State space representation involving derivatives of input

Question

We have the system $y''=-7y'-12y-u'-2u$

If we choose $x_1=y,x_2=y'$ we can write the system as

$x'=Ax + Bu \\ y= Cx$

Finding A is easy, but how do I find expressions for $B$ and $C$ when we have derivatives of the input in the expression?

Answer 1

Set $x_1=y$, $x_2=y'+u$ then \begin{align} x_1'&=y'=x_2-u \\ x_2'&=y''+u'=−7y′−12y−2u\\&=-7(x_2-u)-12x_1-2u\\&=-12x_1-7x_2+5u \end{align}