Question

The block diagram of a control system is shown below,

The control specification are such that the damping ratio of the closed loop system is 0.6 and damped frequency, ${\omega }_d$ is 10 rad/sec. The parameters of PD controller $K_P$ and $K_D$ are

• A. 156.25, 14
• B. 277.78, 19
• C. 2, 4
• D. 11, 100

Answer 1

The correct option is A 156.25, 14

Damped frequency, ${\omega }_d={\omega }_n\sqrt{1-{\xi }^2}$

$\Rightarrow {\omega }_n=\frac{{\omega }_d}{\sqrt{1-{\xi }^2}}=\frac{10}{\sqrt{1-(0.6{)}^2}}=12.5$

The desired second order C.E. is,

$s^2+2\xi {\omega }_{n}s+{\omega }_n^2=0$
$s^2+2\left(0.6\right)\left(12.5\right)s+(12.5{)}^2=0$
$s^2+15s+156.25=0$

The C.E. of given system is,

$1+(K_p+s{K}_D)\left(\frac{1}{s(s+1)}\right)=0$
(or) $s^2+(1+k_D)s+K_P=0$
Hence, $K_p=156.25$
$1+K_D=15$
$\Rightarrow K_D=14$