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

Calculate the following:

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

$(5$ points$).$ 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$.$

Problem #$2.(35$ points$).$ Given the state$-$space representation of the control system from Problem #$1,$ derive the

corresponding transfer function representation.

$(10$ points$).$ Derive the transfer function $G(s)$ from the given state$-$space matrices $A,B,C,$ and $D.$

$(10$ points$).$ Identify the poles and zeros of the transfer function on the s$-$plane.

$(5$ points$).$ Determine the system's natural frequency $({}_n)$ and damping ratio $()$ from the transfer function.

$(5$ points$).$ Plot the step response of the transfer function and compare it with the response obtained from the

state$-$space model in Problem #$1.$

$(5$ points$).$ Discuss any differences, if any, between the transfer function response and the state$-$space response.