Controllability of internal subsystem and input-outpout controllability

Question

I am trying to prove that the state equation $\dot{x} = \begin{bmatrix} A_{11} A_{12}\\A_{21} A_{22}\end{bmatrix}x + \begin{bmatrix} B_{1}\\ 0\end{bmatrix}u$ is controllable if and only if the pair $(A_{22}, A_{21})$ is controllable. Thank you for your help.

Answer 1

We can prove this using one of the many equivalent criteria for controllability:

$(A,B)$ is controllable iff the matrix $[\matrix{A-\lambda I & B}]$ has full row rank for all $\lambda\in\mathbb{C}$

So $(A_{22},A_{21})$ is controllable iff the matrix $[\matrix{A_{22}-\lambda I & A_{21}}]$ has full row rank for all $\lambda\in\mathbb{C}$. This means that all the rows of $[\matrix{A_{22}-\lambda I & A_{21}}]$ are linearly independent for all $\lambda\in\mathbb{C}$. Equivalently all the rows of $[\matrix{A_{21} & A_{22}-\lambda I}]$ are linearly independent for all $\lambda\in\mathbb{C}$ (its an equivalent description of the vectors if you consider a permutation in the numbering of the axes).

Similarly, the full system is controllable iff the matrix $\Big[\matrix{A_{11}-\lambda I & A_{12} & B_1\\A_{21} & A_{22}-\lambda I & 0}\Big]$ has all its rows linearly independent for all $\lambda\in\mathbb{C}$.

Now, if $B_1$ has full row rank i.e. its rows are linearly independent and all the rows of $[\matrix{A_{21} & A_{22}-\lambda I}]$ are linearly independent ($(A_{22},A_{21})$ is controllable) then it's relatively easy to prove that the rows of the whole matrix would be linearly independent (due to the appended zeros in $B$) and therefore the whole system is controllable.