Control Engineering Basics
Raising the gain of the loop transfer function reduced the phase margin from 60° to 30°. Which of the following correctly describes the change in the closed-loop step response?
- AThe overshoot becomes smaller and the oscillation disappears, but in exchange the rise of the response becomes slower
- BThe phase margin is a frequency-response measure, so it does not affect the shape of the step response
- CThe response tends to become faster, but the overshoot becomes larger and the oscillation takes longer to die out
Show answer and explanation
Answer: C. The response tends to become faster, but the overshoot becomes larger and the oscillation takes longer to die out
Raising the gain raises the gain crossover frequency and makes the response faster, but a smaller phase margin lowers the damping ratio of the closed loop. As a rule of thumb, the damping ratio ζ is estimated at about one hundredth of the phase margin in degrees. For the standard system whose loop transfer function is ωn²/(s(s + 2ζωn)), the overshoot is about 9% at a phase margin of 60° and about 42% at 30°. As the phase margin approaches zero the response approaches a sustained oscillation, and when it becomes negative the system is unstable.
- A. This is the trend when the phase margin increases, the opposite.
- B. The phase margin is closely tied to the overshoot of the step response and to how quickly the oscillation dies out.
- C. Correct. Speed and stability margin trade off against each other.