Which of the following systems has maximum peak overshoot due to a step input?

Peak Overshoot Analysis for Step Input

This problem asks us to identify which control system, when subjected to a step input, exhibits the maximum peak overshoot in its response. The systems provided are all second-order systems.

Understanding Second-Order Systems and Peak Overshoot

A standard second-order system's transfer function is generally represented as:

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

where:

- ${\omega }_n$ is the natural frequency.
- $\zeta$ is the damping ratio.

The peak overshoot ($M_p$) is a measure of how much the system's response exceeds the final steady-state value. For a step input, the peak overshoot is primarily determined by the damping ratio ($\zeta$). Specifically, the formula for peak overshoot is:

$M_p=e^{-\frac{\zeta \pi }{\sqrt{1-{\zeta }^2}}}$

From this formula, we can see that a lower value of the damping ratio ($\zeta$) leads to a higher peak overshoot. Our goal is to find the system with the minimum damping ratio.

Calculating Damping Ratio for Each System

We will compare each given transfer function to the standard form to determine its damping ratio ($\zeta$).Comparing Damping Ratios for Maximum Overshoot

We found the damping ratios for the four systems to be:

- System 1: $\zeta =0.5$
- System 2: $\zeta =0.75$
- System 3: $\zeta =0.25$
- System 4: $\zeta =1.0$

As established earlier, the peak overshoot is maximized when the damping ratio ($\zeta$) is minimized. Comparing the values, the minimum damping ratio is $0.25$, which corresponds to the system represented by the transfer function $\frac{100}{s^2+5s+100}$.

Therefore, the system with the maximum peak overshoot due to a step input is the one with the transfer function $\frac{100}{s^2+5s+100}$.