Question

The closed loop frequency response $\left|G\right(j\omega \left)\right|$ versus frequency of a second order prototype system is shown in figure below.


The corresponding unit step response of the system is

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

At resonant frequency, resonant peak is achieved.

$M_r=1.6$,

${\omega }_r=4$

$M_{r}and{\omega }_r$ are given by

$M_r=\frac{1}{2\xi \sqrt{1-{\xi }^2}}$

${\omega }_r={\omega }_n\sqrt{1-2{\xi }^2}$

Now, $1.6=\frac{1}{2\xi \sqrt{1-{\xi }^2}}$

$4{\xi }^4-4{\xi }^2+{\left(\frac{10}{16}\right)}^2=0$

$\xi =0.94,0.33(\omega rtoexist1-2{\xi }^2>0;\xi shouldbelessthan\frac{1}{\sqrt{2}})$

$\xi =0.33$

and ${\omega }_n=\frac{4}{\sqrt{1-2{\left(0.33\right)}^2}}$

${\omega }_n=4.53rad/\sec$

Setting time for 2% tolerance band is,

$t_s=\frac{4}{\xi {\omega }_n}=\frac{4}{0.33\times 4.53}=2.7\sec$