State space representation for the differential equation with input derivative

Question

For a system of differential equations,
$r'' - q' + 2r' + 8r = u_1'-2u_2 $
$q''+5q'-13r+11q=4u_2$
$y_1=2q'-r$
$y_2=r-q$
$y_3=r'$

if we choose state variables as,
$\dot x = [r\quad r'\quad q\quad q']$

the resulting equations would be,
$\dot x_2 = x_4-2x_2+\dot u_1 - 8x_1 - 11x_3$
$\dot x_4 = 4u_2 - 5x_4+13x_1-11x_3$
$y_1 = 2x_4-x_1$
$y_2=x_1-x_3$
$y_3=x_2$

Now, how to go about finding the state space representation of the system with the input derivative $\dot u_1$ ?

Answer 1

HINT: You can avoid the input derivative if, in the state-space vector, you select $x_2=r'-u_1$ i.e. $$x=\left[\matrix{r & r'-u_1 & q & q'}\right]^T$$ I believe you can take it from here.