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In Bayesian estimation, the objective is to calculate the complete density function for an unknown quantity conditioned on noisy observations of that quantity. This work considers recursive estimation of a nonlinear discrete-time system state using successive observations. The formal recursion for the density function is easily written, but generally there is no closed form solution. The numerical solution proposed here is obtained by modifying the recursion and using a simple piece-wise constant approximation to the density functions. The approach also allows detailed analysis of error propagation through the algorithm, yielding a bound on the maximum error growth, and a characterization of the situations with potential for large errors. The stability of the algorithm is demonstrated by comparing its long-term performance to a Kalman filter for a linear system with Gaussian noises. Comparison to the point mass algorithm shows improved estimation accuracy, and, for moderately dense grids, faster computation. As an example, the algorithm is applied to a system identification problem, and the results compared to a popular second order minimum variance estimator. The optimal estimate derived from the approximate density is seen to be far superior. |
