Monte Carlo study of the optimal non-linear estimator: linear systems with non-gaussian initial states †

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Abstract
The optimal, in the mean-square sense, estimate of state vector of a linear discrete system that is excited by white zero mean gaussian noise and that has non-gaussian initial state vector is presented. Both the optimal estimate and the corresponding error covariance matrix are given. It is shown that the optimal estimator consists of two parts : a linear estimator which is a Kalman filter and a non-linear part which is a parameter estimator. In addition, the a posteriori probability density function, p(x(k)λk), is also given. Finally, a suboptimal procedure that reduces the computational requirements is presented. The results of extensive digital computer simulations including Monte Carlo study have been presented to establish that the non-linear filter presented here is far superior to the best linear Kalman filter. A practical filter design criterion for utilizing this non-linear filter with reduced data processing requirements is also given.

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