Consider the following system:
$ \dot{x} = \begin{bmatrix} 0 & 0 \\ 0 & -1 \end{bmatrix} + \begin{bmatrix} 0\\ 1 \end{bmatrix}u $
$ y = \begin{bmatrix} 1& 1 \end{bmatrix}x $
Now, one of the eigenvalues of this system is $0$. Clearly, in the transfer function, this pole is cancelled by a zero. Furthermore, for this example, both modes of the system are observable.
In this source, the following statement is given:
According to this, since there is a zero/pole cancellation, the mode $0$ should be unobservable. What am I missing in here? Can you explain when there is a zero/pole cancellation in a system, how observability and controllability is affected?