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?
$\endgroup$ 52 Answers
$\begingroup$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}
$\endgroup$ 4 $\begingroup$"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:
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.
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