The number of states in a third-order differential equation is

 Q236. The sampling frequency must be at least:

A) Equal to signal frequency
B) Twice the signal frequency
C) Half the signal frequency
D) One-fourth of signal frequency

✅ Answer: B) Twice the signal frequency
Explanation:
According to the Nyquist sampling theorem, $f_s ≥ 2f_{max}$ to avoid aliasing.

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Q237. A system is said to be controllable if:
A) Input cannot influence the state
B) Any state can be reached from any initial state
C) Output is always zero
D) Poles and zeros coincide

✅ Answer: B) Any state can be reached from any initial state
Explanation:
Controllability → possible to drive system to any desired state using input.

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Q238. Observability means:
A) All states are controllable
B) System poles are stable
C) Internal states can be determined from output
D) Output is zero

✅ Answer: C) Internal states can be determined from output
Explanation:
Observability → reconstruct internal states by observing output and input.

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Q239. In a second-order system, increasing damping ratio (ζ) results in:
A) Increased overshoot
B) Decreased overshoot
C) Increased oscillation
D) No effect

✅ Answer: B) Decreased overshoot
Explanation:
Higher damping → less oscillation, smaller overshoot.

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Q240. The steady-state error of a Type 1 system for a ramp input is:
A) Zero
B) Finite
C) Infinite
D) None

✅ Answer: B) Finite
Explanation:
Type 1 → one integrator → finite error for ramp.

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Q241. The phase margin is measured at frequency where:
A) |G(jω)H(jω)| = 1
B) ∠G(jω)H(jω) = 180°
C) |G(jω)H(jω)| = 0
D) ∠G(jω)H(jω) = 0°

✅ Answer: A) |G(jω)H(jω)| = 1
Explanation:
At gain crossover frequency (magnitude 0 dB), phase margin = 180° + phase.

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Q242. Increasing the gain in a closed-loop system generally:
A) Increases steady-state error
B) Reduces bandwidth
C) Reduces steady-state error
D) Increases damping

✅ Answer: C) Reduces steady-state error
Explanation:
Higher gain → more correction → smaller steady-state error (until instability).

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Q243. Integral control eliminates:
A) Overshoot
B) Steady-state error
C) Oscillations
D) Stability

✅ Answer: B) Steady-state error
Explanation:
Integrator accumulates error until zero steady-state error is reached.

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Q244. Nyquist plot of stable system should not encircle:
A) +1 point
B) -1 point
C) Origin
D) jω-axis

✅ Answer: B) -1 point
Explanation:
Nyquist criterion → encirclement of (-1,0) determines closed-loop stability.

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Q245. The phase crossover frequency is the frequency where:
A) |G(jω)H(jω)| = 1
B) ∠G(jω)H(jω) = -180°
C) Both A and B
D) None

✅ Answer: B) ∠G(jω)H(jω) = -180°
Explanation:
Phase crossover → phase = -180°, used for gain margin calculation.

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Q246. A notch filter is used to:
A) Attenuate all frequencies
B) Amplify DC
C) Reject a specific frequency
D) Integrate signal

✅ Answer: C) Reject a specific frequency
Explanation:
Notch filter removes narrow-band unwanted frequency (e.g., 50 Hz noise).

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Q247. Time constant of a first-order system equals:
A) Rise time
B) Time to reach 63.2% of final value
C) Half of settling time
D) Overshoot percentage

✅ Answer: B) Time to reach 63.2% of final value
Explanation:
By definition, $y(τ) = 0.632y_{final}$.

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Q248. Which one increases the order of the system?
A) Lead compensator
B) Lag compensator
C) Integral controller
D) Proportional controller

✅ Answer: C) Integral controller
Explanation:
Integrator adds a pole at origin → increases system order by 1.

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Q249. A proportional controller affects:
A) Transient response
B) Steady-state error
C) Both
D) Neither

✅ Answer: C) Both
Explanation:
Proportional gain affects speed (transient) and reduces steady-state error.

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Q250. State-space model of a system with n state variables is:
A) n equations with one variable
B) n first-order differential equations
C) One nth-order equation
D) None

✅ Answer: B) n first-order differential equations
Explanation:
State-space decomposes higher order system into n first-order equations.

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Q251. Feedback control improves system:
A) Sensitivity
B) Accuracy
C) Noise
D) Instability

✅ Answer: B) Accuracy
Explanation:
Negative feedback enhances accuracy by reducing error.

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Q252. The gain crossover frequency increases when:
A) Gain decreases
B) Gain increases
C) Poles move left
D) Zeros move right

✅ Answer: B) Gain increases
Explanation:
Higher gain shifts magnitude curve upward → crossover at higher frequency.

