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Control System Components, Transfer Functions, and Stability Analysis(16)Frequency Response, Compensators, Controllers, and State Variable Models(11)

Control System Components, Transfer Functions, and Stability Analysis

16 Questions

GATE→2026→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q1

MCQ[1 Mark]

A control system is shown in the Figure. Which option represents the correct transfer function of the system?

(A)

$\frac{1}{(s+4)^{2}}$

(B)

$\frac{1}{(s+4)}$

(C)

$\frac{2}{(s+4)}$

(D)

$\frac{1}{(s^{2}+8s+17)}$

Explanation

GATE→2026→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q2

NAT[2 Marks]

Consider the unity negative feedback control system shown in the Figure. The value of gain $K(>0)$ at which the given system will remain marginally stable is ______ (Answer in integer)

Explanation

GATE→2025→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q3

MCQ[1 Mark]

The Nyquist plot of a system is given in the figure below. Let $\omega_P$, $\omega_Q$, $\omega_R$, and $\omega_S$ be the positive frequencies at the points P, Q, R, and S, respectively. Which one of the following statements is TRUE?

(A)

$\omega_S$ is the gain crossover frequency and $\omega_P$ is the phase crossover frequency

(B)

$\omega_Q$ is the gain crossover frequency and $\omega_R$ is the phase crossover frequency

(C)

$\omega_Q$ is the gain crossover frequency and $\omega_S$ is the phase crossover frequency

(D)

$\omega_S$ is the gain crossover frequency and $\omega_Q$ is the phase crossover frequency

Explanation

GATE→2025→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q4

MSQ[1 Mark]

Consider the unity-negative-feedback system shown in Figure (i) below, where gain $K \ge 0$. The root locus of this system is shown in Figure (ii) below. For what value(s) of K will the system in Figure (i) have a pole at $-1+j1$?

(A)

$K = 5$

(B)

$K = \frac{1}{5}$

(C)

For no positive value of K

(D)

For all positive values of K

Explanation

GATE→2025→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q5

MCQ[2 Marks]

Consider the polynomial $p(s) = s^5 + 7s^4 + 3s^3 - 33s^2 + 2s - 40$. Let $(L, I, R)$ be defined as follows: L is the number of roots of $p(s)$ with negative real parts. I is the number of roots of $p(s)$ that are purely imaginary. R is the number of roots of $p(s)$ with positive real parts. Which one of the following options is correct?

(A)

$L = 2$, $I = 2$, and $R = 1$

(B)

$L = 3$, $I = 2$, and $R = 0$

(C)

$L = 1$, $I = 2$, and $R = 2$

(D)

$L = 0$, $I = 4$, and $R = 1$

Explanation

GATE→2025→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q6

MSQ[2 Marks]

Consider a system represented by the block diagram shown below. Which of the following signal flow graphs represent(s) this system? Choose the correct option(s).

(A)

(A)

(B)

(B)

(C)

(C)

(D)

(D)

Explanation

GATE→2024→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q7

MCQ[1 Mark]

In the feedback control system shown in the figure below $G(s) = \frac{6}{s(s+1)(s+2)}$. If the input $r(t)$ is a unit step function, then

(A)

$\lim_{t\to\infty} e(t) = 0$

(B)

$\lim_{t\to\infty} e(t) = \frac{1}{3}$

(C)

$\lim_{t\to\infty} e(t) = \frac{1}{4}$

(D)

$\lim_{t\to\infty} e(t)$ does not exist, $e(t)$ is oscillatory

Explanation

GATE→2024→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q8

MCQ[2 Marks]

Consider a unity negative feedback control system with forward path gain $G(s) = \frac{K}{(s+1)(s+2)(s+3)}$. The impulse response of the closed-loop system decays faster than $e^{-t}$ if

(A)

$1 \le K \le 5$

(B)

$7 \le K \le 21$

(C)

$-4 \le K \le -1$

(D)

