BEE Control Systems Theory 4 — Questions and Answers
Question 1: The Final Value Theorem states that lim(t→∞) f(t) = lim(s→0) s·F(s), provided that:
- All poles of s·F(s) lie in the closed left-half s-plane (Correct answer)
- F(s) has no zeros in the right-half plane
- The system is underdamped
- The system is second-order or lower
Correct answer: All poles of s·F(s) lie in the closed left-half s-plane
The Final Value Theorem is valid only if all poles of s·F(s) lie in the open left-half plane (stable), ensuring the time-domain limit exists.
Question 2: In signal flow graphs, Mason's gain formula is used to find the:
- Number of feedback loops
- Overall transfer function between a source and sink node (Correct answer)
- Stability of each individual loop
- Frequency response of forward paths
Correct answer: Overall transfer function between a source and sink node
Mason's gain formula computes the overall transfer function T = Σ(Pk·Δk)/Δ, summing contributions of all forward paths weighted by their cofactors.
Question 3: A plant with transfer function G(s) = 1/(s+1) is controlled with proportional gain K. The closed-loop bandwidth _____ as K increases.
- Decreases
- Remains constant
- Increases (Correct answer)
- First decreases then increases
Correct answer: Increases
The closed-loop transfer function is K/[(s+1)+K] = K/(s+1+K), giving a closed-loop pole at −(1+K); bandwidth increases with K.
Question 4: Which of the following is a characteristic of a minimum-phase system?
- It has all poles and zeros in the right-half plane
- It has the minimum possible phase lag for its magnitude response (Correct answer)
- Its Nyquist plot does not encircle the origin
- It cannot be stabilized by a proportional controller
Correct answer: It has the minimum possible phase lag for its magnitude response
A minimum-phase system has all poles and zeros in the left-half plane and exhibits the least phase lag achievable for a given magnitude response.
Question 5: The integral windup problem in a PID controller occurs when:
- The derivative gain is too large, causing oscillations
- The integrator accumulates a large error when the actuator saturates, causing overshoot upon recovery (Correct answer)
- The proportional gain exceeds the stability limit
- The sampling rate is insufficient for the system bandwidth
Correct answer: The integrator accumulates a large error when the actuator saturates, causing overshoot upon recovery
Integral windup happens when actuator saturation prevents error correction, allowing the integral term to grow large and produce significant overshoot when the actuator comes out of saturation.
Question 6: Two systems with transfer functions G₁(s) and G₂(s) are connected in cascade (series). The overall transfer function is:
- G₁(s) + G₂(s)
- G₁(s) / G₂(s)
- G₁(s) · G₂(s) (Correct answer)
- 1 / [G₁(s) + G₂(s)]
Correct answer: G₁(s) · G₂(s)
For cascaded (series) blocks, the overall transfer function is the product of individual transfer functions: G(s) = G₁(s)·G₂(s).
Question 7: The gain crossover frequency of a system is 10 rad/s and the phase of the open-loop transfer function at that frequency is −150°. The phase margin is:
- −30°
- +30° (Correct answer)
- +150°
- −150°
Correct answer: +30°
Phase margin = 180° + ∠G(jωgc) = 180° + (−150°) = +30°, indicating a stable system with 30° of margin.
BEE Control Systems Theory 4
The Final Value Theorem states that lim(t→∞) f(t) = lim(s→0) s·F(s), provided that: