = 1, the system is critically damped.

3) If ζ > 1, the system is overdamped.

__Analysis:__

__\(G(s) = \frac{{(2s + 1)}}{{({s^2} + 2s + 5)}}\)__

Characteristic equation of the system is the denominator of the overall transfer function.

s^{2} + 2s + 5 = 0 ---(1)

Compare it with the standard characteristic equation of 2^{nd} order system:

1 + G(s)H(s) = 0

s^{2} + 2ζω_{n}s + ω^{2}_{n} = 0.....(ii)

On comparing equation (i) and (ii), we get:

ω2n = 5

ω_{n} = √5 rad/sec = 2.23 rad/sec

2ζωn = 2

ζ = \(\frac{1}{{{\omega _n}}}\) = \(\frac{1}{{{2.23}}}\)

=0.44

So, 0 < ζ < 1

Hence the system is underdamped.