Questions: Rise Time, Settling Time, and Overshoot Specifications

Question 1

A control engineer needs to reduce percent overshoot from 25% to 5% without significantly slowing the rise time. Which parameter adjustment achieves this?

• A. Increase ωn while holding ζ constant — higher natural frequency reduces overshoot
• B. Increase ζ while holding ωn approximately constant — damping ratio is the primary handle for overshoot
• C. Decrease both ζ and ωn proportionally — the ratio determines overshoot
• D. Decrease ωn while holding ζ constant — lower natural frequency reduces oscillations

Answer: (B)

Percent overshoot is determined solely by ζ via %OS = 100·exp(−πζ/√(1−ζ²)). Changing ωn does not affect overshoot at all. To go from 25% overshoot (ζ ≈ 0.4) to 5% overshoot (ζ ≈ 0.69), increase ζ. Rise time is primarily controlled by ωn, so if ωn is held roughly constant, rise time is preserved. The independence of these two parameters is the key design insight.

Question 2

For a second-order underdamped system, what is the percent overshoot when the damping ratio ζ = 0.707?

• A. Exactly 0% — ζ = 0.707 is the critically damped condition with no overshoot
• B. Approximately 4.3% — ζ = 0.707 gives modest overshoot with fast response, a common design target
• C. Approximately 16.3% — ζ = 0.707 is the half-damping threshold
• D. Approximately 50% — ζ = 0.707 corresponds to low damping and large overshoot

Answer: (B)

Using %OS = 100·exp(−πζ/√(1−ζ²)) with ζ = 0.707: exp(−π·0.707/√(1−0.5)) = exp(−π·0.707/0.707) = exp(−π) ≈ exp(−3.14) ≈ 4.3%. This is why ζ = 0.707 (often called the 'Butterworth damping') is a common design target — it gives fast response with barely perceptible overshoot. Critical damping (zero overshoot) requires ζ = 1, which is noticeably slower.

Question 3

For an underdamped second-order system, increasing ωn while holding ζ constant will reduce rise time without changing percent overshoot.

• T. True
• F. False

Answer: True

Percent overshoot depends only on ζ (via the formula %OS = 100·exp(−πζ/√(1−ζ²))), so changing ωn while holding ζ constant does not affect overshoot at all. Rise time tr ≈ (π − arccos ζ)/(ωn·√(1−ζ²)) is inversely proportional to ωn — doubling ωn halves the rise time. This approximate independence of the two parameters (overshoot controlled by ζ, speed controlled by ωn) is the core insight for second-order system design.

Question 4

A critically damped system (ζ = 1) responds more slowly than an overdamped system (ζ > 1) because higher damping generally slows the response.

• T. True
• F. False

Answer: False

Counterintuitively, critical damping (ζ = 1) is actually the fastest non-overshooting response — it is faster than any overdamped system (ζ > 1). An overdamped system has two distinct real poles; as ζ increases beyond 1, one pole moves very close to the origin, creating a slow mode that drags out the response. Critical damping balances the two poles at the same location, giving the fastest approach to steady state without overshoot. This is why 'critically damped' is the design target when overshoot is forbidden.

Question 5

A specification requires rise time under 50 ms and overshoot under 5%. Explain how you translate these requirements into constraints on the second-order system parameters ζ and ωn.

Model answer: The overshoot constraint maps to a minimum ζ. Using %OS = 100·exp(−πζ/√(1−ζ²)) ≤ 5%, solving for ζ gives ζ ≥ approximately 0.69. The rise time constraint maps to a minimum ωn. Using tr ≈ (π − arccos ζ)/(ωn·√(1−ζ²)) ≤ 0.05 s, with ζ at its minimum value of 0.69, you can solve for the required ωn. The two constraints together define a region in (ζ, ωn) parameter space: the system's closed-loop poles must satisfy both bounds simultaneously.

The key is recognizing that the two specifications target different parameters: overshoot → minimum ζ, rise time → minimum ωn. Because these parameters are approximately independent (ωn sets speed; ζ sets oscillation level), you can design them separately. Find the minimum ζ from the overshoot bound, then find the minimum ωn from the rise time bound at that ζ. Any (ζ, ωn) pair satisfying both constraints is an acceptable design.