  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.
