I want to design an EKF for a project, and i want to check the observability, in order to decide the sensor layout. I do not know about lie derivatives and lie algebra, but i with some research i think i understood the way to check local weak observability. (The main source is this pdf)
I want to confirm that my understanding is correct. So, here is what i think i have to do. If my system is given by the following equation: $$ \dot{x} = f(x) $$ $$ y = h(x) $$
We compute the Lie Derivatives with the following way ( notation: $J_{g(x)}$ = the jacobian of the column vector function g(x) )? $$ L_{f}h = \frac{ \partial h}{ \partial x} \cdot \dot{x} $$ $$ L_{f}h = J_h \cdot f(x) $$ $$ L_{f}(L_{f}h) = \frac{ \partial ( L_{f}h )}{ \partial x} \cdot \dot{x} $$ $$ L_{f}(L_{f}h) = J_{L_{f}h } \cdot f(x) $$
and then we continue $$ L_{f}\left(L_{f}(L_{f}h) \right) = \frac{ \partial \left(L_{f}(L_{f}h) \right)}{ \partial x} \cdot \dot{x} $$ and so on...
And then check if the rank of Q is equal to m, where Q is:
$$ Q = \left [ \matrix{ J_h \cr J_{L_{f}h } \cr ... } \right] $$
Edit If the dynamics of the system are given by the equation:
$$\dot{x} = f(x,u) $$
Does the process of determining observability change?