Global nonlinear observability Given an arbitrary nonliner system on the form:
\begin{align}
    \dot{x}(t) &=  f(x(t), u(t)), & h(x) &= [y_0, y_1, \dots, y_N]^T
\end{align}
The first Lie derivative is defined as $L_f^0 h(x) = h(x)$.
\begin{equation}
    L_f^{k}h(x) = \frac{\partial}{\partial x}(L_f^{k-1}h(x))f(x)
    \label{eq:Lie_der}
\end{equation}
The observability matrix for weak, nonlinear observability is gradient directions of the Lie derivatives and measurements $h_i(x)$ in $h$.
\begin{equation}
    \mathcal{O}_{local} = 
    \begin{bmatrix}
    \nabla L_f^0h_0 & \nabla L_f^0h_1 & \dots & \nabla L_f^0h_{n-1}\\
    \nabla L_f^1h_0 & \dots & \dots& \nabla L_f^1h_{n-1}\\
    \vdots & \vdots & \vdots & \vdots \\
    \nabla L_f^kh_0 & \nabla L_f^kh_1 & \dots & \nabla L_f^kh_{n-1}
    \end{bmatrix}
\end{equation}
If $\mathcal{O}_{local}$ is full rank we have weak, local observability for the nonlinear system.
The question is: Would full rank $\mathcal{O}_{local}$ for all states imply global observability?
 A: I recommend this paper which deals with some definitions of different kinds of observability and the rank test using Lie derivatives:
R. Hermann and A. Krener, "Nonlinear controllability and observability"
The authors state that from the system being locally observable it follows that the system is both locally weakly observable and observable. From those both kinds of observability it follows that the system is weakly observable. After this discussion it is said that there are no other implications in general.
In my opinion it makes sense that locally weakly observability does not imply global observability, because for locally weakly observability you just consider a neighborhood of each point.
A: R. Hermann and A. Krener, "Nonlinear controllability and observability"
states that the rank condition on $\mathcal{O}$ fullfills local, weak observability (Theorem 3.1). The article was insightful and is probably more useful than the simpler definition I was looking for.
Entirely avoiding linearization and Lie derivatives lead to the definition of algebraic observability, defined in:
Martínez-Guerra R., Cruz-Ancona C.D. (2017) Algebraic Observability for Nonlinear Systems
...Here, the concept of observability is equivalent to the differential field extension
G=K<u, y> being algebraic. This means that the complete differential-algebraic
information is contained on the differential field K<u, y>, since every element in
G can be obtained from K<u, y> via a differential-algebraic equation.
of their time derivatives.
