Hybrid Transition Density Approximation for Efficient Recursive Prediction of Nonlinear Dynamic Systems

For several tasks in sensor networks, such as localization, information fusion, or sensor scheduling, Bayesian estimation is of paramount importance. Due to the limited computational and memory resources of the nodes in a sensor network, evaluation of the prediction step of the Bayesian estimator has to be performed very efficiently. An exact and closed-form representation of the predicted probability density function of the system state is typically impossible to obtain, since exactly solving the prediction step for nonlinear discrete-time dynamic systems in closed form is unfeasible. Assuming additive noise, we propose an accurate approximation of the predicted density, that can be calculated efficiently by optimally approximating the transition density using a hybrid density. A hybrid density consists of two different density types: Dirac delta functions that cover the domain of the current density of the system state, and another density type, e.g. Gaussian densities, that cover the domain of the predicted density. The freely selectable, second density type of the hybrid density depends strongly on the noise affecting the nonlinear system. So, the proposed approximation framework for nonlinear prediction is not restricted to a specific noise density. It further allows an analytical evaluation of the Chapman-Kolmogorov prediction equation and can be interpreted as a deterministic sampling estimation approach. In contrast to methods using random sampling like particle filters, a dramatic reduction in the number of components and a subsequent decrease in computation time for approximating the predicted density is gained.