A system is characterized by the following state equation and output equation ( $u$ : input,

$x$ : state vector, $y$ : output)

$$ \begin{aligned} & \dot{x}=\left[\begin{array}{cc} a & b \\ -a & 0 \end{array}\right] x+\left[\begin{array}{l} 1 \\ 0 \end{array}\right] u \\ & y=\left[\begin{array}{ll} 1 & 2 \end{array}\right] x \end{aligned} $$

What are the values of $a$ and $b$ for which the poles of the transfer function are at $-2+j 3$ and $-2-\beta$ ?

A system is represented in state-space form as follows:

(u: input, $x$ : state vector, $y$ : output)

$$ \begin{aligned} & \dot{x}=\left[\begin{array}{cc} 1 & 2 \\ -3 & 0 \end{array}\right] x+\left[\begin{array}{l} 1 \\ 2 \end{array}\right] u \\ & y=\left[\begin{array}{ll} 1 & 2 \end{array}\right] x \end{aligned} $$

Consider the new state vector $z=\left[\begin{array}{cc}2 & 1 \\ -1 & 0\end{array}\right] x$

What is the state-space representation of the system in terms of the new state vector $z$ ?

The digital circuit shown has 3 inputs $(x, y$ and $z)$.

The simplified logical expression for the output (OUT) is:

The MOSFET switches shown in the circuit are ideal.

Which of the following is the correct option for Boolean logical expression of the output (OUT), and the maximum possible power (P) consumed by the circuit?