BME Control Systems 2 — Questions and Answers
Question 1: The Routh-Hurwitz criterion determines system stability by:
- Plotting the frequency response of the open-loop transfer function
- Examining the signs of elements in the Routh array without computing roots (Correct answer)
- Computing the eigenvalues of the system matrix numerically
- Finding the gain margin from the Bode plot
Correct answer: Examining the signs of elements in the Routh array without computing roots
The Routh-Hurwitz criterion constructs a Routh array from the characteristic equation coefficients and assesses stability by checking for sign changes in the first column, avoiding direct root computation.
Question 2: A second-order system with damping ratio ζ = 0.5 is classified as:
- Overdamped
- Critically damped
- Underdamped (Correct answer)
- Undamped
Correct answer: Underdamped
An underdamped system has 0 < ζ < 1; with ζ = 0.5 the system exhibits oscillatory step response with overshoot that decays to the final value.
Question 3: The gain margin of a control system is measured at the frequency where the open-loop phase angle equals:
- 0°
- −90°
- −180° (Correct answer)
- +180°
Correct answer: −180°
Gain margin is evaluated at the phase crossover frequency ωpc where ∠G(jω)H(jω) = −180°, and it equals the reciprocal of the open-loop magnitude at that frequency.
Question 4: In a Bode magnitude plot, a first-order lag factor (1 + jωT)^(−1) has a high-frequency asymptotic slope of:
- +20 dB/decade
- −20 dB/decade (Correct answer)
- −40 dB/decade
- +40 dB/decade
Correct answer: −20 dB/decade
A single real pole contributes a −20 dB/decade asymptote above its corner frequency (ω = 1/T), reflecting the −1 power in the denominator factor.
Question 5: Root locus plots the trajectories of closed-loop poles as:
- Input frequency varies from 0 to ∞
- Open-loop gain K varies from 0 to ∞ (Correct answer)
- Damping ratio ζ varies from 0 to 1
- Time constant τ varies from 0 to ∞
Correct answer: Open-loop gain K varies from 0 to ∞
Root locus shows how the closed-loop poles migrate in the s-plane as the open-loop gain K increases from 0 to infinity, starting at open-loop poles and ending at open-loop zeros.
Question 6: The phase margin of a stable feedback system is the additional phase lag that would cause:
- The gain to reach 0 dB
- The system to become marginally stable (oscillatory) (Correct answer)
- The steady-state error to become infinite
- The bandwidth to decrease to zero
Correct answer: The system to become marginally stable (oscillatory)
Phase margin is measured at the gain crossover frequency (|G(jω)| = 1) and represents the additional phase lag required to bring the system to the verge of instability (−180° total phase).
Question 7: Which of the following transfer functions represents a pure integrator?
- K·s
- K/s (Correct answer)
- K/(s + a)
- K·s/(s + a)
Correct answer: K/s
A pure integrator has transfer function K/s, corresponding to the time-domain operation of integration, and introduces a pole at the origin of the s-plane.
BME Control Systems 2
The Routh-Hurwitz criterion determines system stability by: