Question

A second-order real system has the following properties:

(a) The damping ratio $\xi =0.5$ and undamped natural frequency ${\omega }_n=10rad/s.$

(b) The steady state value of the output to a unit step input is 1.02.

The transfer function of the system is

• A. $\frac{102}{s^2+5s+100}$
• B. $\frac{102}{s^2+10s+100}$
• C. $\frac{102}{s^2+s+100}$
• D. $\frac{102}{s^2+5s+100}$

Answer 1

The correct option is B $\frac{102}{s^2+10s+100}$

Damping ratio

$\xi$ = 0.5

Undamped natural frequency

${\xi }_n=10rad/sec$

Steady state output to a unit step input

$C_{ss}=1.02$

Hence steady state error = 1.02-1.00

$e_{ss}=0.02$

$\because$ Characteristics equation is,

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

$s^2+2\times 0.5\times 10s+100=0$

$s^2+10s+100=0$

From options, if we take option (b)

$then{C}_{ss}={lim}_{s\to 0}s.C\left(s\right)$

$={lim}_{s\to 0}s\times \frac{1}{s}\times \frac{102}{s^2+10s+100}$

$c_{ss}=1.02$

Hence option (b) is correct answer.