Complete the table using the tables in the class slides.

Determine the missing properties and the phase descriptions in the following table for water:

Problem #$1.(35$ points$).$ Given the state$-$space representation of a control system, calculate the following characteristics

for the response $x_3(t),$ assuming the system input is a unit step function. The initial conditions are set to zero. Use a time

step of $dt=0\mathrm{.}01$ seconds and simulate for a total duration of $10$ seconds.

$\left[\begin{array}{c}{x}_1^{}\\ x_2^{}\\ x_3^{}\end{array}\right]=\left[\begin{array}{ccc}-20& -40& -60\\ 1& 0& 0\\ 0& 1& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]+\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right](t)$

Calculate the following:

$(5$ points$).$ Maximum overshoot; in percentage.

$(5$ poinis$).$ Rise time $(t_r),$ in seconds.

$(5$ points$).$ Peak time $(t_p),$ in seconds.

$(5$ points$).$ Steady$-$state error $(e_{ss}),$ in percentage.

$(5$ points$).$ Settling time $(t_s),$ in seconds.

$(5$ points$).$ Damping ratio $().$

$(5$ points$).$ Natural frequency $({}_n),$ in rad$/$s$.$