Questions: Control Loop Design via Bode Plots and Loop Shaping

Question 1

A plant Bode plot shows the gain crossover frequency occurs where the plant phase is −145°. Your specification requires a phase margin of at least 45°. Which compensator action is most directly needed?

• A. Add a lag compensator at the crossover frequency to increase low-frequency gain
• B. Add a lead compensator centered near the crossover frequency to boost phase by roughly 10°
• C. Reduce the loop gain to lower the crossover frequency to a region with better phase
• D. Add a notch filter at the crossover frequency to suppress the resonant peak

Answer: (B)

Phase margin = 180° + phase at crossover. With phase = −145°, the current phase margin is 35°, which is 10° short of the 45° target. A lead compensator contributes positive phase in a frequency band around its geometric center — placing it near the crossover frequency adds the needed phase boost. Option A (lag) is used for low-frequency gain improvement, not phase margin repair. Option C shifts the crossover but doesn't guarantee better phase unless you know the plant's phase behavior at the new frequency.

Question 2

A control engineer wants to improve the steady-state tracking accuracy of a position servo (reduce velocity error) without significantly changing the closed-loop bandwidth. Which compensator strategy achieves this?

• A. Lead compensator centered at the gain crossover frequency
• B. Increase in proportional gain only
• C. Lag compensator with its pole-zero pair placed well below the crossover frequency
• D. Lead compensator placed above the crossover frequency

Answer: (C)

A lag compensator provides high gain at low frequencies (improving steady-state tracking) while contributing only a small phase lag at crossover — provided it is placed well below the crossover frequency (typically a decade or more). This leaves the crossover frequency and bandwidth essentially unchanged. A pure gain increase (option B) raises the crossover frequency, changing bandwidth and potentially reducing phase margin. A lead compensator adds phase at crossover but doesn't improve low-frequency gain the way lag does.

Question 3

A lead compensator adds positive phase and simultaneously increases the magnitude of the loop gain in the frequency band around crossover, which raises the gain crossover frequency.

• T. True
• F. False

Answer: True

Both effects are real and coupled: a lead compensator (zero above its pole) contributes positive phase in the band between its pole and zero, and its magnitude increases in that same region. This magnitude increase means the 0 dB crossover point moves to a higher frequency — raising the bandwidth. This is why lead design requires checking that the new, higher crossover frequency still sits in a phase-favorable region of the plant's Bode plot.

Question 4

Increasing open-loop gain generally improves both closed-loop bandwidth and phase margin.

• T. True
• F. False

Answer: False

Increasing gain raises the gain crossover frequency (higher bandwidth), but if the plant's phase decreases steeply with frequency (as is typical), the new crossover frequency falls in a region of worse phase, reducing phase margin. In many practical systems, the gain that maximizes bandwidth also drives phase margin dangerously low. Loop shaping with compensators allows bandwidth and phase margin to be addressed more independently than a simple gain change permits.

Question 5

Why is loop shaping described as 'inverting' the skill of reading Bode plots, and what is the engineer's primary degree of freedom in the design?

Model answer: Reading a Bode plot extracts stability margins from a given system. Loop shaping goes in reverse: you specify the desired margins and crossover frequency, then design a compensator whose Bode plot, added to the plant's, produces the target shape. The engineer's degree of freedom is the compensator transfer function — its poles, zeros, and gain add directly to the plant's magnitude and phase on the log-scale plot. Since the plant cannot be changed, the compensator is the only design variable.

This inversion framing helps clarify the goal. The design workflow — determine required ωc and PM from transient specs, evaluate the plant at ωc, compute the deficit in phase and gain, design lead or lag to close the gap, verify — is a structured way of inverting the analysis problem. The key insight is that on a Bode plot, compensator magnitude and phase add directly to plant magnitude and phase, so the combined open-loop response is literally the superposition of two Bode plots.