The asymptotic Bode magnitude plot of a system is shown. Which one of the following options best represents the transfer function of the system?

The correct answer is:
$$G\left(s\right)=\frac{\frac{s}{{\omega }_0}}{1+\frac{s}{{\omega }_0}}$$

Analysis of the Bode Plot:
By inspecting the provided asymptotic Bode magnitude plot, we observe two main frequency regions separated by the corner frequency, which is labeled as ${\omega }_0$:
1. For high frequencies ($\omega >{\omega }_0$), the magnitude plot is flat at a constant level of $0\text{dB}$.
2. For low frequencies ($\omega <{\omega }_0$), the asymptote has a slope of $-20\text{dB/decade}$ when moving from low frequencies up towards ${\omega }_0$, representing a transition region where the magnitude changes with frequency.

Mathematical Derivation:
Let us evaluate the frequency response of the transfer function $G\left(s\right)$ by substituting $s=j\omega$:
$$G\left(j\omega \right)=\frac{\frac{j\omega }{{\omega }_0}}{1+\frac{j\omega }{{\omega }_0}}$$

Now, we look at the asymptotic behavior in the two frequency limits:
Case 1: High-Frequency region ($\omega \gg {\omega }_0$)
When the frequency is much higher than the corner frequency, the term $\frac{j\omega }{{\omega }_0}$ dominates over $1$ in the denominator:
$$G\left(j\omega \right)\approx \frac{\frac{j\omega }{{\omega }_0}}{\frac{j\omega }{{\omega }_0}}=1$$
The magnitude in decibels in this region is:
$${\left|G\right(j\omega \left)\right|}_{\text{dB}}=20{log}_{10}\left(1\right)=0\text{dB}$$
This perfectly matches the flat asymptote of $0\text{dB}$ for frequencies above ${\omega }_0$ shown in the plot.

Case 2: Low-Frequency region ($\omega \ll {\omega }_0$)
When the frequency is much lower than the corner frequency, the term $\frac{j\omega }{{\omega }_0}$ is much smaller than $1$, so the denominator is approximately $1$:
$$G\left(j\omega \right)\approx \frac{j\omega }{{\omega }_0}$$
The magnitude in decibels in this region is:
$${\left|G\right(j\omega \left)\right|}_{\text{dB}}\approx 20{log}_{10}\left(\frac{\omega }{{\omega }_0}\right)$$
This mathematical form represents a straight line on a logarithmic scale with a slope of $+20\text{dB/decade}$ as frequency increases towards ${\omega }_0$. Correspondingly, as frequency decreases away from ${\omega }_0$, the magnitude rolls off at a rate of $20\text{dB/decade}$. Thus, the transfer function matching this behavior is indeed:
$$G\left(s\right)=\frac{\frac{s}{{\omega }_0}}{1+\frac{s}{{\omega }_0}}$$