A FILTER WITH A GUARANTEED ASYMPTOTIC PERFORMANCE

In the case of low noise levels the optimal probability density functi on summarizing the available information about the state of a system c an be accurately approximated by the product of a gaussian function an d a linear function. The approximation preserves the ability to estima te to an accuracy of O(lambda-2) the expected value of any twice conti nuously differentiable function defined on the state space. The parame ter lambda depends on the noise level. If the noise level in the syste m is low then lambda is large. A new filtering method based on this ap proximation is described. The approximating function is updated recurs ively as the system evolves with time, and as new measurements of the system state are obtained. The updates preserve the ability to estimat e the expected values of functions to an accuracy of O(lambda-2). The new filter does not store previous measurements or previous approximat ions to the optimal probability density function. The new filter is ca lled the asymptotic filter, because the definition of the filter and t he analysis of its properties are based on the theory of asymptotic ex pansion of integrals of Laplace type. An analysis of the state propaga tion equations shows that the asymptotic filter performs better than a particular widely used suboptimal approximation to the optimal filter , the extended Kalman filter. The extended Kalman filter does not, in general, preserve the ability to estimate expected values to an accura cy of O(lambda-2). The computational cost of the asymptotic filter is comparable to that of the iterated extended Kalman filter.