GATE ECE 2024 | State Space Analysis Question 1 | Control Systems

GATE ECE 2024

MCQ (More than One Correct Answer)

-1.33

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

GATE ECE 2016 Set 3

MCQ (Single Correct Answer)

-0.6

A second-order linear time-invariant system is described by the following state equations $$$\eqalign{& {d \over {dt}}{x_1}\left( t \right) + 2{x_1}\left( t \right) = 3u\left( t \right) \cr & {d \over {dt}}{x_2}\left( t \right) + {x_2}\left( t \right) = u\left( t \right) \cr} $$$

Where x₁(t), then the system is

controllable but not observable

observable but not controllable

both controllable and observable

neither controllable nor observable