GATE ECE 2025 | State Space Analysis Question 1 | Control Systems

GATE ECE 2025

MCQ (Single Correct Answer)

-0.67

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

$$ \frac{d}{d t}\left[\begin{array}{l} x_1(t) \\ x_2(t) \\ x_3(t) \end{array}\right]=\left[\begin{array}{ccc} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{array}\right]\left[\begin{array}{l} x_1(t) \\ x_2(t) \\ x_3(t) \end{array}\right]+\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right] u(t) $$

Which of the following statements is/are TRUE?

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

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.

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

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

GATE ECE 2024

MCQ (More than One Correct Answer)

-0

Consider a system $S$ represented in state space as

$$\frac{dx}{dt} = \begin{bmatrix} 0 & -2 \\ 1 & -3 \end{bmatrix}x + \begin{bmatrix} 1 \\ 0 \end{bmatrix}r , \quad y = \begin{bmatrix} 2 & -5 \end{bmatrix}x.$$

Which of the state space representations given below has/have the same transfer function as that of $S$?

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

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

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

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

GATE ECE 2018

MCQ (Single Correct Answer)

-0.67

The state equation and the output equation of a control system are given below:

$$\mathop x\limits^. = \left[ {\matrix{ { - 4} & { - 1.5} \cr 4 & 0 \cr } } \right]x + \left[ {\matrix{ 2 \cr 0 \cr } } \right]u,$$

$$y = \left[ {\matrix{ {1.5} & {0.625} \cr } } \right]x.$$

The transfer function representation of the system is

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

$${{3s - 1.875} \over {{s^2} + 4s + 6}}$$

$${{4s + 1.5} \over {{s^2} + 4s + 6}}$$

$${{6s + 5} \over {{s^2} + 4s + 6}}$$

GATE ECE 2017 Set 2

MCQ (Single Correct Answer)

-0.6

A second order LTI system is described by the following state equation. $$$\eqalign{ & {d \over {dt}}{x_1}\left( t \right) - {x_2}\left( t \right) = 0 \cr & {d \over {dt}}{x_2}\left( t \right) + 2{x_1}\left( t \right) + 3{x_2}\left( t \right) = r\left( t \right) \cr} $$$

When x₁(t) and x₂(t) are the two state variables and r(t) denotes the input. The output c(t)=X₁(t). The systyem is

undamped (oscillatory)

under damped

critically damped

over damped