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Q253. The major advantage of state-space approach:
A) Frequency domain only
B) Time-domain analysis for MIMO systems
C) Applicable only to linear systems
D) None

✅ Answer: B) Time-domain analysis for MIMO systems
Explanation:
State-space handles multi-input multi-output systems efficiently.

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Q254. The Routh array helps in determining:
A) Damping ratio
B) Frequency response
C) Stability and number of RHP poles
D) Steady-state error

✅ Answer: C) Stability and number of RHP poles
Explanation:
Routh array → number of sign changes = number of unstable poles.

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Q255. If the Nyquist plot encircles (-1,0) twice in clockwise direction, system has:
A) 2 unstable poles
B) 2 stable poles
C) 2 encirclements → unstable
D) Depends on open-loop poles

✅ Answer: D) Depends on open-loop poles
Explanation:
Nyquist → $N = Z - P$, encirclement count + open-loop poles → closed-loop poles.

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Q256. ZOH (Zero Order Hold) in digital control holds signal:
A) Constant
B) Ramp
C) Sine
D) Zero

✅ Answer: A) Constant
Explanation:
ZOH holds each sample value constant until next sample arrives.

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Q257. Discretization converts:
A) Digital → analog
B) Continuous → discrete
C) Time-invariant → variant
D) None

✅ Answer: B) Continuous → discrete
Explanation:
Sampling and quantization transform continuous signals into discrete-time equivalents.

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Q258. Transfer function of pure integrator:
A) 1/s
B) s
C) 1 + s
D) 1/(1+s)

✅ Answer: A) 1/s
Explanation:
Integrator has one pole at origin.

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Q259. The output response of a stable system to bounded input is:
A) Unbounded
B) Bounded
C) Oscillatory infinite
D) Undefined

✅ Answer: B) Bounded
Explanation:
Definition of stability (BIBO) → bounded input → bounded output.

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Q260. A pole-zero cancellation in open-loop transfer function can:
A) Always stabilize
B) Sometimes cause hidden instability
C) Never affect stability
D) None

✅ Answer: B) Sometimes cause hidden instability
Explanation:
If unstable pole cancelled by unstable zero → internal instability.

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Q261. An overdamped system has:
A) ζ < 1
B) ζ = 1
C) ζ > 1
D) ζ = 0

✅ Answer: C) ζ > 1
Explanation:
Overdamping → slow response, no oscillation.

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Q262. Sensitivity function $S = \frac{1}{1+GH}$ shows:
A) System accuracy
B) System robustness
C) System sensitivity
D) All

✅ Answer: D) All
Explanation:
Lower |S| → less sensitivity to disturbances → more accurate and robust.

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Q263. In discrete-time domain, aliasing occurs due to:
A) Undersampling
B) Oversampling
C) Filtering
D) Feedback

✅ Answer: A) Undersampling
Explanation:
Sampling below Nyquist rate → overlapping frequency components (aliasing).

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Q264. In state feedback, eigenvalues of (A - BK) represent:
A) Open-loop poles
B) Closed-loop poles
C) Transmission zeros
D) Observer poles

✅ Answer: B) Closed-loop poles
Explanation:
State feedback modifies A matrix → moves closed-loop eigenvalues.

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Q265. Bode plot phase asymptote for a single pole starts changing at:
A) ω = 0
B) ω = ωₙ/10
C) ω = ωₙ
D) ω = 10ωₙ

✅ Answer: B) ω = ωₙ/10
Explanation:
Phase starts changing one decade before corner frequency.

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Q266. The final value theorem is valid only if:
A) Poles on imaginary axis
B) System stable (no poles in RHP)
C) System unstable
D) None

✅ Answer: B) System stable (no poles in RHP)
Explanation:
Otherwise final value diverges.

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Q267. A PID controller includes:
A) Proportional + Integral + Differential actions
B) Only proportional
C) Lag + Lead
D) None

✅ Answer: A) Proportional + Integral + Differential actions
Explanation:
PID combines advantages: fast, accurate, minimal error.

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Q268. The number of states in a third-order differential equation is:
A) 1
B) 2
C) 3
D) 4

✅ Answer: C) 3
Explanation:
Each derivative adds one state variable.

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Q269. An increase in phase margin makes the system:
A) More oscillatory
B) Less oscillatory
C) Unstable
D) Critically stable

✅ Answer: B) Less oscillatory
Explanation:
Higher phase margin → greater damping → smoother response.

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Q270. A pole near jω-axis indicates:
A) Fast response
B) Slow response
C) No effect
D) Oscillation

✅ Answer: B) Slow response
Explanation:
Poles near jω-axis → small negative real part → slower decay.