Hello Manuel

The parameter of interest is the states of the system (X(t)).

Thanks for your recomended references.

Is there any analytical solutions to this problem?

Hello Habib

This system is linear but uncertainty in h(t) makes it impossible to use the standard Kalman filter. h(t) in an unknown stochastic of binary random variables so the matrices of the system is unknown. Does Kalman fit in this situation?

Thanks

Hello. 

You should look for hybrid systems. Check for instance 

State estimation for hybrid systems: applications to aircraft tracking

http://www.eecs.berkeley.edu/~tomlin/papers/journals/hbt06_iee.pdf

Article State estimation for hybrid systems: Applications to aircraft tracking

Hi Rafael

Thanks for your contribution. It helped me alot but I only found systems with markov property for its switching state. The parameters(h(t)) of the model mentioned here is binary random variables. Are there any hybrid system with gaussian and binary random variables?

Thanks

Hello

I believe that Rafael Rui is on the right track ;-) It's a type of hybrid stochastic system. Normally the formulation is for h(t) to be a Markov chain, hence scalar process with finite state space. In your case you have a vector valued process with binary components. To use hybrid filtering methods like IMM, GPB, etc, you need to convert to a scalar valued Markov chain. This is feasible by defining a state space with 2^N states where N is the dimension of h.You then need to set up the transition probabilities, which must be known.

A link to some approaches to manoeuvring target tracking, which is often solved using hybrid systems, is attached.

A warning is in order. The algorithm complexity will be high even for moderate N.

Technical Report A Survey of Manoeuvring Target Tracking Methods

Hi Graham

Do you mean that there are not any method to solve this type of problems without converting it to standard hybrid systems?

As Manuel chiachio said, it can only be possible to solve it with monte carlo filtering?

Is it true that this type of problems can be considered as a nonlinear non-gaussian tracking category? 

Thanks

Hello Ramin,

The way that seems most natural is to convert to standard hybrid system form and use the theory developed for these systems. If you have another way, then that is fine too. Monte Carlo filtering is certainly not the only way to solve this state estimation problem.

Dear Graham

I like to know your opinion about the best solution for this problem as you are the professional in the filed of target tracking and state estimation.

Thanks

Hello Ramin,

The approach I outlined in my first post is the one I would use. IMM is an easy way to do it once you have defined the appropriate Markov chain. Upvotes and acknowledgements are always nice if you write a paper on it :-)

Hi Graham

I'll try to solve this problem with your method.

It's my pleasure that you are one the writers of this paper if it is sussed:)

Thanks alot

Hello Ramin,

I will be happy just to get an acknowledgement. Authorship usually requires a significant contribution rather than just an initial suggestion. Good luck with the algorithm development.

Dear Graham

How can it be possible to apply directly the Bayesian theorem

p(Xk|y1:k) = ....

to extract likelihood, prediction and normalization densities?

Do you know any good references about the extraction of this type of equations in probability theorem?

 Thanks