A vanilla Kalman Filter allows for a time varying observation matrix $H_k$. Is it allowable for $H_k$ to be a function of the system state $x_k$ in a vanilla Kalman filter?
First, am I correct that this is not allowed in the vanilla Kalman filter? If it is allowed, then I don't understand how measurements would then act to correct the parameters of $x$ that predict them via $H_k$. I'd appreciate a more explicit description of this than my own hunch though!
Second, am I correct that the Extended Kalman Filter & the Unscented Kalman Filter do explicitly allow for this situation? If I am, do these filters allow observations to correct the terms of $x$ that are predicting them?