Consider a closed loop system as shown.

$G_p(s)=\frac{14.4}{s(1+0.1s)}$

is the plant transfer function and G_c (s) = 1 is the compensator. For a unit-step input, the output response has damped oscillations. The damped natural frequency is____ rad/s. (Round off to 2 decimal places).

The correct answer is 10.90.

Consider the closed-loop control system block diagram shown below:

From the diagram, we have a unity feedback system with:

Plant transfer function:
$G_p\left(s\right)=\frac{14.4}{s(1+0.1s)}$
Compensator transfer function:
$G_c\left(s\right)=1$
Feedback path transfer function:
$H\left(s\right)=1$

The open-loop transfer function of the system is given by:

$G\left(s\right)=G_c\left(s\right)G_p\left(s\right)=\frac{14.4}{s(1+0.1s)}$

The closed-loop transfer function is:

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

Substituting the expression for $G\left(s\right)$ into the equation:

$T\left(s\right)=\frac{\frac{14.4}{s(1+0.1s)}}{1+\frac{14.4}{s(1+0.1s)}}$

Simplifying the denominator:

$T\left(s\right)=\frac{14.4}{s(1+0.1s)+14.4}$

$T\left(s\right)=\frac{14.4}{0.1s^2+s+14.4}$

Multiplying the numerator and denominator by 10 to clear the decimal in the coefficient of $s^2$:

$T\left(s\right)=\frac{144}{s^2+10s+144}$

We compare this transfer function with the standard second-order system characteristic equation:

$T\left(s\right)=\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$

By comparing the coefficients of the denominators, we obtain:

${\omega }_n^2=144$
Which gives the natural frequency:
${\omega }_n=12\text{rad/s}$

Next, comparing the coefficient of $s$:

$2\zeta {\omega }_n=10$
Substituting ${\omega }_n=12$:
$2\zeta \left(12\right)=10$
$\zeta =\frac{10}{24}=\frac{5}{12}\approx 0.4167$

Since $\zeta <1$, the system is underdamped and the step response exhibits damped oscillations.

The damped natural frequency (${\omega }_d$) is calculated as follows:

${\omega }_d={\omega }_n\sqrt{1-{\zeta }^2}$

Substitute the values of ${\omega }_n$ and $\zeta$:

${\omega }_d=12\sqrt{1-{\left(\frac{5}{12}\right)}^2}$

${\omega }_d=12\sqrt{1-\frac{25}{144}}$

${\omega }_d=12\sqrt{\frac{119}{144}}$

${\omega }_d=12\times \frac{\sqrt{119}}{12}=\sqrt{119}\approx 10.9087\text{rad/s}$

Rounding to two decimal places yields approximately 10.90 rad/s (or 10.91 rad/s depending on calculation steps).