The Gaussian Mixture Consider Kalman Filter

The consider Kalman filter, or Schmidt-Kalman filter, is a tool developed by S.F. Schmidt at NASA Ames in the 1960s to account for uncertain parameters or biases within the system and observational models of a tracking algorithm. Its novelty is in that it considers the effects of the uncertain parameters rather than other Kalman-filter-based approaches, which instead estimate these parameters directly. Avoiding this online estimation of parameters allows, in many cases, for a more computationally feasible algorithm to be acquired, making it amenable to real-time applications. The consider Kalman filter, however, is an approach that works solely with the mean and covariance of the posterior distribution. In many problems, mean and covariance are often insufficient statistical descriptions of the filtering state. This work presents a consider formulation that works with a Gaussian sum approximation of the true distribution, permitting the Gaussian mixture consider Kalman filter and enabling an operator to maintain a more complete description of the true posterior state density while still working within a consider framework