The damping ratio and undamped natural frequency of a closed loop system as shown in the figure, are denoted as ξ and ω_n , respectively. The values of ξ and ω_n are

To find the damping ratio ($\xi$) and undamped natural frequency (${\omega }_n$) of the given closed-loop system, we can analyze the block diagram step-by-step:

Step 1: Simplify the inner feedback loop
Let $X\left(s\right)$ be the output of the first block $\frac{10}{s}$. The inner feedback loop is a negative feedback loop where the feedback path has a gain of 1 (unity feedback) and the forward path transfer function is:
$G_1\left(s\right)=\frac{10}{s}$
The closed-loop transfer function of this inner loop, which we can call $T_{\text{inner}}\left(s\right)$, is given by:
$T_{\text{inner}}\left(s\right)=\frac{G_1\left(s\right)}{1+G_1\left(s\right)}=\frac{\frac{10}{s}}{1+\frac{10}{s}}=\frac{10}{s+10}$

Step 2: Find the overall open-loop transfer function
The inner loop block $T_{\text{inner}}\left(s\right)$ is in cascade with the second block $\frac{10}{s}$. Therefore, the overall forward path transfer function $G\left(s\right)$ is:
$G\left(s\right)=T_{\text{inner}}\left(s\right)\times \frac{10}{s}=\frac{10}{s+10}\times \frac{10}{s}=\frac{100}{s(s+10)}$

Step 3: Find the overall closed-loop transfer function
The outer feedback loop is also a negative feedback loop with a unity feedback path ($H\left(s\right)=1$). The overall transfer function $T\left(s\right)=\frac{Y\left(s\right)}{R\left(s\right)}$ is:
$T\left(s\right)=\frac{G\left(s\right)}{1+G\left(s\right)}=\frac{\frac{100}{s(s+10)}}{1+\frac{100}{s(s+10)}}=\frac{100}{s^2+10s+100}$

Step 4: Compare with the standard second-order system equation
The standard transfer function for a second-order system is:
$T\left(s\right)=\frac{{\omega }_n^2}{s^2+2\xi {\omega }_{n}s+{\omega }_n^2}$

By comparing the denominator terms:
1) For the undamped natural frequency:
${\omega }_n^2=100\Rightarrow {\omega }_n=10\text{rad/s}$

2) For the damping ratio:
$2\xi {\omega }_n=10$

Substituting ${\omega }_n=10$ into the equation:
$2\times \xi \times 10=10$
$20\xi =10\Rightarrow \xi =0.5$

Thus, the damping ratio is $\xi =0.5$ and the undamped natural frequency is ${\omega }_n=10\text{rad/s}$.