$-24 \le K \le -6$

Explanation

GATE→2023→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q9

MCQ[1 Mark]

The open loop transfer function of a unity negative feedback system is $G(s) = \frac{k}{s(1+sT_1)(1+sT_2)}$, where k, $T_1$ and $T_2$ are positive constants. The phase cross-over frequency, in rad/s, is

(A)

$\frac{1}{\sqrt{T_1 T_2}}$

(B)

$\frac{1}{T_1 T_2}$

(C)

$\frac{1}{T_1 \sqrt{T_2}}$

(D)

$\frac{1}{T_2 \sqrt{T_1}}$

Explanation

GATE→2023→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q10

MCQ[2 Marks]

Consider a unity negative feedback control system with forward path gain $G(s) = \frac{K}{(s+1)(s+2)(s+3)}$ as shown. The impulse response of the closed-loop system decays faster than $e^{-t}$ if

(A)

$1 \le K \le 5$

(B)

$7 \le K \le 21$

(C)

$-4 \le K \le -1$

(D)

$-24 \le K \le -6$

Explanation

GATE→2023→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q11

MCQ[2 Marks]

A closed loop system is shown in the figure where $k > 0$ and $\alpha > 0$. The steady state error due to a ramp input ($R(s) = \alpha/s^2$) is given by

(A)

$\frac{2\alpha}{k}$

(B)

$\frac{\alpha}{k}$

(C)

$\frac{\alpha}{2k}$

(D)

$\frac{\alpha}{4k}$

Explanation

GATE→2023→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q12

MCQ[2 Marks]

In the following block diagram, $R(s)$ and $D(s)$ are two inputs. The output $Y(s)$ is expressed as $Y(s) = G_1(s)R(s) + G_2(s)D(s)$. $G_1(s)$ and $G_2(s)$ are given by

(A)

Option A

(B)

Option B

(C)

Option C

(D)

Option D

Explanation

GATE→2022→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q13

MCQ[1 Mark]

Consider a closed-loop control system with unity negative feedback and $KG(s)$ in the forward path, where the gain $K=2$. The complete Nyquist plot of the transfer function $G(s)$ is shown in the figure... The number of poles of the closed-loop transfer function in the closed right-half of the complex plane is

(A)

0

(B)

1

(C)

2

(D)

3

Explanation

GATE→2022→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q14

MCQ[2 Marks]

Consider an even polynomial $p(s)$ given by $p(s) = s^4 + 5s^2 + 4 + K$, where K is an unknown real parameter. The complete range of K for which $p(s)$ has all its roots on the imaginary axis is

(A)

$-4 \le K \le \frac{9}{4}$

(B)

$-3 \le K \le \frac{9}{2}$

(C)

$-6 \le K \le \frac{5}{4}$

(D)

$-5 \le K \le 0$

Explanation

GATE→2022→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q15

MSQ[2 Marks]

Two linear time-invariant systems with transfer functions $G_1(s) = \frac{10}{s^2 + s + 1}$ and $G_2(s) = \frac{10}{s^2 + s\sqrt{10} + 10}$ have unit step responses $y_1(t)$ and $y_2(t)$, respectively. Which of the following statements is/are true?

(A)

$y_1(t)$ and $y_2(t)$ have the same percentage peak overshoot.

(B)

$y_1(t)$ and $y_2(t)$ have the same steady-state value.

(C)

$y_1(t)$ and $y_2(t)$ have the same damped frequency of oscillation.

(D)

$y_1(t)$ and $y_2(t)$ have the same 2% settling time.

Explanation

GATE→2022→Control Systems→Control System Components, Transfer Functions, and Stability Analysis

Q16

NAT[2 Marks]

The block diagram of a closed-loop control system is shown in the figure... Assuming the reference input $r(t) = 0$ for all t, the steady-state error $e(\infty)$, due to a unit step disturbance $d(t)$ is _______ (rounded off to two decimal places).

