The initial error covariance matrix should be defined based on your initialization error.  I.e., if you think your initial state is not very close, the P0 value should be large, whereas if the initialization is very good (high confidence that your states are close to the correct values) you can assume a smaller P0 value.  I would not recommend setting P0 to zero, as this assumes there is no initialization error and your initial states are perfect.  This is almost never the case.  Many times the P0 matrix is diagonal, with the diagonal components corresponding to the expected variance in the corresponding state, i.e. how much deviation you might expect in the initialization of that state.  If you have no idea where to start, I recommend using an identity matrix rather than the zero matrix. 

The Q matrix on the other hand corresponds to the uncertainty that you expect in your state equations.  This could include modeling errors or other uncertainties in the equations themselves.  Some formulations consider input measurements in the state equations which introduces process noise.  Sometimes discretization error could be considered here as well.  If you are very confident in your equations, you could set Q to zero, though typically you will have some form of uncertainty in the equations due to modeling errors, measurement errors, discretization, approximations in the derivation, etc.

To address your second question, it should not matter which you update first.  Either way, both of these predictions occur based on the previous states, and then are used for the update step.

I hope this helps some.  Good luck with your work!

Thank you Matthew,

Now I am able to visualize the difference between error and process noise covariance.

Thanks a Lot!

Yes Michael...I am using non-zero P & Q matrix. I have tried to put Q in line with my knowledge of the model and seems correct. As you said...I am also planning to put some conditional use of the filter which will keep the output estimates good enough for usage on an Automated Transmission. I will trigger the filter only when the vehicle speed and acceleration moves above a threshold and try to ensure sufficient excitation. Thanks again but i want the filter to be fast too. Is there any way by which I can ensure a good filter which is adequately fast and accurate??

One has to understand the meaning of the Q and P matrices in order to answer your question. First, the P matrix is just a covariance matrix associated with the errors in the state vector. The Q matrix, has nothing to do with any errors. It represents any driving inputs that cause your state vector to deviate from your noise free state vector transition model. For example, in a constant velocity model, the Q matrix represents accelerations that allows the tracked object to deviate from constant velocity. This will expand the predicted P matrix to allow for slight deviations in the track when the object maneuvers. The Q matrix can NEVER be filled with zeros. If you do that the filter will use the noise free model to predict the state vector and will ignore any measurement data since your model is assumed perfect. Generally, one would set the diagonals of the Q matrix to be proportional to twice the expected maneuver accelerations (for a constant velocity model). So, in your case, you have to look at your noise free model and figure out what deviations can occur due to unaccounted differences between your model and the real world. That's what goes into the Q matrix.

If you model is really a no noise model, consider using a nonlinear least squares method, because they assume ONLY measurement noise and no dynamic noise.

As for the initial P matrix, you should put variance along the diagonal that represent the uncertainties in your INITIAL state vector (NEVER zeros along the diagonal). Also, remember that the diagonals of any covariance matrix must always be positive since they represent squared errors.

For your second question, it doesn't matter in what order you predict the state vector or the Jacobian. They are both based on the state vector from the last update.

I have a book out that explains a lot of this with several worked out problems.

only the relative importance between Q and R matrices brings significant meaning, not the absolute values. As far as I know, there is no mathematical method to identify Q and R. We need do a lot of experiments to figure out this relative importance. Q should be smaller than R if we feel that there are so much noise in measurements and vice versa.

Try, Square Root Kalman Filter. It has been used to solve the problem of numerical round-off diverge.

While the question is already a little bit dated, I would like to point interested followers to the attached tutorial, which may help to understand the role played by the different matrices. Moreover, it also gives some tuning guidelines and comments on improved numerical stability.

Concerning the comment of Tu Ha, it is indeed not true that there exist no systematic mathematical methods to identify Q and R. In fact, a recent one is called the Autocovariance least-squares method, and its development started in the group of Prof. Rawlings. While there are more recent developments, a good start into this topic may be their original publication, which can be found here: http://jbrwww.che.wisc.edu/tech-reports/twmcc-2003-04.pdf

Article How To NOT Make the Extended Kalman Filter Fail

Please see our two papers in the December 2016 issue of the journal SADHANA attached herewith. Prof.M.R.Ananthasayanam.