BEE Control System Principles 4 β Questions and Answers
Question 1: In a feedback control system, what is the effect of increasing proportional gain Kp on steady-state error for a step input in a Type 0 system?
- Increases steady-state error
- Has no effect on steady-state error
- Decreases but does not eliminate steady-state error (Correct answer)
- Completely eliminates steady-state error
Correct answer: Decreases but does not eliminate steady-state error
For a Type 0 system, steady-state error = 1/(1+Kp), so increasing Kp reduces error but it approaches zero only as Kp β β.
Question 2: Which condition must be satisfied for a system to be both controllable and observable?
- The system must be BIBO stable
- The controllability matrix and observability matrix must both have full rank (Correct answer)
- All eigenvalues must be in the left-half plane
- The transfer function must have no pole-zero cancellations
Correct answer: The controllability matrix and observability matrix must both have full rank
Controllability requires rank(C) = n and observability requires rank(O) = n, where n is the system order.
Question 3: If a system's Bode phase plot crosses -180Β° at 10 rad/s and the magnitude at that frequency is -10 dB, what is the gain margin?
- 10 dB (Correct answer)
- -10 dB
- 20 dB
- 0 dB
Correct answer: 10 dB
Gain margin is the negative of the magnitude at the phase crossover frequency: GM = 0 - (-10 dB) = +10 dB.
Question 4: In discrete-time control systems, what is the z-transform equivalent of an integrator (1/s)?
- z/(z-1) (Correct answer)
- T/(z-1)
- 1/(z-1)
- (z+1)T/[2(z-1)]
Correct answer: z/(z-1)
The bilinear or exact z-domain equivalent of 1/s using the forward Euler method maps to z/(z-1), while Tustin's method gives (z+1)T/[2(z-1)].
Question 5: What is the percent overshoot for a second-order system with damping ratio ΞΆ = 0.5?
- 16.3% (Correct answer)
- 4.3%
- 25.4%
- 9.5%
Correct answer: 16.3%
Percent overshoot = exp(-ΟΞΆ/β(1-ΞΆΒ²)) Γ 100 = exp(-ΟΓ0.5/β0.75) Γ 100 β 16.3%.
Question 6: A lag compensator is typically used to:
- Increase the phase margin at high frequencies
- Improve transient response speed
- Increase the low-frequency gain to reduce steady-state error (Correct answer)
- Add a zero near the origin
Correct answer: Increase the low-frequency gain to reduce steady-state error
Lag compensators boost low-frequency gain (improving steady-state accuracy) while minimally affecting phase near crossover.
Question 7: When applying the final value theorem to find steady-state output, which condition must hold?
- The system must be minimum phase
- All poles of sY(s) must be in the open left-half plane (Correct answer)
- The input must be a unit impulse
- The transfer function must have relative degree of 1
Correct answer: All poles of sY(s) must be in the open left-half plane
The final value theorem lim(tββ) y(t) = lim(sβ0) sY(s) is valid only if all poles of sY(s) are in the open left-half plane (system is stable and converges).
BEE Control System Principles 4
In a feedback control system, what is the effect of increasing proportional gain Kp on steady-state error for a step input in a Type 0 system?