I have a vector field (unable to provide the details, unfortunately) $$\frac{dx}{dt} = f(x,u)$$ and measurements $$y = g(x,u)$$
Linearizing around any $\bar{x}$ and $\bar{u}$, I verified that the linearized pair $(A,C)$ is observable, as long as the first state, $x_1$, is non-zero. In practice, this occurs if the system is perpetually excited in some manner.
However, when actually implementing the EKF, I found that the states do not converge to the correct values, even when the aforementioned condition is met. Any idea how this could be? How could the EKF, which is an optimal observer, fail to observe the states when the linearized pair is observable?