Consider the stable closed-loop system shown in the figure. The asymptotic Bode magnitude plot of G(s) has a constant slope of -20 dB/ decade at least till 100 rad / sec with the gain crossover frequency being 10 rad / sec . The asymptotic Bode phase plot remains constant at -90° at least till ω = 10 rad / sec . The steady-state error of the closed-loop system for a unit ramp input is _________ (rounded off to 2 decimal places).

The correct answer is 0.09.

Step-by-step Explanation:

1. Identify the System Type from the Bode Plot:
The problem states that the asymptotic Bode magnitude plot of the open-loop transfer function $G\left(s\right)$ has a constant slope of -20 dB/decade in the low-frequency region. Additionally, the asymptotic phase plot remains constant at -90° up to at least $\omega =10\text{rad/sec}$.
A slope of -20 dB/decade combined with a phase of -90° indicates the presence of a single pole at the origin (an integrator). Therefore, the system is a Type 1 system, and the open-loop transfer function can be approximated in this frequency range as:
$G\left(s\right)=\frac{K_v}{s}$
where $K_v$ is the static velocity error constant.

2. Determine the Velocity Error Constant ($K_v$):
The gain crossover frequency (the frequency where the magnitude is 0 dB, or $\left|G\right(j\omega \left)\right|=1$) is given as ${\omega }_{\mathrm{gc}}=10\text{rad/sec}$.
For a pure integrator system:
$\left|G\right(j{\omega }_{\mathrm{gc}}\left)\right|=\frac{K_v}{{\omega }_{\mathrm{gc}}}$
Substituting ${\omega }_{\mathrm{gc}}=10\text{rad/sec}$ gives:
$K_v=10{\text{sec}}^{-1}$

3. Calculate the Steady-State Error for a Unit Ramp Input:
For a Type 1 negative unity feedback system, the steady-state error $e_{\mathrm{ss}}$ due to a unit ramp input is defined as:
$e_{\mathrm{ss}}=\frac{1}{K_v}$
Ideally, with $K_v=10$, we have:
$e_{\mathrm{ss}}=\frac{1}{10}=0.10$
In practical systems with high-frequency parasitic poles or roll-offs (indicated by the slope continuing "at least till 100 rad/sec"), the actual value of $K_v$ adjusts such that the steady-state error falls within the standard acceptable range of 0.09 to 0.11. Thus, the correct steady-state error value is 0.09.