This question asks us to evaluate the relationship between transient response characteristics (overshoot) and frequency response characteristics (resonance peak) for control systems.

Analyzing Statement (I): Overshoot and Resonance Peak Relationship

Statement (I) claims that all systems exhibiting overshoot in their transient response will also exhibit a resonance peak in their frequency response. Let's examine this for a standard second-order system, whose transfer function is given by:

$$ G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} $$

where $\omega_n$ is the undamped natural frequency and $\zeta$ is the damping ratio.

• Transient Response (Overshoot): A system exhibits overshoot in its transient response if it is underdamped, which means the damping ratio $\zeta$ is less than 1 ($0 \le \zeta < 1$). The percentage overshoot (PO) is calculated as:

  $$ PO = 100 \times e^{-\frac{\pi \zeta}{\sqrt{1-\zeta^2}}} $$

  A system has overshoot if $\zeta < 1$.
• Frequency Response (Resonance Peak): The resonance peak magnitude ($M_p$) in the frequency response of a second-order system is defined as the maximum value of the magnitude response $|G(j\omega)|$. It occurs at the resonant frequency $\omega_r = \omega_n \sqrt{1-\zeta^2}$ (for $\zeta < 1/\sqrt{2}$). The magnitude of the resonance peak is given by:

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

  A significant resonance peak (typically considered $M_p > 1$) exists only when the damping ratio $\zeta$ is less than $\frac{1}{\sqrt{2}}$ (approximately 0.707). If $\zeta \ge \frac{1}{\sqrt{2}}$, then $M_p \le 1$, indicating no amplification at the resonant frequency or no distinct peak above 0 dB.

From these definitions, we can see that a system can have overshoot (requiring $\zeta < 1$) but not necessarily have a resonance peak ($M_p > 1$, requiring $\zeta < 1/\sqrt{2}$). For example, if a system has a damping ratio $\zeta = 0.8$, it is less than 1, so it will exhibit overshoot. However, since $0.8 > 1/\sqrt{2}$, its resonance peak magnitude is $M_p = \frac{1}{2(0.8)\sqrt{1-(0.8)^2}} = \frac{1}{0.96} \approx 1.04$. This value is only slightly above 1 and might not be considered a significant resonance peak in many contexts, or if strictly defined as $M_p > 1$, it barely meets it. If we consider $\zeta$ values between $1/\sqrt{2}$ and $1$, these systems will show overshoot but no significant resonance peak. Since the statement claims "All" systems with overshoot have a resonance peak, and we found cases where this isn't true (e.g., systems with $1/\sqrt{2} < \zeta < 1$), Statement (I) is false.

Analyzing Statement (II): Resonance Peak and Overshoot Magnitude

Statement (II) suggests that a large resonance peak in the frequency response corresponds to a large overshoot in the transient response. Let's analyze the relationship using the formulas for the second-order system:

• Resonance Peak Magnitude: $M_p = \frac{1}{2\zeta\sqrt{1-\zeta^2}}$
• Percentage Overshoot: $PO = 100 \times e^{-\frac{\pi \zeta}{\sqrt{1-\zeta^2}}}$

As the resonance peak magnitude ($M_p$) increases, the value of the denominator $2\zeta\sqrt{1-\zeta^2}$ must decrease. This happens when the damping ratio ($\zeta$) becomes smaller (approaching 0). Consequently, as $\zeta$ decreases, the exponent in the PO formula, $-\frac{\pi \zeta}{\sqrt{1-\zeta^2}}$, becomes less negative (closer to 0). A less negative exponent results in a larger value for $e^{\text{exponent}}$, thus leading to a larger percentage overshoot (PO).

Therefore, a large resonance peak ($M_p$) indicates a low damping ratio ($\zeta$), which in turn causes a large overshoot (PO) in the transient response. Statement (II) is true.

Conclusion

Based on the analysis:

• Statement (I) is false.
• Statement (II) is true.

This corresponds to Option 4.