Question

Unit step response of second order system is shown in figure.


Transfer function of the system is

• A. $\frac{5}{s^2+2s+5}$
• B. $\frac{9.3}{s^2+2s+9.3}$
• C. $\frac{2.7}{s^2+2s+2.7}$
• D. $\frac{10}{s^2+2s+10}$

Answer 1

The correct option is C $\frac{2.7}{s^2+2s+2.7}$

From time response curve,
$t_s=\frac{4}{\xi {\omega }_n}$
(For 2% tolerance band)

$4=\frac{4}{\xi {\omega }_n}$

$\xi {\omega }_n=1$

and $M_p=e^{-\pi \xi /\sqrt{1-{\xi }^2}}$

$0.09=e^{-\pi \xi /\sqrt{1-\xi 2}}$

$In\left(0.09\right)=-\frac{\pi \xi }{\sqrt{1-{\xi }^2}}$

${\left(2.41\right)}^2(1-{\xi }^2)={\pi }^2{\xi }^2$

${\left(2.41\right)}^2={\xi }^2({\pi }^2+{\left(2.41\right)}^2)$

$\sqrt{\frac{{\left(2.41\right)}^2}{{\pi }^2+{\left(2.41\right)}^2}}=\xi$

Damping ratio, $\xi =0.61$

${\omega }_n=\frac{1}{0.61}=1.64$
Standard second order transfer function:

$T\left(s\right)=\frac{{\omega }_n^2}{s^2+2\xi {\omega }_{n}s+{\omega }_n^2}$

$T\left(s\right)=\frac{2.7}{s^2+2s+2.7}$