Explanation

Frequency Response, Compensators, Controllers, and State Variable Models

11 Questions

GATE→2026→Control Systems→Frequency Response, Compensators, Controllers, and State Variable Models

Q17

MCQ[2 Marks]

For the control system shown in the Figure, the transfer function of a plant, $G(s)=\frac{1}{(s+1)(s+2)}$ is connected in cascade with a compensator $C(s)=K(s+\alpha)$, where K and a are positive real valued constants. Which of the following pairs $(K,\alpha)$ represent the correct values for the closed loop system to have poles at $(-3\pm j\sqrt{5})$?

(A)

2,3

(B)

3,4

(C)

2,4

(D)

3,3

Explanation

GATE→2026→Control Systems→Frequency Response, Compensators, Controllers, and State Variable Models

Q18

MCQ[2 Marks]

The state and output equations for a control system are: $\dot{x}=[\begin{matrix}-4&-1.5\\ 4&0\end{matrix}]x+[\begin{matrix}2\\ 0\end{matrix}]u$ $y=[\begin{matrix}1.5&0.625\end{matrix}]x$ Which of the following expressions correctly represents the transfer function $\frac{Y(s)}{U(s)}$ of the system with zero initial conditions?

(A)

$\frac{3s}{s^{2}+4s-6}$

(B)

$\frac{3s+5}{s^{2}+4s-6}$

(C)

$\frac{3s+5}{s^{2}+4s+6}$

(D)

$\frac{3s}{s^{2}+4s+6}$

Explanation

GATE→2025→Control Systems→Frequency Response, Compensators, Controllers, and State Variable Models

Q19

MCQ[2 Marks]

Let $G(s) = \frac{1}{10s^2}$ be the transfer function of a second-order system. A controller $M(s)$ is connected to the system $G(s)$ in the configuration shown below. Consider the following statements. (i) There exists no controller of the form $M(s) = \frac{K_I}{s}$, where $K_I$ is a positive real number, such that the closed loop system is stable. (ii) There exists at least one controller of the form $M(s) = K_P + s K_D$, where $K_P$ and $K_D$ are positive real numbers, such that the closed loop system is stable. Which one of the following options is correct?

(A)

(i) is TRUE and (ii) is FALSE

(B)

(i) is FALSE and (ii) is TRUE

(C)

Both (i) and (ii) are FALSE

(D)

Both (i) and (ii) are TRUE

Explanation

GATE→2025→Control Systems→Frequency Response, Compensators, Controllers, and State Variable Models

Q20

MSQ[2 Marks]

Consider a system where $x_1(t)$, $x_2(t)$, and $x_3(t)$ are three internal state signals and $u(t)$ is the input signal. The differential equations governing the system are given by $\frac{d}{dt} \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} = \begin{bmatrix} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} + \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} u(t)$. Which of the following statements is/are TRUE?

(A)

The signals $x_1(t)$, $x_2(t)$ and $x_3(t)$ are bounded for all bounded inputs

(B)

There exists a bounded input such that at least one of the signals $x_1(t)$, $x_2(t)$ and $x_3(t)$ is unbounded

(C)

There exists a bounded input such that the signals $x_1(t)$, $x_2(t)$, and $x_3(t)$ are unbounded

(D)

The signals $x_1(t)$, $x_2(t)$ and $x_3(t)$ are unbounded for all bounded inputs

Explanation

GATE→2024→Control Systems→Frequency Response, Compensators, Controllers, and State Variable Models

Q21

MCQ[1 Mark]

In the context of Bode magnitude plots, 40 dB/decade is the same as ______

(A)

12 dB/octave

(B)

6 dB/octave

(C)

20 dB/octave

(D)

10 dB/octave

Explanation

GATE→2024→Control Systems→Frequency Response, Compensators, Controllers, and State Variable Models

Q22

MCQ[2 Marks]

