Correlated Estimation Problems and the Ensemble Kalman Filter

摘要

The Kalman flter is a recursive algorithm that estimates the state of a linear dynamical system
\nfrom a sequence of noisy sensor measurements. Due to its relative simplicity, numerical efficiency
\nand optimality, the Kalman flter and its variants have been applied to a wide range of problems
\nin technology, notably in the areas of guidance, navigation, and control. The traditional
\ndefinition of the Kalman flter is based on the assumption that at any given time, the errors
\nassociated with the predicted state estimate and the observation are statistically independent.
\nHowever, in many practical problems, this assumption is not satisfied, and as such the Kalman
\nfilter may provide overconfident state estimates and diverge. This can have serious consequences
\nin the context of safety-critical systems.
\nAlthough there are modifications of the Kalman filter that accommodate various types of
\ncorrelation in the process and observation noises, these are not suitable in the situation where
\nthe correlation between the errors associated with the predicted state estimate and the observation
\nis caused by the presence of common past information between the state estimate and
\nthe observation, which is characteristic of distributed sensor networks. On the contrary, existing
\nmethods that deal with the common past information problem either provide overly conservative
\nestimates, or have too strict assumptions on the structure of the problem, such as the
\ncommunication topology of the sensor network.
\nThis thesis presents two new filters to address various correlated estimation problems that
\nare based on the Ensemble Kalman filter, a Monte Carlo variant of the Kalman filter, which
\nrepresents the state estimates and observations using sets of random samples instead of the
\nconventional mean vectors and covariance matrices. Specifically, both of these filters provide a
\nnew generalised update rule that computes consistent state estimates even in the presence of
\ncorrelation between the errors associated with the state estimate and the observation. This is
\nonly possible due to the fact that in the context of the Ensemble Kalman filter, the magnitude
\nof such a correlation can be estimated from the random samples The new filters retain all of the important features of the Ensemble Kalman filter, such as
\nscaling linearly with the number of state-space dimensions, and supporting non-linear process
\nand observation models. An analysis of the numerical properties of the filters is provided, including
\na comparison with state-of-the-art methods in several benchmark scenarios. Furthermore, in
\norder to demonstrate their practical utility, the new filters have been applied to three different
\nreal-world problems in the larger field of robot localisation: cooperative vehicle localisation,
\nsimultaneous localisation and mapping, and global satellite-based positioning.