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Mathematical Modeling & LTI System Analysis(16)Stability Criteria, Compensators & Controllers(11)

Mathematical Modeling & LTI System Analysis

16 Questions

GATE→2026→Control Systems→Mathematical Modeling & LTI System Analysis

Q1

MCQ[1 Mark]

The step response Laplace transform is Y(s) = 100 / (s(s+100)). Find T_r, T_s (0.98 final value), and T_c.

(A)

T_r = 0.022, T_s = 0.04, T_c = 0.01

(B)

T_r = 0.22, T_s = 0.404, T_c = 0.01

(C)

T_r = 2.2, T_s = 4.04, T_c = 1.01

(D)

T_r = 22, T_s = 40.4, T_c = 10.1

Explanation

GATE→2026→Control Systems→Mathematical Modeling & LTI System Analysis

Q2

MCQ[2 Marks]

Given the state-space model with matrix A = [a, b; -a, 0], find a and b such that system poles are at -2 ± j3.

(A)

a = 4, b = 3.25

(B)

a = -4, b = 3.25

(C)

a = 4, b = -3.25

(D)

a = -4, b = -3.25

Explanation

GATE→2026→Control Systems→Mathematical Modeling & LTI System Analysis

Q3

MCQ[2 Marks]

Perform a state transformation using z matrix to find the new state-space representation.

(A)

z_dot = [-1 4; -1 -2]z + [4; -1]u, y = [2 3]z

(B)

z_dot = [2 3; 0 3]z + [3; 5]u, y = [2 3]z

(C)

z_dot = [4 9; -2 -3]z + [4; -1]u, y = [2 3]z

(D)

z_dot = [2 1; -4 1]z + [4; -1]u, y = [4 -1]z

Explanation

GATE→2025→Control Systems→Mathematical Modeling & LTI System Analysis

Q4

MCQ[1 Mark]

Given step response data points for a stable first-order LTI system, find the closest time constant in seconds.

(A)

1

(B)

2

(C)

3

(D)

4

Explanation

GATE→2025→Control Systems→Mathematical Modeling & LTI System Analysis

Q5

NAT[2 Marks]

Given state-space matrix A, find the sum of the magnitudes of the system poles.

Explanation

GATE→2024→Control Systems→Mathematical Modeling & LTI System Analysis

Q6

MCQ[1 Mark]

For the given block diagram, find the closed-loop transfer function C(s)/R(s).

(A)

G(s) / (1 + 2G(s))

(B)

-G(s) / (1 + 2G(s))

(C)

G(s) / (1 - 2G(s))

(D)

G(s) / (1 - 2G(s))

Explanation

GATE→2024→Control Systems→Mathematical Modeling & LTI System Analysis

Q7

MCQ[1 Mark]

Compare settling times of System 1 (omega_n = 3, angle theta = 60°) and System 2 (omega_n = 1, angle theta = 70°).

(A)

Settling time of System 1 is more than that of System 2.

(B)

Settling time of System 2 is more than that of System 1.

(C)

Settling times of both the systems are the same.

(D)

Settling time cannot be computed from the given information.

Explanation

GATE→2024→Control Systems→Mathematical Modeling & LTI System Analysis

Q8

MCQ[1 Mark]

For the cascaded transfer function (1/(s+1)) * ((s+40)/(s+20)), find a first-order pole-only approximation with matching steady-state step response.

(A)

1/(s+1)

(B)

2/(s+1)

(C)

1/(s+20)

(D)

2/(s+20)

Explanation

GATE→2024→Control Systems→Mathematical Modeling & LTI System Analysis

Q9

NAT[2 Marks]

For the stable closed-loop system with given Bode plot characteristics, find the steady-state error for a unit ramp input.

Explanation

GATE→2023→Control Systems→Mathematical Modeling & LTI System Analysis

Q10

MCQ[1 Mark]

For the given block diagram, find the closed-loop transfer function Y(s)/R(s).

(A)

(s + 1) / (3s + 2)

(B)

(3s + 2) / (s - 1)

(C)

(s + 1) / (3s + 2)

(D)

(3s + 2) / (s + 1)

Explanation

GATE→2023→Control Systems→Mathematical Modeling & LTI System Analysis

Q11

MCQ[1 Mark]

For a unity-gain feedback system with G(s) = 1/(s-1) and PI controller (Kp=3, Ki=1), find final values of controller output and plant output for unit step input.

(A)

(8, 0)

(B)

(1, 0)

(C)

(1, -1)

(D)

(-1, 1)

Explanation

GATE→2023→Control Systems→Mathematical Modeling & LTI System Analysis

Q12

MCQ[2 Marks]

From the given magnitude and phase plots of an LTI system, identify its transfer function.

(A)

2.51 e^{-0.032s} / (s + 1)

(B)

e^{-2.514s} / (s + 1)

(C)

1.04 e^{-2.514s} / (s + 1)

(D)

2.51 e^{-1.047s} / (s + 1)

Explanation

GATE→2023→Control Systems→Mathematical Modeling & LTI System Analysis

Q13

NAT[2 Marks]

For the given state-space system, find the input frequency omega > 0 for which the steady-state sinusoidal output is zero.

Explanation

GATE→2022→Control Systems→Mathematical Modeling & LTI System Analysis

Q14

MCQ[1 Mark]

A first-order stable system has flat Bode magnitude and high-frequency phase of -180°. What are its pole-zero features?

