An approximation of the Bayesian state observer with Markov chain Monte Carlo propagation stage

The state estimation problem for nonlinear systems with stochastic uncertainties can be formulated in the Bayesian framework, where the objective is to replace the state completely by its probability density function. Without the restriction to selected system classes and disturbance properties, the Bayesian estimator is particularly interesting for highly nonlinear systems with non-Gaussian noise. The main limitations of Bayesian filters are the significant computational costs and the implementation problems for higher dimensional systems. The present paper introduces a piecewise linear approximation of the Bayesian state observer with Markov chain Monte Carlo propagation stage and kernel density estimation. These methods are suitable for the prediction of multivariate probability density functions. The piecewise linear approximation and the proposed algorithms can increase the estimation performance at reasonable computational cost. The estimation performance is demonstrated in a benchmark comparing the Bayesian state observer with an extended Kalman filter and a particle filter.