Question

Given figure shows a closed loop unity feedback system. The controller block has transfer function denoted by $G_c\left(s\right)$. The controller is cacaded to plant, which is denoted by $G_p\left(s\right)$



The peak percentage overshoot of the closed loop system is

• A. 5.0%
• B. 10.0%
• C. 16.3%
• D. 1.63%

Answer 1

The correct option is C 16.3%

The simplified block diagram is :


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

Standard second order equation is
$\frac{C\left(s\right)}{R\left(s\right)}=\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}...\left(ii\right)$

Compare equation (i) with equation (ii),

we get
${\omega }_n=1rad/sec$
$2\zeta {\omega }_n=1$
$\therefore \zeta =0.5$

Percentage of peak overshoot
$M_p=100\times e^{\frac{-\pi \zeta }{\sqrt{1-{\zeta }^2}}}$

$=100\times e^{\frac{-\pi 0.5}{\sqrt{1-(0.5{)}^2}}}$

$=100\times 0.163=16.3\%$