In the feedback control system shown in the figure below $G(s) = \dfrac{6}{s(s+1)(s+2)}$.

$R(s), Y(s),$ and $E(s)$ are the Laplace transforms of $r(t), y(t),$ and $e(t)$, respectively. If the input $r(t)$ is a unit step function, then __________

Consider a unity negative feedback control system with forward path gain $G(s) = \frac{K}{(s + 1)(s + 2)(s + 3)}$ as shown.

The impulse response of the closed-loop system decays faster than $e^{-t}$ if ________.

A satellite attitude control system, as shown below, has a plant with transfer function $G(s) = \frac{1}{s^2}$ cascaded with a compensator $C(s) = \frac{K(s +\alpha)}{s + 4}$, where $K$ and $\alpha$ are positive real constants.

In order for the closed-loop system to have poles at $-1 \pm j \sqrt{3}$, the value of $\alpha$ must be ______.

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 $S$?