Control system miscellaneous Easy Questions and Answers | Page - 29

Control system miscellaneous

By a suitable choice of the scalar parameter K, the system shown in the figure, can be made t o oscillate continuously at a fr equency of _____rad/s

1. 1
1. 4
1. 63
1. 16

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Characteristic equation is 1 + G (s) H (s) = 0
∴ s (s² + 10s + 16) + K = 0
⇒ jω (– ω2 + 10jω + 16) K = 0
For this system to oscillate, j terms should be set to zero.
∴ – jω³+ 16 jω = 0
⇒ ω = 4 r ad/ sec.
Then, – 10 ω² + K = 0
⇒ K = 160.

Correct Option: B

Characteristic equation is 1 + G (s) H (s) = 0
∴ s (s² + 10s + 16) + K = 0
⇒ jω (– ω2 + 10jω + 16) K = 0
For this system to oscillate, j terms should be set to zero.
∴ – jω³+ 16 jω = 0
⇒ ω = 4 r ad/ sec.
Then, – 10 ω² + K = 0
⇒ K = 160.

The state variable description of an LTI system is given by

where y is the output and u is the input. The system is controllable for

1. a₁ ≠ 0, a₂ = 0, a₃ ≠ 0
1. a₁ = 0, a₂ ≠ 0, a₃ ≠ 0
1. a₁ = 0, a₂ ≠ 0, a₃ = 0
1. a₁ ≠ 0, a₂ ≠ 0, a₃ = 0

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Q_c = [B AB A²B]

If rank of Q_c = 3 = order of matrix, then
Q_c is controllable , then |Q_c | ≠ 0

Correct Option: D

Q_c = [B AB A²B]

If rank of Q_c = 3 = order of matrix, then
Q_c is controllable , then |Q_c | ≠ 0

For the transfer function
G(s) H(s) = 1
s (s + 1)(s + 0.5)

the phase cross-over frequency is_____ rad/sec

1. 0.707
1. 707
1. 7.07
1. None of the above

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G(s) H(s) = 1
s (s + 0.5) (s + 1)

G(jω) H(jω) = 1
jω (jω + 0.5) (jω + 1)

φ = ∠ G (jω) H (jω)
= - 90° - tan^-1 2 ω - tan^-1 ω
At phase cross-over point, = – 180°
∴ tan^{– 1} ω + tan^{– 1} 2ω = 90°
⇒ tan^{– 1} 2ω² = 90°
1 - 2ω²

⇒ 2ω² = tan 90° = ∞
1 - 2ω²

∴ 1 - 2ω² = 0
⇒ ω = 1 = 0.707 rad /sec
√2

Correct Option: A

G(s) H(s) = 1
s (s + 0.5) (s + 1)

G(jω) H(jω) = 1
jω (jω + 0.5) (jω + 1)

φ = ∠ G (jω) H (jω)
= - 90° - tan^-1 2 ω - tan^-1 ω
At phase cross-over point, = – 180°
∴ tan^{– 1} ω + tan^{– 1} 2ω = 90°
⇒ tan^{– 1} 2ω² = 90°
1 - 2ω²

⇒ 2ω² = tan 90° = ∞
1 - 2ω²

∴ 1 - 2ω² = 0
⇒ ω = 1 = 0.707 rad /sec
√2

The gain margin of the transfer function
G(s) = 0.75
(s + 1)(s + 2)

will be _____dB

1. 2 dB
1. 12 dB
1. 16 dB
1. 32 dB

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GM = K T₁ T₂ ^-1
T₁ + T₂

Here, K = 0.75, T₁ = 1, T₂ = 0.5
∴ GM = 0.75 × 1 × 0.5 ^-1 = 2.4
1 + 0.5

∴ GM = 20 log a = 12 dB

Correct Option: B

GM = K T₁ T₂ ^-1
T₁ + T₂

Here, K = 0.75, T₁ = 1, T₂ = 0.5
∴ GM = 0.75 × 1 × 0.5 ^-1 = 2.4
1 + 0.5

∴ GM = 20 log a = 12 dB

Consider a unity feedback system having an open-loop transfer function
G(jω) = K
jω (j 0.2 ω + 1)(j 0.05 ω + 1)

For K = 1, GM = 28 dB. When GM at 20 dB, K will be equal to______

1. 0.25
1. 25
1. 5
1. 2.5

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For a GM of 20 db, the Nyquist plot should intersect the real axis at a,
where, 20 log 1 = 20
a

⇒ a = 0.1
This is achieved if the system gain is increased by a factor of 0.1 = 2.5
0.04

Thus K = 2.5

Correct Option: D

For a GM of 20 db, the Nyquist plot should intersect the real axis at a,
where, 20 log 1 = 20
a

⇒ a = 0.1
This is achieved if the system gain is increased by a factor of 0.1 = 2.5
0.04

Thus K = 2.5