Can you answer these questions I have about using linear Kalman filters and extended Kalman filters with a nonlinear system?
1. Does using a linear Kalman filter mean that I must have a time-invariant linear system? Or can a linear Kalman filter also be used with a linear system varying in time?
2. Linearization of this nonlinear system around the estimated state value is like linearization around a reference trajectory, except that now the matrices $A$ and $B$ vary with time. Is this correct?
3. Can I say that a linear Kalman filter is a time-invariant filter and an extended Kalman filter is a time-varying filter?
Context: I have some ideas but I am not sure if my understanding is correct.
I know that linearization of a nonlinear system around an equilibrium point means that the linearized system is linear and time-invariant, because $A$ and $B$ are constant. Therefore the nonlinear system can be used with a linear time-discrete Kalman filter after a discretization.