State-space Analysis - Control Systems - Electrical Engineering Previous Year Questions

1

2009 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2009

The system transfer function is

A

\(\frac{s + 2}{s^2 + 5s - 6}\)

B

\(\frac{s + 3}{s^2 + 5s + 6}\)

C

\(\frac{2s + 5}{s^2 + 5s + 6}\)

D

\(\frac{2s - 5}{s^2 + 5s - 6}\)

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2

2009 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2009

The state-transition matrix of the above system is

A

\(\begin{bmatrix} e^{-3t} & 0 \\ e^{-2t} + e^{-3t} & e^{-2t} \end{bmatrix}\)

B

\(\begin{bmatrix} e^{-3t} & e^{-2t} - e^{-3t} \\ 0 & e^{-2t} \end{bmatrix}\)

C

\(\begin{bmatrix} e^{-3t} & e^{-2t} + e^{-3t} \\ 0 & e^{-2t} \end{bmatrix}\)

D

\(\begin{bmatrix} e^{3t} & e^{-2t} - e^{-3t} \\ 0 & e^{-2t} \end{bmatrix}\)

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3

2010 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2010

The system \(\dot{x} = Ax + Bu\) with \(A = \begin{bmatrix} -1 & 2 \\ 0 & 2 \end{bmatrix}, \; B = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\) is

A

stable and controllable

B

stable but uncontrollable

C

unstable but controllable

D

unstable and uncontrollable

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4

2012 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2012

The state variable description of an LTI system is given by (see image for state equations) where y is the output and u is the input. The system is controllable for

A

a₁ ≠ 0, a₂ = 0, a₃ ≠ 0

B

a₁ = 0, a₂ ≠ 0, a₃ ≠ 0

C

a₁ = 0, a₂ ≠ 0, a₃ = 0

D

a₁ ≠ 0, a₂ ≠ 0, a₃ = 0

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5

2013 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2013 [Session 1]

The system is

A

controllable but not observable

B

not controllable but observable

C

both controllable and observable

D

both not controllable and not observable

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6

2013 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2013 [Session 2]

Common Data for Questions 48 and 49: The state variable formulation of a system is given as [x₁̇; x₂̇] = [-2, 0; 0, -1] [x₁; x₂] + [1; 1] u, x₁(0) = 0, x₂(0) = 0 and y = [1, 0] [x₁; x₂]. The system is

A

controllable but not observable

B

not controllable but observable

C

both controllable and observable

D

both not controllable and not observable

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7

2014 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2014 [Session 2]

The state transition matrix for the system \(\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} 1 \\ 1 \end{bmatrix} u\) is

A

\(\begin{bmatrix} e^t & 0 \\ e^t & e^t \end{bmatrix}\)

B

\(\begin{bmatrix} e^t & 0 \\ t e^t & e^t \end{bmatrix}\)

C

\(\begin{bmatrix} e^t & 0 \\ t e^t & 0 \end{bmatrix}\)

D

\(\begin{bmatrix} e^t & t e^t \\ 0 & e^t \end{bmatrix}\)

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8

2014 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2014 [Session 2]

The second order dynamic system \( \frac{dX}{dt} = PX + Qu \), \( y = RX \) has the matrices P, Q and R as follows: \( P = \begin{bmatrix} -1 & 1 \\ 0 & -3 \end{bmatrix} \), \( Q = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \), \( R = \begin{bmatrix} 0 & 1 \end{bmatrix} \). The system has the following controllability and observability properties:

A

Controllable and observable

B

Not controllable but observable

C

Controllable but not observable

D

Not controllable and not observable

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9

2016 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2016 [Session 1]

Consider the following state-space representation of a linear time-invariant system.
\(\dot{\boldsymbol{x}}(t) = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} \boldsymbol{x}(t), \quad y(t) = \boldsymbol{c}^T \boldsymbol{x}(t), \quad \boldsymbol{c} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \text{ and } \boldsymbol{x}(0) = \begin{bmatrix} 1 \\ 1 \end{bmatrix}\)
The value of \(y(t)\) for \(t = \log_e 2\) is ________.

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10

2016 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2016 [Session 2]

Consider a linear time invariant system \(\dot{x} = Ax\), with initial condition \(x(0)\) at \(t = 0\). Suppose \(\alpha\) and \(\beta\) are eigenvectors of (2 x 2) matrix A corresponding to distinct eigenvalues \(\lambda_1\) and \(\lambda_2\) respectively. Then the response \(x(t)\) of the system due to initial condition \(x(0) = \alpha\) is

A

\(e^{\lambda_1 t} \alpha\)

B

\(e^{\lambda_2 t} \beta\)

C

\(e^{\lambda_1 t} \beta\)

D

\(e^{\lambda_1 t} \alpha + e^{\lambda_2 t} \beta\)

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11

2017 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2017 [Session 1]

The transfer function of the system Y(s)/U(s) whose state-space equations are given below is:
\[ \begin{bmatrix} \dot{x}_1(t) \\ \dot{x}_2(t) \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 2 & 0 \end{bmatrix} \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} + \begin{bmatrix} 1 \\ 2 \end{bmatrix} u(t) \]
\[ y(t) = [1 \; 0] \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} \]

A

\(\frac{s+2}{s^2-2s-2} \)

B

\(\frac{s-2}{s^2+s-4} \)

C

\(\frac{s-4}{s^2+s-4} \)

D

\(\frac{s+4}{s^2-s-4} \)

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12

2017 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2017 [Session 2]

Consider the system described by the following state space representation \[\begin{bmatrix} \dot{x}_1(t) \\ \dot{x}_2(t) \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & -2 \end{bmatrix} \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} u(t)\] \[y(t) = \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix}\] If \(u(t)\) is a unit step input and \(\begin{bmatrix} x_1(0) \\ x_2(0) \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\), the value of output \(y(t)\) at \(t = 1\) sec (rounded off to three decimal places) is __________.

