Stochastic filtering in jump systems with state dependent mode transitions

We introduce a new methodology to construct a Gaussian mixture approximation to the true filter density in hybrid Markovian switching systems. We relax the assumption that the mode transition process is a Markov chain and allow it to depend on the actual and unobservable state of the system. The main feature of the method is that the Gaussian densities used in the approximation are selected as the solution of a convex programming problem which trades off sparsity of the solution with goodness of fit. A meaningful example shows that the proposed method can outperform the widely used interacting multiple model (IMM) filter in terms of accuracy at the expenses of an increase in computational time.