When phase-lag compensation is applied to a control system, how do the gain crossover frequency, the closed-loop bandwidth, and the undamped natural frequency typically change?
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AIncreased, Increased, Increased
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BIncreased, Increased, Decreased
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CIncreased, Decreased, Decreased
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DDecreased, Decreased, Decreased
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EUnchanged, Increased, Decreased
Answer
Correct Answer: Decreased, Decreased, Decreased
Explanation
Introduction / Context:Phase-lag compensators are used to improve steady-state accuracy by boosting low-frequency loop gain, at the cost of high-frequency performance. This question asks you to evaluate how key frequency-domain metrics shift when lag compensation is introduced to a control loop.
Given Data / Assumptions:
- Classic lead–lag compensator theory for linear time-invariant systems.
- Lag compensator has magnitude |Glag(jω)| < 1 for higher frequencies and provides additional phase lag.
- Design objective prioritizes steady-state error reduction, not bandwidth expansion.
Concept / Approach:Lag compensation increases low-frequency gain to reduce steady-state error while attenuating higher-frequency gain, thereby lowering the gain crossover frequency. Reduced gain crossover frequency generally reduces closed-loop bandwidth and undamped natural frequency, improving noise immunity but slowing response.
Step-by-Step Solution:
Lag compensator magnitude is below unity beyond its corner frequency → loop gain is smaller at mid/high frequencies.Smaller loop gain at higher ω → gain crossover frequency shifts left (decreases).Lower gain crossover frequency correlates with reduced closed-loop bandwidth.With less bandwidth, dominant pole location moves closer to the imaginary axis → undamped natural frequency decreases.Verification / Alternative check:
Bode plot intuition: adding lag decreases magnitude slope after its zero–pole pair; cross-over and bandwidth thus drop. Time-domain response becomes slower but with improved steady-state accuracy.Why Other Options Are Wrong:
Options showing increases contradict the known attenuation at higher frequencies from lag compensation.Mixed increase/decrease options do not match standard lag behavior in classical control design.Common Pitfalls:
Confusing lag with lead compensation; assuming any compensator always increases bandwidth; overlooking the trade-off between steady-state error and speed.Final Answer:
Decreased, Decreased, Decreased