A control system has a phase margin of only 15° at its gain crossover frequency, causing oscillatory step responses. The steady-state error to a ramp input is acceptable. Which compensator is most appropriate, and where should its maximum contribution be placed?
ALag compensator — place its corner frequencies at the gain crossover frequency to maximize phase drag reduction
BLead compensator — place its maximum phase contribution at the desired gain crossover frequency to boost phase margin
CLag compensator — place its pole at the origin to eliminate steady-state error and indirectly reduce oscillation
DLead compensator — place its zero at DC to maximize low-frequency gain and reduce the steady-state error that is causing the oscillation
The diagnosis here is clear: insufficient phase margin (15°) causes oscillation; steady-state accuracy is not the problem. A lead compensator adds positive phase in the band between its zero and pole. The design procedure places the maximum phase contribution (at ωm = √(zp)) at the desired gain crossover frequency, directly increasing phase margin. A lag compensator would be wrong here — it subtracts a small amount of phase near crossover and addresses low-frequency gain, not phase margin. Option C confuses a lag compensator with a pure integrator (pole at origin).
Question 2 Multiple Choice
Why must the corner frequencies of a lag compensator be placed far below (approximately 1/10 of) the gain crossover frequency, rather than at or near it?
ABecause placing them near crossover would cause the lag compensator to add too much phase, overshooting the design target
BBecause a lag compensator subtracts phase in its transition region — placing its corners near crossover would reduce phase margin and worsen stability
CBecause the lag compensator only provides benefit at frequencies above its pole, not below
DBecause placing the corners at crossover maximizes the noise amplification at high frequencies
A lag compensator is not a pure gain element — in the transition region between its pole and zero, it subtracts phase. This phase lag is the cost of the low-frequency gain benefit. If the transition region overlaps with the gain crossover frequency, this phase subtraction reduces phase margin and can destabilize the system. By placing both corner frequencies at 1/10 of ωgc or below, the phase contribution at crossover is limited to less than 5° — an acceptable cost. The lag compensator's benefit (increased low-frequency gain → reduced steady-state error) is entirely at frequencies well below crossover.
Question 3 True / False
A lag compensator improves steady-state accuracy by adding positive phase near the gain crossover frequency, which reduces tracking error.
TTrue
FFalse
Answer: False
This inverts the actual mechanism. A lag compensator does *not* add positive phase — it subtracts a small amount near its transition region, which is why its corners must be placed far below crossover. The lag compensator improves steady-state accuracy by increasing the *magnitude* of the loop gain at low frequencies (DC gain increases), which reduces the steady-state error coefficient. Phase is not the mechanism; the benefit is purely a gain increase at the frequencies where steady-state tracking matters. Adding positive phase is the job of a lead compensator.
Question 4 True / False
A lead compensator increases high-frequency loop gain as a side effect of adding phase margin, which can amplify sensor noise.
TTrue
FFalse
Answer: True
This is a genuine engineering tradeoff with lead compensators. On the Bode magnitude plot, a lead compensator rises by +20 dB/decade between its zero and pole frequencies, then levels off — so it increases gain in the frequency band where the phase is boosted and above. Since sensor noise typically occurs at high frequencies, increasing high-frequency gain directly amplifies that noise into the control signal. In practice, this limits how much lead can be added (typically no more than 60° of maximum phase lead), and lead compensators are often paired with low-pass filters in noise-sensitive applications.
Question 5 Short Answer
Explain why a lag compensator must have its pole and zero placed well below the gain crossover frequency, and what goes wrong if they are placed at or near it.
Think about your answer, then reveal below.
Model answer: A lag compensator produces a phase lag (negative phase contribution) in the frequency region between its pole and zero — the transition band where gain is changing. If this band overlaps with the gain crossover frequency, the lag reduces phase margin at exactly the point that determines stability, potentially causing oscillation or instability. By placing both the pole and zero at about 1/10 of ωgc, the transition band is well below crossover, and the phase contribution at ωgc is negligible (under 5°). The low-frequency gain increase that improves steady-state accuracy is preserved while the stability margin is not compromised.
The asymmetry between lead and lag design is important: for a lead compensator, you want its maximum phase contribution *at* crossover (by design). For a lag compensator, you want its phase contribution to be essentially zero at crossover — so you push it far away. The two compensators are solving different problems in different frequency regions, and their placement rules reflect this.