Question

A conttrol system with a PD controller is shown in the figure. If the velocity error constant $K_v=1000$ and the damping ratio $\zeta =0.5$, then the values of $K_P$ and $K_D$ are

• A. $K_P=100,K_D=0.09$
• B. $K_P=100,K_D=0.9$
• C. $K_P=10,K_D=0.09$
• D. $K_P=10,K_D=0.9$

Answer 1

The correct option is B $K_P=100,K_D=0.9$

$K_v=\underset{s\to 0}{lim}sG\left(s\right)H\left(s\right)$
$1000=\underset{s\to 0}{lim}s\times \frac{(K_P+K_{D}s)100}{s(s+10)}$
$\Rightarrow K_P=100$
Now characteristics eq. $(1+G(s\left)H\right(s\left)\right)=0$
$1+\frac{(K_P+K_{D}s)100}{s(s+10)}=0$
Putting $K_P=100$
$s^2+10s+{10}^4+100K_{D}s=0$
$s^2+(10+100K_D)s+{10}^4=0$
Comparing with standard second order eq.
i.e $s^2+2\zeta {\omega }_{n}s+{\omega }_n^2=0$
So, ${\omega }_n=100;2\zeta {\omega }_n=10+100K_D$
Given,$\zeta =0.5$
$2\times 0.5\times 100=10+100K_D$
$K_D=0.9$