Question

A double integrator plant, $G\left(s\right)=\frac{K}{s^2},H\left(s\right)=1$ is to be compensated to acheive the damping ratio $\zeta =0.5$ and an undamped natural frequency, ${\omega }_n=5rad/sec$. Which one of the following compensator $G_c\left(s\right)$ will be suitable?

• A. $\frac{s+3}{s+9.9}$
• B. $\frac{s+9.9}{s+3}$
• C. $\frac{s-6}{s+8.33}$
• D. $\frac{s+6}{s}$

Answer 1

The correct option is A $\frac{s+3}{s+9.9}$


$\zeta =0.5$
$co{s}^{-1}0.5={60}^{\circ }$
$\therefore \theta ={60}^{\circ }$
$\angle G\left(s\right)=\frac{K}{s^2}{|}_{s=-2.5+4.33j}$
$=-2ta{n}^{-1}\frac{4.33}{-2.5}={120}^{\circ }$
$\therefore$ For compensated system $\angle =180-120;{60}^{\circ }$ (b) & (d) are lag network & for compensating lag network $K/s^2$ , a lead network is required
$\therefore$ Putting $s=-2.5+4.33J$ in (a) gives
$\frac{K(s+3)}{s^2(s+9.9)}=\frac{0.5+j4.33}{7.4+j4.33}={53}^{\circ },{60}^{\circ }$
$\therefore$ option (a) is the correct answer.