Second-order underdamped control system: identify the correct relationship between decay ratio and overshoot, and the qualitative effect of damping coefficient on overshoot (choose the only correct statement).

Introduction / Context:
This question tests fundamental time-domain terminology for a standard second-order, underdamped control system responding to a step input. Two commonly used descriptors are overshoot (how much the first peak exceeds the final steady value) and decay ratio (how rapidly successive peaks die out). Understanding their mathematical link helps in tuning and interpreting system performance.

Given Data / Assumptions:

• Standard second-order, underdamped system (0 < ζ < 1).
• Response considered is the unit-step response.
• Overshoot is the first peak above the final value expressed as a fraction of the final value.
• Decay ratio is the ratio of the second peak magnitude to the first peak magnitude.

Concept / Approach:
For an underdamped second-order system, overshoot M_p is given by M_p = exp(−πζ / sqrt(1 − ζ^2)). The second peak occurs one damped period later, so its amplitude relative to the first is exp(−2πζ / sqrt(1 − ζ^2)). Thus, decay ratio DR = exp(−2πζ / sqrt(1 − ζ^2)) = [exp(−πζ / sqrt(1 − ζ^2))]^2 = (M_p)^2.

Step-by-Step Solution:
Write M_p = exp(−πζ / sqrt(1 − ζ^2)).Compute the second peak attenuation over one period: factor = exp(−2πζ / sqrt(1 − ζ^2)).Recognize decay ratio DR equals that attenuation = (M_p)^2.Infer qualitative trend: increasing ζ decreases M_p (less overshoot).

Verification / Alternative check:
Pick ζ = 0.5. Then M_p ≈ exp(−π*0.5/√0.75) ≈ 0.163. Squaring gives ≈ 0.0266. Numerical simulation of a standard second-order loop yields the same decay ratio between first two peaks, confirming the relationship.

Why Other Options Are Wrong:
Decay ratio = overshoot: Incorrect; DR compares peak-to-peak, not peak-to-final.Overshoot increases when damping increases: Opposite of true behavior; more damping → less overshoot.A larger damping coefficient implies smaller damping: Contradictory statement.Decay ratio = 1 / overshoot: No such general relation exists.

Common Pitfalls:
Confusing decay ratio with logarithmic decrement; mixing percent overshoot with fractional overshoot; forgetting that both M_p and DR depend monotonically on ζ for 0 < ζ < 1.

Final Answer:
Decay ratio = (overshoot)^2