GATE 2024 Electronics & Communication Q53

Question

Consider a system $S$ represented in state space as
$\frac{dx}{dt} = \begin{bmatrix} 0 & -2 \ 1 & -3 \end{bmatrix}x + \begin{bmatrix} 1 \ 0 \end{bmatrix}r, y=[2 \ -5]x$.
Which of the state space representations given below has/have the same transfer function as that of $S$?

Answer

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

Answer

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

Answer

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

Answer

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