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 $S$?
The electrical system shown in the figure converts input source current $i_s(t)$ to output voltage $\theta_O(t)$.
Current $i_L(t)$ in the inductor and voltage $\vartheta_C(t)$ across the capacitor ate taken as the state variables, both assumed to be initially equal to Zero, i.e., $i_L(0)=0$ and $\vartheta_c(0)=0$. The system is
$$\mathop x\limits^. = \left[ {\matrix{ { - 4} & { - 1.5} \cr 4 & 0 \cr } } \right]x + \left[ {\matrix{ 2 \cr 0 \cr } } \right]u,$$
$$y = \left[ {\matrix{ {1.5} & {0.625} \cr } } \right]x.$$
The transfer function representation of the system is
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
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