Questions: Bandwidth and Frequency Domain Specifications

Question 1

A control engineer needs to reduce the rise time of a servo system from 100 ms to 50 ms. What must happen to the closed-loop bandwidth?

• A. The bandwidth must be doubled, since rise time is approximately inversely proportional to bandwidth
• B. The bandwidth must be halved, since a faster rise means fewer high-frequency components are needed
• C. The bandwidth must increase by exactly √2, matching the change in damping ratio
• D. Rise time and bandwidth are independent design parameters that can be adjusted separately

Answer: (A)

For a second-order system, tr ≈ 1.8/BW, so halving the rise time (100 ms → 50 ms) requires doubling the bandwidth. This is a fundamental physical constraint: a faster step response requires the system to pass higher-frequency content, which is exactly what wider bandwidth means. Option 3 is the key misconception to avoid — rise time and bandwidth are tightly coupled by the physics of frequency response. You cannot make a system respond faster without giving it higher-frequency tracking capability.

Question 2

A position control system uses an encoder with significant high-frequency electrical noise. An engineer considers increasing the closed-loop bandwidth from 5 Hz to 50 Hz to improve tracking speed. What is the most likely consequence?

• A. Improved tracking with no adverse effects, since encoder noise averages out over time
• B. High-frequency encoder noise will be amplified and appear as actuator chatter and position error
• C. The system will become unconditionally unstable because no practical system can operate above 50 Hz
• D. The motor will run more smoothly because wider bandwidth averages over more frequencies simultaneously

Answer: (B)

A higher-bandwidth controller faithfully responds to all frequencies within its passband — including noise. Encoder electrical noise at high frequencies, previously attenuated by the narrow-bandwidth design, now passes through the control loop and commands the actuator to follow the noise. The result is increased actuator activity (chatter, wear, high current), not better performance. This is the core bandwidth-noise tradeoff: every frequency added to the passband adds both useful signal tracking and noise sensitivity. Option 0 is wrong — noise does not average away in a closed-loop system; it drives actuator commands.

Question 3

The -3 dB bandwidth is the frequency at which the closed-loop output power falls to half its DC value, corresponding to the magnitude ratio dropping to approximately 0.707.

• T. True
• F. False

Answer: True

Power is proportional to the square of amplitude. When amplitude drops to 1/√2 ≈ 0.707, power becomes (1/√2)² = 0.5 — exactly half. In decibels: 20·log₁₀(0.707) ≈ -3 dB (amplitude), or equivalently 10·log₁₀(0.5) = -3 dB (power). This is the standard -3 dB point definition used consistently in signals and systems, control theory, and RF engineering. Signals below this frequency pass with minimal attenuation; above it, they are progressively suppressed.

Question 4

Increasing the bandwidth of a closed-loop control system typically improves its performance, since higher bandwidth means the system can respond to a wider range of input frequencies.

• T. True
• F. False

Answer: False

More bandwidth means more noise sensitivity — the system responds faithfully to sensor noise, quantization errors, and high-frequency disturbances in addition to the desired input. In physical systems, this manifests as actuator chatter, mechanical wear, and excessive energy consumption. Additionally, increasing loop gain to widen bandwidth reduces phase margin, pushing the system toward instability. The engineering goal is the *minimum* bandwidth that satisfies tracking and disturbance rejection specifications, not the maximum achievable. The noise-bandwidth tradeoff is fundamental and cannot be avoided.

Question 5

Explain the fundamental tradeoff between bandwidth and noise sensitivity, and why an engineer cannot simply maximize bandwidth to achieve the fastest possible tracking.

Model answer: Bandwidth determines which input frequencies the closed-loop system responds to. All sensors have noise, so the control signal always contains both useful tracking error and noise at every frequency. Below the bandwidth, the controller attempts to correct both genuine tracking error and noise-driven deviations. Maximizing bandwidth therefore maximizes noise amplification: the actuator is commanded to correct for every noise fluctuation, not just real errors. In practice this causes chattering, actuator wear, and possibly instability near the phase crossover frequency. The engineer selects the minimum bandwidth that tracks desired reference trajectories within spec while keeping noise-induced actuator activity within acceptable bounds — a tradeoff requiring knowledge of both the reference spectrum and the sensor noise spectrum.

This tradeoff is why practical control design involves both frequency-domain specifications (bandwidth, gain/phase margins) and noise characterization. A motor control system for a precision robot needs to track smooth position commands but not encoder count noise; a sensor filtering bandwidth below encoder noise frequencies achieves acceptable tracking without chattering. The same principle applies in electrical circuit design (op-amp bandwidth limits), communication receivers (noise-bandwidth tradeoffs), and signal processing.