Let´s study some control properties of the following system with $a,b,d,c > 0$ :
$$ \pmatrix{x\\y\\z\\v}'=\pmatrix{-a&a&0&0\\0&0&1&0\\d&-d&0&0\\c&0&0&0}\pmatrix{x\\y\\z\\v} + \pmatrix{0\\0\\bd\\0}u $$
My attempt:
a) In terms of controllability, we can check that the controllability matrix $R(A,b) = [b|Ab|A^2b|A^3b]$ is invertible. Is there a quicker way to do this?
b) Regarding the controller form $A^{\flat}$, one could compute the eigenvalues of matrix $A$. However, we could also use the fact that $AR(A,b)=R(A,b)\hat{A}$ and $A^{\flat}R(A^{\flat}, b^{\flat})=R(A^{\flat}, b^{\flat})\hat{A}$ since $A^{\flat}$ and $A$ have the same characteristic polynomial. As a result, $A^{\flat} = T^{-1}AT$ where $T=R(A,b)R(A^{\flat}, b^{\flat})^{-1}$. But I don't know about $A^{\flat}$.
I also wonder whether $R(A^{\flat}, b^{\flat})=I$ because it is known that $R(\hat{A},\hat{b})=I.$
c) If I am interested in finding a feedback $u = fx$ so that all eigenvalues of $A + bf$ equal $−1$, I can just impose the conditions. The system is controllable, the Pole Shifting Theorem helps with this purpose.