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?
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 $S$?
$$\mathop x\limits^. = \left[ {\matrix{ { - 4} & { - 1.5} \cr 4 & 0 \cr } } \right]x + \left[ {\matrix{ 2 \cr 0 \cr } } \right]u,$$
$$y = \left[ {\matrix{ {1.5} & {0.625} \cr } } \right]x.$$
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