GATE Control Systems 2 — Questions and Answers
Question 1: The steady-state error of a Type-1 system to a unit ramp input is:
- Zero
- 1/Kv (Correct answer)
- 1/Ka
- Infinite
Correct answer: 1/Kv
For a Type-1 system, the velocity error constant Kv is finite, so the steady-state error to a ramp is 1/Kv.
Question 2: Which of the following is the correct expression for the Gain Margin (GM) of a system?
- GM = 20 log|G(jω_pc)H(jω_pc)|
- GM = -20 log|G(jω_pc)H(jω_pc)| (Correct answer)
- GM = 20 log(1/|G(jω_gc)H(jω_gc)|)
- GM = |G(jω_pc)H(jω_pc)|
Correct answer: GM = -20 log|G(jω_pc)H(jω_pc)|
Gain margin in dB is -20 log|G(jω)H(jω)| evaluated at the phase crossover frequency where the phase is -180°.
Question 3: For a second-order underdamped system, the peak overshoot depends on:
- Natural frequency only
- Damping ratio only (Correct answer)
- Both damping ratio and natural frequency
- Neither damping ratio nor natural frequency
Correct answer: Damping ratio only
Peak overshoot Mp = exp(-πζ/√(1-ζ²)) × 100%, which is a function of the damping ratio ζ only.
Question 4: A system has poles at s = -1 ± j2 and a zero at s = -3. The damped natural frequency is:
- 1 rad/s
- 2 rad/s (Correct answer)
- √5 rad/s
- 3 rad/s
Correct answer: 2 rad/s
The damped natural frequency ωd is the imaginary part of the complex poles, which equals 2 rad/s.
Question 5: In a Bode plot, a pure integrator (1/s) contributes a slope of:
- -20 dB/decade and -90° phase (Correct answer)
- +20 dB/decade and +90° phase
- -40 dB/decade and -180° phase
- -20 dB/decade and +90° phase
Correct answer: -20 dB/decade and -90° phase
A pure integrator 1/s has a gain of -20 dB/decade and a constant phase of -90° across all frequencies.
Question 6: The number of asymptotes in the root locus of a system with 4 poles and 1 zero is:
- 1
- 3 (Correct answer)
- 4
- 5
Correct answer: 3
The number of asymptotes equals P - Z = 4 - 1 = 3, where P is the number of poles and Z is the number of zeros.
Question 7: The transfer function of an ideal proportional-integral (PI) controller is:
- Kp(1 + Tds)
- Kp(1 + 1/(Tis)) (Correct answer)
- Kp(1 + Tds + 1/(Tis))
- Kp/(1 + Tis)
Correct answer: Kp(1 + 1/(Tis))
A PI controller adds an integrator to eliminate steady-state error, with transfer function Kp(1 + 1/(Ti·s)).
GATE Control Systems 2
The steady-state error of a Type-1 system to a unit ramp input is: