Understanding the Concept:
A standard prototype second-order closed-loop control system function is conventionally modeled as:
\[
T(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}
\]
Where \(\omega_n\) represents the undamped natural frequency and \(\zeta\) is the damping ratio parameter.
The settling time (\(t_s\)) represents the total time required for the system's response transient output to sink and stabilize within a specified error band around its steady-state value.
For a standard 2% error tolerance band criterion, the settling time formula is given by:
\[
t_s = \frac{4}{\zeta\omega_n}
\]
Where the term \(\zeta\omega_n\) is also known as the attenuation factor (\(\alpha\)).
Step 1: Identify system equations.
The problem states that the system has an open-loop transfer function \(G(s)\) with unity feedback (\(H(s)=1\)). However, let us examine the provided expression layout: \(G(s) = \frac{100}{s^2+10s+100}\). Typically, when a second-order system formula is written directly in this layout with a quadratic denominator, it represents the overall closed-loop transfer function \(T(s)\). Let us perform a comparison against the standard form:
\[
T(s) = \frac{100}{s^2 + 10s + 100}
\]
Step 2: Extract \(\omega_n\) and \(\zeta\).
Comparing the denominator coefficients with \(s^2 + 2\zeta\omega_n s + \omega_n^2 = 0\):
From the constant term:
\[
\omega_n^2 = 100 \implies \omega_n = \sqrt{100} = 10 \text{ rad/s}
\]
From the coefficient of \(s\):
\[
2\zeta\omega_n = 10
\]
We can directly observe that the complete term \(\zeta\omega_n\) (which represents the real part of the system poles) is:
\[
\zeta\omega_n = \frac{10}{2} = 5
\]
Step 3: Compute the 2% settling time.
Using the standard 2% error performance criteria formula:
\[
t_s = \frac{4}{\zeta\omega_n}
\]
Substitute the calculated factor value \(\zeta\omega_n = 5\) directly into the denominator:
\[
t_s = \frac{4}{5} = 0.8 \text{ seconds}
\]
This perfectly matches option (B).