Question

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 _________ %.

Answer 1

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 + 2ζω_ns + ω²_n ^{= 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

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