Approximate nonlinear filtering with applications to navigation

In this dissertation we address nonlinear techniques in filtering, estimation, and detection that arise in satellite based navigation. Here, we emphasize the theoretical aspect of these techniques, and we also address their applications.; We first introduce particle filtering for an exponential family of densities. We prove that under certain conditions the approximated conditional density converges to the true conditional density. For the case where the conditional density does not lie in an exponential family but stays close to it, we show that under certain assumptions the error of the estimate given by this approximate nonlinear filtering, projection particle filtering, is bounded. We give similar results for a family of mixture densities. We use projection particle filtering for an exponential family of densities to estimate the position of a mobile platform that has a combination of inertial navigation system (INS) and global positioning system (GPS), referred to as an integrated INS/GPS. We show via numerical experiments that projection particle filtering exceeds regular particle filtering methods in navigation performance.; Using carrier phase measurements enables the differential GPS to reach centimeter level accuracy. The phase lock loop of a GPS receiver cannot measure the full cycle part of the carrier phase. This unmeasured part is called integer ambiguity, and it should be resolved through other means. Here, we present a new integer ambiguity resolution method. In this method we treat the integer ambiguity as a random digital vector. Using particle filtering, we approximate the conditional probability mass function of the integer ambiguity given the observation. The resolved integer is the MAP estimate of the integer given the observation.; Reliability of a positioning system is of great importance for navigation purposes. Therefore, an integrity monitoring system is an inseparable part of any navigation system. Failures or changes due to malfunctions in sensors and actuators should be detected and repaired to keep the integrity of the system intact. Since in most practical applications, sensors and actuators have nonlinear dynamics, this nonlinearity should be reflected in the corresponding change detection methods. In this dissertation we present a change detection method for nonlinear stochastic systems based on projection particle filtering. The statistic for this method is chosen in such a way that it can be calculated recursively, while the computational complexity of the method remains constant with respect to time. We present some simulation results that show the advantages of this method compared to linearization techniques.