TITLE: State space representation involving derivatives of input
QUESTION [9 upvotes]: 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?

REPLY [2 votes]: "Suppose if the physical meaning of $y =$ position and $u =$ force. What is the physical meaning of $x_2$? We know that the $x_1$ will be position, what is the physical meaning of $x_2$? velocity + force? – Unknown123"
You can find a very neat approach here:
https://www.ram.ewi.utwente.nl/aigaion/attachments/single/597
From that it is self-evident that the physical dimension of $y'+u$ comes from the physical dimension of the coefficients of variables.
From the equation  $$y′′=−7y′−12y−u′−2u,$$ if $y$ is a length, then $-u'$ and $-2u$ are accelerations, supposed the derivative is with respect to time; in case $u$ is intended to be a force, the coefficient $-2$ should rearrange dimensionality, and its physical meaning is the inverse of a mass. The coefficient of $u'$ has the dimensions of time/mass.