Question

Consider a unity gain closed-loop transfer function with forward path gain, $G\left(s\right)=\frac{K(s+1)}{s^3+0.5s^2+3s+1}$
If the system is producing undamped oscillations, then value of K and corresponding frequency of oscillations are respectively

• A. 2.5 and 1 rad/s
• B. 1 and 2rad/s
• C. 1 and 2.5 rad/s
• D. 2 and 1 rad/s

Answer 1

The correct option is B 1 and 2rad/s

The characteristic equation is,

$q\left(s\right)=s^3+0.5s^2+(K+3)s+(K+1)=0$

For a system to oscillate a row should become zero.

$\therefore K+3-2K-2=0$
K = 1
Given system is third order system $(s+a)(s^2+bs+c)=0$

For a marginally stable system $\xi =0$

$s^2+2\xi {\omega }_{n}s+{\omega }_n^2=0$

$s^2+{\omega }_n^2=0$
Take the coefficients of $s^2$ row.
$0.5s^2+(K+1)=0$
$0.5s^2+2=0$
$s=\pm j2$
$\omega =2rad/s$