(A)

one LHP pole and one RHP zero at the same frequency.

(B)

one LHP pole and one LHP zero at the same frequency.

(C)

two LHP poles and one RHP zero.

(D)

two RHP poles and one LHP zero.

Explanation

GATE→2022→Control Systems→Mathematical Modeling & LTI System Analysis

Q15

MCQ[2 Marks]

For G(s) = 100/(s^2 + 0.1s + 10) and input r(t) = 1 + 0.1 sin(10t), find steady-state output parameters a and b.

(A)

a = 1, b = 10

(B)

a = 10, b = 1

(C)

a = 1, b = 100

(D)

a = 100, b = 1

Explanation

GATE→2022→Control Systems→Mathematical Modeling & LTI System Analysis

Q16

MCQ[2 Marks]

Find damping ratio zeta and natural frequency omega_n for the closed-loop system shown in the block diagram.

(A)

zeta = 0.5 and omega_n = 10 rad/s

(B)

zeta = 0.1 and omega_n = 10 rad/s

(C)

zeta = 0.707 and omega_n = 10 rad/s

(D)

zeta = 0.707 and omega_n = 100 rad/s

Explanation

Stability Criteria, Compensators & Controllers

11 Questions

GATE→2026→Control Systems→Stability Criteria, Compensators & Controllers

Q17

MCQ[1 Mark]

An asymptotic Bode magnitude plot has a slope of -20 dB/decade starting from omega_0. Which transfer function represents it?

(A)

G(s) = (1 + s/omega_0) / (s/omega_0)

(B)

G(s) = (s/omega_0) / (1 + s/omega_0)

(C)

G(s) = (1 + s/omega_0) / (1 - s/omega_0)

(D)

G(s) = (1 - s/omega_0) / (1 + s/omega_0)

Explanation

GATE→2025→Control Systems→Stability Criteria, Compensators & Controllers

Q18

MCQ[1 Mark]

The Nyquist plot of strictly stable G(s) with numerator (s-3) encircles -1 once anti-clockwise. The closed-loop system is

(A)

The system stability cannot be ascertained.

(B)

The system is marginally stable.

(C)

The system is stable.

(D)

The system is unstable.

Explanation

GATE→2025→Control Systems→Stability Criteria, Compensators & Controllers

Q19

MSQ[1 Mark]

For G(s) = Ks(s+2)/((s+5)(s+7)), which real axis points are on the root locus for K >= 0?

(A)

-1

(B)

-4

(C)

-6

(D)

-10

Explanation

GATE→2025→Control Systems→Stability Criteria, Compensators & Controllers

Q20

MCQ[2 Marks]

For G(s) = 1/((s+1)(s+2)) and forward/feedback transfer function K(s-1)G(s), determine closed-loop stability for K.

(A)

stable for all K > 2

(B)

unstable for all K > 2

(C)

unstable for all K > 1

(D)

stable for all K > 1

Explanation

GATE→2025→Control Systems→Stability Criteria, Compensators & Controllers

Q21

NAT[2 Marks]

For plant G(s) and controller D(s) = 1 + K_D s, find K_D to yield 45° phase margin at gain crossover frequency omega_gc = 10 rad/s.

Explanation

GATE→2024→Control Systems→Stability Criteria, Compensators & Controllers

Q22

NAT[2 Marks]

Find the angle of departure of the root locus branch from the complex pole (-1 + j2) for G(s) = K(s^2 - 2s + 2)/(s^2 + 2s + 5).

Explanation

GATE→2024→Control Systems→Stability Criteria, Compensators & Controllers

Q23

NAT[2 Marks]

Using the tabulated frequency response of G(s), find the integral gain K_I for a phase margin of 50°.

Explanation

GATE→2023→Control Systems→Stability Criteria, Compensators & Controllers

Q24

MCQ[1 Mark]

For G(s)H(s) = (3s+5)/(s-1), where does the infinite semi-circular arc of the Nyquist contour map in the G(s)H(s) plane?

(A)

G(s)H(s) = infinity

(B)

G(s)H(s) = 0

(C)

G(s)H(s) = 3

(D)

G(s)H(s) = -5

Explanation

GATE→2023→Control Systems→Stability Criteria, Compensators & Controllers

Q25

MCQ[2 Marks]

A lead compensator produces maximum phase lead at omega_m = 4 rad/s with a gain of 6 dB at that frequency. Find parameters a and beta.

(A)

(1, 16)

(B)

(2, 4)

(C)

(3, 5)

(D)

(2.66, 2.25)

Explanation

GATE→2022→Control Systems→Stability Criteria, Compensators & Controllers

Q26

MCQ[1 Mark]

For G(s) = k / (s^2 + 4s - 5) in a unity feedback system, find the range of k for closed-loop stability.

(A)

k > 3

(B)

k < 3

(C)

k > 5

(D)

k < 5

Explanation

GATE→2022→Control Systems→Stability Criteria, Compensators & Controllers

Q27

MCQ[2 Marks]

For G(s) = 1/(s(s+1)) with the Nyquist contour indenting around the origin, find the number of encirclements N of (-1+j0).

(A)

0

(B)

1

(C)

2

(D)

3

Explanation