A system with transfer function $G\left(s\right)=\frac{1}{s^2+1}$ has zero initial conditions. The percentage overshoot in its step response is _________ %.

Concept:

General expression for the transfer function of a second order system

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

When denominator is equated to zero it gives characteristic equation i.e.

s² + 2ζωₙs + ω²ₙ${}^{=0}$

$\%overshoot=e^{-\frac{\zeta \pi }{\sqrt{1-{\zeta }^2}}}\times 100$

Calculation:

$\frac{C\left(s\right)}{R\left(s\right)}=\frac{1}{s^2+1}$

Comparing the denominator ζ = 0

$\%overshoot=e^{-\frac{\zeta \pi }{\sqrt{1-{\zeta }^2}}}\times 100=100$