In a Bode plot, what is the phase margin when the gain crossover frequency equals the phase crossover frequency?
Answer: A
At the gain crossover frequency, magnitude is 0 dB. When this equals phase crossover frequency, the phase is -180°, giving phase margin = -180° - (-180°) = 0°
Q.23Medium
For a second-order system with ζ = 0.5 and ωn = 4 rad/s, what is the peak overshoot?
What is the effect of adding a pole at the origin to a stable open-loop system?
Answer: C
Adding a pole at origin adds -90° phase shift at all frequencies, decreasing phase margin. It also reduces high-frequency gain, decreasing gain margin
Q.25Medium
A lead compensator Gc(s) = K(s+a)/(s+b) where b > a provides maximum phase lead at frequency:
Answer: A
For lead compensator, maximum phase lead occurs at ωm = 1/√(τ₁τ₂) where τ₁ = 1/a and τ₂ = 1/b, giving ωm = √(ab)
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Q.26Medium
In state-space representation, if eigenvalues of A matrix are at s = -1, -2, -3, the system is:
Answer: B
Stability depends only on eigenvalue locations (all in LHP = stable). Controllability requires rank[B AB A²B] = n, which eigenvalues alone don't determine
Q.27Medium
For an underdamped second-order system, the relationship between settling time ts and damping ratio ζ is:
Answer: C
Settling time ts ≈ 4/(ζωn) for 2% criterion, thus ts ∝ 1/(ζωn)
Q.28Medium
Which compensator is preferred for improving steady-state error without significantly affecting transient response?
Answer: B
Lag compensator increases DC gain significantly, improving steady-state error, while its phase lag is restricted to lower frequencies, minimizing transient effects
Q.29Medium
A system has poles at -2±j3. The natural frequency and damping ratio are approximately:
Answer: A
ωn = √(4+9) = √13 ≈ 3.6 rad/s, ζ = 2/√13 ≈ 0.55
Q.30Medium
In a unity feedback system, increasing loop gain K generally:
Answer: B
Higher K reduces ess proportionally but shifts root locus rightward, potentially crossing into RHP, thus reducing stability margins
Q.31Medium
In a compensated system, if phase margin PM = 30° and gain margin GM = 8 dB, this indicates:
Answer: C
PM = 30° is acceptable (typically 30-60°), GM = 8 dB (>6 dB threshold) indicates good gain stability. Both margins suggest satisfactory performance
Q.32Medium
For a unity feedback control system with G(s) = 10/[s(s+5)], the static velocity error constant Kv is:
A system exhibits steady-state error of 0.2 for unit ramp input with loop gain K = 50. The static velocity error constant is:
Answer: A
For ramp input, ess = 1/Kv. Given ess = 0.2, therefore Kv = 01.2 = 5 sec⁻¹
Q.34Medium
The Nyquist stability criterion states that for stability, the Nyquist plot should not encircle:
Answer: B
The Nyquist criterion checks encirclements of the critical point (-1, 0) in the complex plane. No encirclement indicates stability for minimum phase systems.
Q.35Medium
A control system with gain margin of 6 dB means:
Answer: A
Gain margin in dB = 20log₁₀(GM). Therefore, 6 = 20log₁₀(GM), GM = 10^(206) ≈ 2
Q.36Medium
The settling time of a second-order underdamped system is approximately given by:
Answer: A
For 2% criteria, settling time ts ≈ 4/(ζωₙ). This is the standard formula for second-order systems.
Q.37Medium
In state-space representation, observability matrix rank should be equal to:
Answer: A
For a system to be completely observable, the observability matrix [C; CA; CA²; ...] must have rank equal to n (number of states).
Q.38Medium
A lead compensator with transfer function Gc(s) = (1 + aTs)/(1 + Ts) where a > 1 provides:
Answer: B
Lead compensator (a > 1) provides phase lead in mid-frequency range, improving transient response and system speed, hence used for transient improvement.
Q.39Medium
A system with transfer function H(s) = 100/(s² + 10s + 100) has natural frequency ωₙ and damping ratio ζ respectively as:
Answer: A
Standard form: ωₙ² = 100, so ωₙ = 10 rad/s. 2ζωₙ = 10, so ζ = 10/(2×10) = 0.5
Q.40Medium
Peak overshoot of underdamped second-order system depends on:
Answer: A
Peak overshoot Mp = e^(-πζ/√(1-ζ²)). It depends only on damping ratio ζ, not on ωₙ.
