So i understand that given a Matrix A, and a vector B, you can work out a controllability matrix G such that

$\displaystyle G = [B , AB, A^2B, A^3B,...]$

and then the controllability depends on whether G is of full rank or not etc.

But what happens when A is an nxn matrix but B is not a 1xn vector, but instead a 2xn vector?

For example (and this is a completely made up example) -

$\displaystyle A=\begin{bmatrix} 2 & 1& 1 \\4 &3 &1\\ 2& 8 &9 \end{bmatrix}$

and

$\displaystyle B= \begin{bmatrix} 1 & 3\\ 2&1 \\ 1&5\end{bmatrix}$

how is G formed from this?

to save you having to work out:

$\displaystyle AB =\begin{bmatrix} 5 & 12\\ 11& 20\\ 27 & 59\end{bmatrix} \mathrm{and} A^2B=\begin{bmatrix} 48 & 103\\ 80 & 167 \\ 341 & 715\end{bmatrix}$

Then the same question for observability

$\displaystyle \theta = \begin{bmatrix} C \\ CA \\ CA^2 \\ :\end{bmatrix}

$

using the same A above, and $\displaystyle C= \begin{bmatrix} 2 &2 &4 \\ 1&2&1\end{bmatrix}$

how is

$\displaystyle \theta$ formed?

Again, to save you having to work out

$\displaystyle CA= \begin{bmatrix} 20 & 40& 40\\ 12& 15& 12\end{bmatrix} \mathrm{and} CA^2 = \begin{bmatrix}280 & 460 &420\\ 108 & 153& 135\end{bmatrix} $