BEE Control Systems Theory 3 β Questions and Answers
Question 1: The phase margin of a stable feedback system is measured at the frequency where the open-loop magnitude is:
- 0 dB (unity gain) (Correct answer)
- β3 dB
- +3 dB
- Equal to the phase crossover frequency
Correct answer: 0 dB (unity gain)
Phase margin is the additional phase lag at the gain crossover frequency (where |G(jΟ)H(jΟ)| = 1, or 0 dB) before the system becomes unstable.
Question 2: Which of the following statements about the root locus is correct?
- Branches start at open-loop zeros and end at open-loop poles
- The number of branches equals the number of zeros
- Branches start at open-loop poles and end at open-loop zeros (or infinity) (Correct answer)
- The root locus is always symmetric about the real axis
Correct answer: Branches start at open-loop poles and end at open-loop zeros (or infinity)
Root locus branches begin at open-loop poles (K=0) and terminate at open-loop zeros or travel to infinity along asymptotes as Kββ.
Question 3: A second-order system with ΞΆ = 0.5 and Οn = 10 rad/s has a damped natural frequency (Οd) of:
- 5 rad/s
- 8.66 rad/s (Correct answer)
- 10 rad/s
- 11.55 rad/s
Correct answer: 8.66 rad/s
The damped natural frequency is Οd = Οnβ(1βΞΆΒ²) = 10β(1β0.25) = 10β0.75 β 8.66 rad/s.
Question 4: In the Nyquist stability criterion, encirclements of the β1+j0 point in the clockwise direction are counted to determine:
- The number of right-half-plane zeros
- The number of unstable closed-loop poles (Correct answer)
- The number of gain crossover frequencies
- The relative stability only
Correct answer: The number of unstable closed-loop poles
By the Nyquist criterion, N = Z β P, where N is clockwise encirclements of β1, P is open-loop RHP poles, and Z is closed-loop RHP poles (instability count).
Question 5: A system is said to be observable if:
- All states can be driven to the origin using an input
- The initial state can be determined from the output over a finite time interval (Correct answer)
- The system matrix A has no repeated eigenvalues
- The transfer function has no pole-zero cancellations
Correct answer: The initial state can be determined from the output over a finite time interval
Observability means that every initial state x(0) can be uniquely determined by observing the output y(t) over a finite time interval.
Question 6: Which compensator is most effective at improving the steady-state accuracy of a control system without significantly affecting transient response?
- Lead compensator
- Lag compensator (Correct answer)
- PD controller
- Notch filter
Correct answer: Lag compensator
A lag compensator increases low-frequency gain to reduce steady-state error while placing its pole-zero pair far below the crossover frequency, minimally affecting transient response.
Question 7: For a unity-feedback system with open-loop transfer function G(s) = K/[s(s+4)], what value of K places both closed-loop poles at s = β2?
- K = 4 (Correct answer)
- K = 8
- K = 2
- K = 16
Correct answer: K = 4
Closed-loop characteristic equation: sΒ² + 4s + K = 0; for double root at s = β2, we need (s+2)Β² = sΒ² + 4s + 4, so K = 4.
BEE Control Systems Theory 3
The phase margin of a stable feedback system is measured at the frequency where the open-loop magnitude is: