Hello I have a rather specific question.
I am trying to follow a derivation and can't work out the last step. Given: $$s\,X(s) = (A-B\,K\,C) X(s) + B\,K\,V(s)$$ and $$Y(s) = CX(s)$$
Multiplying $Y(s) = CX(s)$ through by s, then substituting in $sX(s)$....
Here is where I am at: $$ sY(s) = C{((A-BKC)X(s) + BKV(s))} $$
A is an n x n matrix, K is 1 x n, X(s) is n x 1.
However, the final form is given as this: $$ Y(s) = C(sI-A+BKC)^{-1}BK V(s) $$
Hoping someone can help me with the intermediate steps.
Starting from the equation expressed in $X(s)$ and $V(s)$ and solving it for $X(s)$ yields,
$$ s\, X(s) - \left(A - B\, K\, C\right) X(s) = \left(s\, I - A + B\, K\, C\right) X(s) = B\, K\, V(s), $$
$$ X(s) = \left(s\, I - A + B\, K\, C\right)^{-1} B\, K\, V(s). $$
Now using the definition of $Y(s)$ yields,
$$ Y(s) = C\, X(s) = C \left(s\, I - A + B\, K\, C\right)^{-1} B\, K\, V(s). $$