I am running an unscented Kalman filter on my system and am able to estimate the states within 4% of their true values. This is true with a Monte Carlo simulation consisting of 1000 runs. However, when checking the observability Gramian of the system, I am finding that the matrix is not full rank.
Is it possible to estimate states well using a UKF without all the states being observable?
My system is nonlinear, so to calculate the Gramian I first calculate the state transition matrix of the dynamics (Phi) for each time step. I then compute Phi' * H' * H * Phi' at each time step, where H is the Jacobian of the measurement matrix. I sum these matrices to obtain the observability Gramian at the end of the simulation and find that it is not full rank.
Any insight into my problem would be much appreciated.