Approximate Nonlinear Filtering with Applications to Navigation

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In this dissertation we address nonlinear techniques in filtering,estimation, and detection that arise in satellite basednavigation. Here, we emphasize the theoretical aspect of thesetechniques, and we also address their applications.

We first introduce particle filtering for an exponential family ofdensities. We prove that under certain conditions the approximatedconditional density converges to the true conditional density. Forthe case where the conditional density does not lie in anexponential family but stays close to it, we show that undercertain assumptions the error of the estimate given by thisapproximate nonlinear filtering, {em projection particlefiltering}, is bounded. We give similar results for a family ofmixture densities. We use projection particle filtering for anexponential family of densities to estimate the position of amobile platform that has a combination of inertial navigationsystem (INS) and global positioning system (GPS), referred to asan integrated INS/GPS. We show via numerical experiments thatprojection particle filtering exceeds regular particle filteringmethods in navigation performance.

Using carrier phase measurements enables the differential GPS toreach centimeter level accuracy. The phase lock loop of a GPSreceiver cannot measure the full cycle part of the carrier phase.This unmeasured part is called {em integer ambiguity}, and itshould be resolved through other means. Here, we present a newinteger ambiguity resolution method. In this method we treat theinteger ambiguity as a random digital vector. Using particlefiltering, we approximate the conditional probability massfunction of the integer ambiguity given the observation. Theresolved integer is the MAP estimate of the integer given theobservation.

Reliability of a positioning system is of great importance fornavigation purposes. Therefore, an integrity monitoring system isan inseparable part of any navigation system. Failures or changesdue to malfunctions in sensors and actuators should be detectedand repaired to keep the integrity of the system intact. Since inmost practical applications, sensors and actuators have nonlineardynamics, this nonlinearity should be reflected in thecorresponding change detection methods. In this dissertation wepresent a change detection method for nonlinear stochastic systemsbased on projection particle filtering. The statistic for thismethod is chosen in such a way that it can be calculatedrecursively, while the computational complexity of the methodremains constant with respect to time. We present some simulationresults that show the advantages of this method compared tolinearization techniques.

Keywords: Nonlinear Filtering, Particle Filtering, Monte Carlo Method, Abrupt Change Detection, Navigation, INS, GPS, Integer Ambiguity Resolution, Cycle Slip Detection