Question

The block diagram of a closed loop control system is given by figure. The values of k and P such that the system has a damping ratio of 0.7 and an undamped natural frequency ${\omega }_n$ of 5 rad/sec, are respectively equal to

• A. 20 and 0.3
• B. 20 and 0.2
• C. 25 and 0.3
• D. 25 and 0.2

Answer 1

The correct option is D 25 and 0.2

$G\left(s\right)=\frac{k}{s(s+2)}andH\left(s\right)=1+sP$

Closed-loop transfer fucntion

$T\left(s\right)=\frac{G\left(s\right)}{1+G\left(s\right)H\left(s\right)}$

$=\frac{k/s(s+2)}{1+\frac{k}{s(s+2)}.(1+sP)}$

$=\frac{k}{s(s+2)+k(1+sP)}$

$\Rightarrow T\left(s\right)=\frac{k}{s^2+(2+kP)s+k}$

So, characteristic equation
$=s^2+(2+kP)s+P$

comparing with standard equation
$=s^2+2\xi {\omega }_{n}s+{\omega }_n^2$
$k={\omega }_n^n=5^2=25$

(where ${\omega }_n=$undamped natural frequency)

$2\xi {\omega }_n=2+kP$
$\Rightarrow 2\times 0.7\times 5=2+25P$
(where $\xi =$damping ratio)
$P=0.2$