A satellite attitude control system... has a plant with transfer function $G(s) = \frac{1}{s^2}$ cascaded with a compensator $C(s) = \frac{K(s+\alpha)}{s+4}$ where K and $\alpha$ are positive real constants. In order for the closed-loop system to have poles at $-1 \pm j\sqrt{3}$, the value of $\alpha$ must be

(A)

0

(B)

1

(C)

2

(D)

3

Explanation

GATE→2024→Control Systems→Frequency Response, Compensators, Controllers, and State Variable Models

Q23

MSQ[2 Marks]

Consider a system S represented in state space as given. Which of the state space representations given below has/have the same transfer function as that of S?

(A)

$\frac{dx}{dt} = \begin{bmatrix} 0 & 1 \\ -2 & -3 \end{bmatrix} x + \begin{bmatrix} 0 \\ 1 \end{bmatrix} r$, $y = \begin{bmatrix} 1 & 2 \end{bmatrix} x$

(B)

Option B

(C)

$\frac{dx}{dt} = \begin{bmatrix} -1 & 0 \\ 0 & -2 \end{bmatrix} x + \begin{bmatrix} -1 \\ 3 \end{bmatrix} r$, $y = \begin{bmatrix} 1 & 1 \end{bmatrix} x$

(D)

Option D

Explanation

GATE→2023→Control Systems→Frequency Response, Compensators, Controllers, and State Variable Models

Q24

MCQ[2 Marks]

A satellite attitude control system... has a plant with transfer function $G(s) = \frac{1}{s^2}$ cascaded with a compensator $C(s) = \frac{K(s+\alpha)}{s+4}$ where K and $\alpha$ are positive real constants. In order for the closed-loop system to have poles at $-1 \pm j\sqrt{3}$, the value of $\alpha$ must be

(A)

0

(B)

1

(C)

2

(D)

3

Explanation

GATE→2023→Control Systems→Frequency Response, Compensators, Controllers, and State Variable Models

Q25

MCQ[2 Marks]

The state equation of a second order system is $\dot{x}(t) = A x(t)$, $x(0)$ is the initial condition. Suppose $\lambda_1$ and $\lambda_2$ are two distinct eigenvalues of A and $v_1$ and $v_2$ are the corresponding eigenvectors. For constants $\alpha_1$ and $\alpha_2$, the solution, $x(t)$, of the state equation is

(A)

$\sum_{i=1}^2 \alpha_i e^{\lambda_i t} v_i$

(B)

$\sum_{i=1}^2 \alpha_i e^{2\lambda_i t} v_i$

(C)

$\sum_{i=1}^2 \alpha_i e^{3\lambda_i t} v_i$

(D)

$\sum_{i=1}^2 \alpha_i e^{4\lambda_i t} v_i$

Explanation

GATE→2023→Control Systems→Frequency Response, Compensators, Controllers, and State Variable Models

Q26

NAT[2 Marks]

The asymptotic magnitude Bode plot of a minimum phase system is shown in the figure. The transfer function of the system is $G(s) = \frac{k(s+z)^a}{s^b(s+p)^c}$, where k, z, p, a, b and c are positive constants. The value of $(a+b+c)$ is _______ (rounded off to the nearest integer).

Explanation

GATE→2022→Control Systems→Frequency Response, Compensators, Controllers, and State Variable Models

Q27

MCQ[1 Mark]

The root-locus plot of a closed-loop system with unity negative feedback and transfer function $KG(s)$ in the forward path is shown in the figure... Select the transfer function $G(s)$ that results in the root-locus plot of the closed-loop system as shown in the figure.

(A)

$G(s) = \frac{1}{(s+1)^5}$

(B)

$G(s) = \frac{1}{s^5+1}$

(C)

$G(s) = \frac{s-1}{(s+1)^6}$

(D)

$G(s) = \frac{s+1}{s^6+1}$

Explanation