Consider the a system which is a combination of two systems connected in series, $S_2$ and $S_1$, as shown in the dynamic diagram below:
You will notice that there is no output - that is, the output of $S_1$ feeds into $S_2$
By definition, a system is observable if the rank of its observability matrix $\textbf{O}$ equals the number of system variables, $n$, where the observability matrix is:
$\begin{align} \textbf{O} &= \begin{bmatrix} \textbf{C} \\ \textbf{CA} \\ \textbf{CA^2}\\ \vdots \\ \textbf{ CA}^{n-1} \end{bmatrix} \end{align}$
and where in turn, $C$ is the co-efficient matrix of $x(t)$ contained in the output $\textbf{y}(k) = \textbf{Cx}(t) + \textbf{Du}(t)$.
Am I correct in saying that in the example given above, the value of $\textbf{y}(t)$ is zero, therefore the value of $\textbf{C}$ is zero, therefore the rank of $\textbf{O}$ is less than $n$, and therefore the system is not observable?