The Laplace transform of the step response of a system is given by

$$ Y(s)=\frac{100}{s(s+100)} $$

The rise time is defined as the time taken for the response to go from 0.1 to 0.9 of its final value. The settling time is defined as the time taken for the response to reach 0.98 of its final value.

For this system, the rise time ( $T_r$ ), settling time ( $T_s$ ), and time constant ( $T_c$ ), all expressed in seconds, are

The asymptotic Bode magnitude plot of a system is shown.

Which one of the following options best represents the transfer function of the system?

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 $z$ ?