The state and output equations for a control system are:

$$ \begin{aligned} & \dot{x}=\left[\begin{array}{cc} -4 & -1.5 \\ 4 & 0 \end{array}\right] x+\left[\begin{array}{l} 2 \\ 0 \end{array}\right] u \\ & y=\left[\begin{array}{ll} 1.5 & 0.625 \end{array}\right] x \end{aligned} $$

Which of the following expressions correctly represents the transfer function $\frac{Y(s)}{U(s)}$ of the system with zero initial conditions?

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 $S$?

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