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13

2018 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2018

Consider a system governed by the following equations \(\frac{dx_1(t)}{dt} = x_2(t) - x_1(t)\) \(\frac{dx_2(t)}{dt} = x_1(t) - x_2(t)\) The initial conditions are such that \(x_1(0) < x_2(0) < \infty\). Let \(x_{1f} = \lim_{t\to\infty} x_1(t)\) and \(x_{2f} = \lim_{t\to\infty} x_2(t)\). Which one of the following is true?

A

\(x_{1f} < x_{2f} < \infty\)

B

\(x_{2f} < x_{1f} < \infty\)

C

\(x_{1f} = x_{2f} < \infty\)

D

\(x_{1f} = x_{2f} = \infty\)

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14

2019 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2019

Consider a state-variable model of a system \(\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -\alpha & -2\beta \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} 0 \\ \alpha \end{bmatrix} r\) and \(y = \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\), where \(y\) is the output, and \(r\) is the input. The damping ratio \(\xi\) and the undamped natural frequency \(\omega_n\) (rad/sec) of the system are given by

A

\(\xi = \frac{\beta}{\sqrt{\alpha}} ; \omega_n = \sqrt{\alpha}\)

B

\(\xi = \sqrt{\alpha} ; \omega_n = \frac{\beta}{\sqrt{\alpha}}\)

C

\(\xi = \frac{\alpha}{\beta} ; \omega_n = \sqrt{\beta}\)

D

\(\xi = \sqrt{\beta} ; \omega_n = \frac{\alpha}{\sqrt{\beta}}\)

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15

2021 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2021

The state space representation of a first-order system is given as $\dot{x} = -x + u$ $y = x$ where, $x$ is the state variable, $u$ is the control input and $y$ is the controlled output. Let $u = -Kx$ be the control law, where K is the controller gain. To place a closed-loop pole at -2, the value of K is ______.

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16

2026 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2026

A system is characterized by the following state equation and output equation (\(u\): input, \(\mathbf{x}\): state vector, \(y\): output) \[\dot{\mathbf{x}} = \begin{bmatrix} a & b \\ -a & 0 \end{bmatrix}\mathbf{x} + \begin{bmatrix} 1 \\ 0 \end{bmatrix}u\] \[y = \begin{bmatrix} 1 & 2 \end{bmatrix}\mathbf{x}\] What are the values of \(a\) and \(b\) for which the poles of the transfer function are at \(-2 + j3\) and \(-2 - j3\)?

A

\(a=4, b=3.25\)

B

\(a=-4, b=3.25\)

C

\(a=4, b=-3.25\)

D

\(a=-4, b=-3.25\)

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17

2026 · Electrical Engineering · Control Systems · State-space Analysis

Electrical Engineering (EE) 2026

A system is represented in state-space form as follows:
(u: input, x: state vector, y: output) \[ \[\dot{\mathbf{x}} = \begin{bmatrix} 1 & 2 \\ -3 & 0 \end{bmatrix} \mathbf{x} + \begin{bmatrix} 1 \\ 2 \end{bmatrix} u \\\] \[y = \begin{bmatrix} 1 & 2 \end{bmatrix} \mathbf{x}\] \] Consider the new state vector \(\mathbf{z} = \begin{bmatrix} 2 & 1 \\ -1 & 0 \end{bmatrix} \mathbf{x}\)
What is the state-space representation of the system in terms of the new state vector z ?

A

\(\dot{\mathbf{z}} = \begin{bmatrix} -1 & 4 \\ -1 & -2 \end{bmatrix} \mathbf{z} + \begin{bmatrix} 4 \\ -1 \end{bmatrix} u\)
\(y = \begin{bmatrix} 2 & 3 \end{bmatrix} \mathbf{z}\)

B

\(\dot{\mathbf{z}} = \begin{bmatrix} 2 & 3 \\ 0 & 3 \end{bmatrix} \mathbf{z} + \begin{bmatrix} 3 \\ 5 \end{bmatrix} u\)
\(y = \begin{bmatrix} 2 & 3 \end{bmatrix} \mathbf{z}\)

C

\(\dot{\mathbf{z}} = \begin{bmatrix} 4 & 9 \\ -2 & -3 \end{bmatrix} \mathbf{z} + \begin{bmatrix} 4 \\ -1 \end{bmatrix} u\)
\(y = \begin{bmatrix} 2 & 3 \end{bmatrix} \mathbf{z}\)

D

\(\dot{\mathbf{z}} = \begin{bmatrix} 2 & 1 \\ -4 & 1 \end{bmatrix} \mathbf{z} + \begin{bmatrix} 4 \\ -1 \end{bmatrix} u\)
\(y = \begin{bmatrix} 4 & -1 \end{bmatrix} \mathbf{z}\)

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