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I am studying for a linear system theory exam later on this week. The professor has recommended some problems in order to practice and prepare for the exam. This is one of them that I'm trying to prove:

Examine the controllability for a linear time-invariant system whose “A-matrix” is in Jordan form as follows: (a) Let $A=\left( \begin{array}{ccc} λ & 1 & 0 &0 \\ 0 & λ & 1 &0\\ 0 & 0 & λ &1\\ 0 & 0 & 0 &λ\end{array} \right)$ , $B =\left( \begin{array}{ccc} b_1\\b_2\\b_3\\b_4\end{array} \right)$

Find necessary and sufficient conditions on ${bi}$ so that the system is controllable.

(b) Generalize your result from part (a) to

$A=\left( \begin{array}{ccc} J_1&0&...&0\\ 0 &J_2&...&0\\ :& & &:\\ 0& ...&...&J_r \end{array} \right)$ , $B =\left( \begin{array}{ccc} B_1\\B_2\\:\\Br\end{array} \right)$ where each $J_i$ is a Jordan block of the $A$ matrix, and the $B_i$ are the corresponding vector blocks of the B matrix.

For part (a) I have constructed the controllability matrix:

$C=\left( \begin{array}{ccc} b_1 & λb_1+b_2 & λ^2b_1+2λb_2+b_3 &λ^3b_1+3λ^2b_2+3λb_3+b_4 \\ b_2 & λb_2+b_3 & λ^2b_2+2λb_3+b_4 &λ^3b_2+3λ^2b_3+3λb_4\\ b_3 & λb_3+b_4 & λ^2b_3+2λb_4+b_4 &λ^3b_3+3λ^2b_4\\ b_4 & λb_4 & λ^2b_4 &λ^3_b4\end{array} \right)$

It is clear how C is only full rank if $b_4$ is different from $0$. Therefore that is the necessary and sufficient condition for the system to be controllable. In order to prove it is enough to calculate the determinant of $C$ and see that it is:

$$\det(C)=b_4^4$$

For part (b) I have been reading some literature that states that the condition for a matrix formed by jordan blocks of this type is that the last row of each $B_i$ matrix has to be different from zero and linearly independent for the $B_j$, with $j$ different from $i$, and going from $1$ to $r$. I have been giving thought to this but I can't seem to find a way to go and prove it right.

Besides proving the following statement, depending on the number of inputs, each $B$ matrix will have a number of columns equal to it. Therefore, if it has $m$ inputs, there are $m$ linearly independent vectors that should compose the lower rows of each $B_i$.

This is all that I have thought until now, any help or guidance would be greatly appreciated. Thanks for your time reading and help in advance.

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A useful test to consider is that $(A,B)$ is cc. iff the Popov Belevitch Hautus criterion holds.

That is, $(A,B)$ is cc. iff $\operatorname{rk} \begin{bmatrix} sI-A & B \end{bmatrix} = n$ for all $s \in \mathbb{C}$.

Part (a) is immediate from this (note that you only need to consider $s$ in the spectrum of $A$).

A criterion for Part (b) can be deduced by considering blocks with the same eigenvalue (in particular, $B$ may need to have multiple columns to be cc.).

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