The centroid of asymptotes is given by

 

11. The open-loop transfer function of a unity feedback system is $G(s) = \frac{K}{s(s+3)}$. The system is stable if:

a) K < 0
b) K > 0
c) K = 0
d) K < 10
Answer: b) K > 0
Explanation: For unity feedback with poles on the left half-plane, positive gain ensures stability.

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12. A proportional controller increases:

a) Rise time
b) Overshoot
c) Settling time
d) Steady-state error
Answer: b) Overshoot
Explanation: Increasing proportional gain reduces rise time but increases overshoot.

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13. Derivative control action improves:

a) Steady-state response
b) Transient response
c) Both steady and transient response
d) Frequency response only
Answer: b) Transient response
Explanation: Derivative control reduces overshoot and improves damping.

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14. Integral control primarily reduces:

a) Rise time
b) Overshoot
c) Steady-state error
d) Stability
Answer: c) Steady-state error
Explanation: Integral action continuously adds up the error and forces it toward zero.

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15. The root locus starts from:

a) Zeros
b) Poles
c) Imaginary axis
d) Origin only
Answer: b) Poles
Explanation: Root locus plots start from the open-loop poles and end at the zeros.

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16. The condition for a system to be marginally stable is:

a) All poles on right half
b) Poles on imaginary axis
c) One pole at origin
d) All poles on left half
Answer: b) Poles on imaginary axis
Explanation: Marginal stability occurs when poles lie on the imaginary axis (s = ±jω).

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17. A system with transfer function $G(s)H(s) = \frac{10}{s(s+2)(s+5)}$ has type:

a) Type 0
b) Type 1
c) Type 2
d) Type -1
Answer: b) Type 1
Explanation: One factor of ‘s’ in denominator indicates one integrator → Type 1.

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18. Bode plot is a plot of:

a) Frequency vs Gain
b) Gain and Phase vs Log frequency
c) Gain vs Time
d) None
Answer: b) Gain and Phase vs Log frequency
Explanation: Bode plots show magnitude and phase with logarithmic frequency scale.

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19. Nyquist criterion is used to determine:

a) Gain margin
b) Phase margin
c) Stability of feedback systems
d) Frequency response
Answer: c) Stability of feedback systems
Explanation: Nyquist criterion predicts closed-loop stability from open-loop frequency response.

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20. The addition of a zero to a system:

a) Increases overshoot
b) Decreases speed
c) Reduces damping
d) Makes system sluggish
Answer: a) Increases overshoot
Explanation: A zero adds a phase lead, reducing damping, hence increasing overshoot.

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21. The damping ratio (ζ) of a second-order system affects mainly:

a) Steady-state error
b) Rise time
c) Overshoot
d) Gain margin
Answer: c) Overshoot
Explanation: Damping ratio controls oscillation level, thus affecting overshoot.

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22. A phase lead compensator is used to:

a) Increase phase lag
b) Improve phase margin
c) Reduce gain margin
d) Increase steady-state error
Answer: b) Improve phase margin
Explanation: Lead compensator adds phase lead and increases system stability.

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23. In a unity feedback system, if open-loop gain K increases, steady-state error for step input:

a) Increases
b) Decreases
c) Remains same
d) Becomes infinite
Answer: b) Decreases
Explanation: Higher gain increases sensitivity to input, reducing steady-state error.

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24. The Routh-Hurwitz criterion is used to determine:

a) Relative stability
b) Frequency response
c) Phase margin
d) Steady-state error
Answer: a) Relative stability
Explanation: Routh criterion determines the number of poles in right-half s-plane (stability).

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25. In state-space representation, the number of state variables equals:

a) Number of poles
b) Number of zeros
c) Number of feedback loops
d) None
Answer: a) Number of poles
Explanation: Each independent energy storage element represents one pole and one state variable.

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26. Controllability of a system means:

a) Output can be controlled
b) State can be driven from any initial to final value
c) Input can be varied
d) Output is measurable
Answer: b) State can be driven from any initial to final value
Explanation: A system is controllable if proper input can move it anywhere in state space.

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27. Observability of a system means:

a) Internal states can be inferred from outputs
b) Input can be varied
c) Output can be made zero
d) System is stable
Answer: a) Internal states can be inferred from outputs
Explanation: Observability ensures that internal states can be determined from output measurements.

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28. The number of asymptotes in a root locus equals:

a) Number of poles
b) Number of zeros
c) Difference between poles and zeros
d) Sum of poles and zeros
Answer: c) Difference between poles and zeros
Explanation: Asymptotes guide the root locus to infinity where no zeros exist.

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29. The centroid of asymptotes is given by:

a) $(\sum \text{poles} - \sum \text{zeros})/n$
b) $(\sum \text{zeros} - \sum \text{poles})/n$
c) $(\sum \text{poles} + \sum \text{zeros})/n$
d) None
Answer: a) $(\sum \text{poles} - \sum \text{zeros})/n$
Explanation: The centroid is computed from pole-zero locations to locate asymptote center.

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30. The Nyquist plot encircles the point (-1, 0) in the complex plane to indicate:

a) Unstable system
b) Stable system
c) Marginal stability
d) Steady-state error
Answer: a) Unstable system
Explanation: Encircling (-1,0) corresponds to closed-loop poles in the right half-plane.