EGR $386,$ Summer $2020$

Homework $22$

A system is described by the differential equation

$($d$^2$ y$)/($dt$^2)+5$ dy$/$dt$+4$y$=10$u $.$

Write a state$-$space description of the system in the form

$($x$)=$A$$x$+$Bu $,$ y$=$C$$x$+$Du ;

Write a second state$-$space description of the system;

Write a third state$-$space description of the system;

Use Octave to find the eigenvalues of the A matrix in each case; are the eigenvalues the same?

Determine the transfer function of the system, $($Y$($s$))/($U$($s$))$ ; what are the poles of the system? Comment.

Repeat problem $1)$ above for the system with differential equation

$($d$^2$ y$)/($dt$^2)+6$ dy$/$dt$+8$y$=16$u $.$

Repeat problem $1)$ above $($except for parts b$)$ and c$))$ for the system with differential equation

$($d$^2$ y$)/($dt$^2)+2$ dy$/$dt$+10$y$=5$u $.$