Questions: Resonance, Peaking, and Bandwidth Relationships

Question 1

A control engineer reduces the damping ratio of a second-order system from ζ = 0.7 to ζ = 0.4 in order to increase bandwidth. What else unavoidably happens?

• A. Only the cutoff frequency changes; step response overshoot is unaffected by damping ratio
• B. The resonance peak M_r increases and step response percent overshoot increases — all three are manifestations of the same pole locations
• C. The natural frequency ω_n automatically decreases to compensate, keeping overshoot constant
• D. Bandwidth and overshoot change in opposite directions, so overall response quality remains constant

Answer: (B)

Bandwidth, peaking (M_r), and step-response overshoot are not independent — they are all governed by the same damping ratio ζ and the same closed-loop pole locations. Moving poles closer to the imaginary axis (reducing ζ) simultaneously widens bandwidth, raises M_r, and increases percent overshoot. There is no free lunch: you cannot widen bandwidth without also accepting more peaking and overshoot. This interconnection is the central design trade-off in second-order system synthesis.

Question 2

At ζ = 0.707 (the 'Butterworth' or 'critically flat' condition), what is special about the system's behavior?

• A. The system is at the boundary of stability — any further reduction in ζ causes instability
• B. The resonance peak M_r reaches its maximum value, providing the most amplification
• C. There is no resonance peaking in the frequency response, step overshoot is minimized (~4%), and the response is maximally flat before rolloff
• D. The bandwidth is exactly equal to the natural frequency ω_n, providing the simplest design relationship

Answer: (C)

At ζ = 0.707, M_r ≈ 1 (no magnitude peak above the DC gain), and step overshoot is approximately 4% — often a good starting point for design. This is called the 'maximally flat' or Butterworth condition. It is not the stability boundary (that is ζ = 0); stability requires ζ > 0. Option B is exactly backwards: M_r is maximized as ζ → 0, not at ζ = 0.707.

Question 3

A control system designer can independently choose to increase bandwidth (for faster tracking) while also decreasing percent overshoot (for less peaking), by selecting different values of the natural frequency ω_n and damping ratio ζ.

• T. True
• F. False

Answer: False

Bandwidth and overshoot are both tied to the same parameter ζ, so they cannot be tuned independently by changing ζ alone. Reducing ζ raises both bandwidth and overshoot simultaneously. The partial workaround is to increase ω_n while holding ζ fixed — this speeds up the response without changing the overshoot percentage — but even this strategy eventually hits limits from actuator saturation, noise amplification, and higher-order dynamics. The fundamental trade-off between speed and damping cannot be eliminated.

Question 4

An undamped second-order system (ζ = 0) driven continuously at its natural frequency will grow to an unbounded amplitude over time.

• T. True
• F. False

Answer: True

From the resonance peak formula M_r = 1/(2ζ√(1−ζ²)), as ζ → 0, M_r → ∞. With zero damping, every cycle of forcing adds energy that cannot escape, so oscillation amplitude grows without bound. In physical systems this manifests as structural failure — the classic example being a bridge or aircraft component driven at its resonant frequency. In control systems, a plant with a lightly damped resonant mode can cause the loop to saturate or oscillate destructively if the controller excites that frequency.

Question 5

A spec requires fast settling time AND small overshoot. Using the resonance-damping-bandwidth relationship, explain the fundamental design tension this creates and how you would reason about resolving it.

Model answer: Fast settling requires wide bandwidth, which requires low ζ. Small overshoot requires high ζ. Since both bandwidth and overshoot decrease together as ζ increases, the two specs pull in opposite directions on the same parameter. The standard resolution is to first select ζ to satisfy the overshoot constraint (e.g., ζ ≈ 0.6 for ~10% overshoot), then increase ω_n to meet the speed requirement — because raising ω_n scales up the response speed without changing the overshoot percentage. This separates the two specs onto different parameters: ζ controls overshoot shape, ω_n controls absolute speed.

This is the core two-parameter design procedure for second-order systems: (1) set ζ from the overshoot (or M_r) specification, (2) set ω_n from the settling time or bandwidth specification. The trade-off is real but manageable because the two parameters control largely orthogonal aspects of response quality. Recognizing that simultaneous tight specs on both speed and overshoot may require accepting some compromise — or adding a compensator — is the practical skill this topic develops.