Question

The open-loop transfer function of a unity-feedback control system is given by
$G\left(s\right)=\frac{K}{s(s+2)}$
For the peak overshoot of the closed-loop system to a unit step input to be $10\%,$ the value of $K$ is

• A. 2.8

Answer 1

The correct option is A 2.8

$G\left(s\right)=\frac{K}{s(s+2)};H\left(s\right)=1$

Characteristic equation
$=1+G\left(s\right)H\left(s\right)=0$

$1+\frac{K}{s(s+2)}=0$

$s^2+2s+K=0\Rightarrow {\omega }_n=\sqrt{K}$

$2\xi {\omega }_n=2;\xi =\frac{1}{\sqrt{K}}$

$M_P=e^{-\pi \xi \sqrt{1-{\xi }^2}}=0.1$

$-\frac{\pi \xi }{\sqrt{1-{\xi }^2}}=\ln \left(0.1\right)$

$\Rightarrow \frac{\pi \xi }{\sqrt{1-{\xi }^2}}=2.3$

${\pi }^2{\xi }^2=\left(2.3{)}^2\right(1-{\xi }^2)$

$15.16{\xi }^2=(2.3{)}^2$

$\Rightarrow \xi =0.59$

$\Rightarrow \text{Also},K=\frac{1}{{\xi }^2}=2.8$