I am working on a nonlinear system and i want to know how well its states can be estimated by a time-discrete estimator. Which kind of observability, i.e. global or local (for every possible state and input) would the system need to fullfill such that an estimator would converge to the true states for every possible trajectory?
I learned that local observability at a point can be sufficiently proved by the rank test using Lie derivatives. Is it possible to make a statement about the ability of the estimator to estimate the correct states if i use this rank test just at every timestep for the current states and inputs?
Alternatively, do i gain any information about this ability by linearizing the system at every timestep and using the linear rank test? I already read that a nonlinear system can be observable at a point while its linearization is not.
For an Extended Kalman Filter which uses the linearization of the system in its algorithm, is it necessary that the linearized system is observable at every time?
To evaluate the observability of the nonlinear system, you must use Lie derivatives. However, you might be interested to look at the following paper:
The linearized system is time-varying due to the computation of the Jacobians, so the local observability might be useful depending on your application. In some cases, the local observability of the linearized system is equivalent to the observability of the nonlinear system if the system is linearized using the true states. Unfortunately, the true state are not available in practice, so the local observability likely does not match the observability of the nonlinear system.
Nevertheless, the local observability is still a useful, but these things should be kept in mind.