Research On Some Problems In Bayesian Filtering

In the last decades, Bayesian Filtering achieved great progress, and has been successfully applied in many different fields, e.g. signal processing, target tracking, and finance. However, it still exists many problems need to be solved. In this dissertation, some problems on Bayesian Filtering are studied. The main contributions are as follows:(1) Despite many theoretical advances have been reported in the last decade, the study of the convergence properties of Particle Filters (PF) is still an open problem. In this dissertation, the almost sure convergence of the Generic Particle Filter (GPF) is discussed in a outfrank way. First, a Modified Generic Particle Filter (MGPF) is constructed. Different from the GPF, the MGPF will determine whether it is necessary to rerun the Importance Sampling (IS) step according to a conditional criterion after performing the IS step at each time. Then the almost sure convergence of MGPF will be concerned. Later, when the recursive time is finite and the interesting function with extended state as independent variable is 4th power integrable with respect to the posterior probability distribution of the extended state, the sufficient condition for GPF estimation converges almost surely to the optimal estimation is discussed. A simulation experiment is designed to illustrate the almost sure convergence of GPF(2) A novel Particle Filter, in which the Unscented Transformation (UT) on the noise space is adopted to construct the Proposal Distribution, is proposed in the frame of the Auxiliary Particle Filter (APF), named Noise Space Unscented Transformation Auxiliary Particle Filter (NSUTAPF). Each particle in traditional UT based PF such as Unscented Particle Filter (UPF) and Unscented Transformation based Auxiliary Particle Filter (UTAPF), denotes a sample of the state sequence. However, in NSUTAPF, each particle denotes a sample of the extended process noise sequence, which is combined by the initial state and the process noise sequence. NSUTAPF has three advantages over the UPF and the UTAPF. Firstly, NSUTAPF doesn't need to make the assumption that the state transition probability distribution is available. Hence, NSUTAPF has wider application scope than UPF and UTAPF. Secondly, NSUTAPF has lower computational cost. Thirdly, each particle in UPF or UTAPF is assumed to have a state covariance, which is inherited from its parent particle, but it is still uncertain whether this assumption is reasonable or not. This assumption can be avoided in NSUTAPF. NSUTAPF achieves better performance when compared with Sampling Importance Resampling (SIR), Gaussian Sum Particle Filter (GSPF), UPF and UTAPF in two simulation experiments.(3) In order to solve the problem of Nonlinear filtering with unknown continuous system parameter, Interacting Multiple Region Model (IMRM) is proposed. IMRM regards the continuous system parameter space as a set of disjoint sub-regions, and each sub-region is assigned to a sub-model respectively. Enlightened by the framework of Interacting Multiple Model (IMM), the interaction and parallel filtering operations are adopted in IMRM for the purpose of responding to the system parameter changing as quickly as possible. IMRM runs a bank of sub-filters in parallel. At each time step, IMRM computes the mixed initial condition for each sub-model by interaction operation, and each sub-filter estimates the state and system parameter on the condition that the system parameter belongs to a unique sub-region. Different from that the system parameter is a fixed value in each sub-filter of IMM, in each sub-filter of IMRM, system parameter is a random variable need to be estimated. Hence, IMRM can achieve better estimation performance than IMM when the true system parameter does not belong to the model set of IMM. In order to implement the IMRM efficiently when the dynamic system is nonlinear, Unscented Transformation based IMRM (UT-IMRM) is proposed. In each sub-filter of UT-IMRM, UKF is adopted to estimate the state and system parameter together. Experiment results show that UT-IMRM achieves higher performance when compared with IMM, SIR and UKF in bearings only tracking problem(4) Since the estimation performance of bayesian filters is largely affected by the observations, sensor management has attracted many researchers in bayesian filtering. State Prior Probability Distribution (SPPD) at each time step plays an important role in the dynamic optimization of sensor parameter. However, before all measurements came, SPPD at each time step is not available in the static optimization. In order to solve this problem, an objective function, which is based on the observation density, is built to evaluate the effect of the sensor parameter, named Minimum Expected Relative Observation Density (MEROD). Considering the problem when the objective function does not have the analytical solution, a numerical method, which is based on Monte Carlo Integration, is proposed. The validity of this numerical method is proved here. A bearings only tracking experiment is implemented to compare the optimized sensor parameter with another four randomly selected sensor parameters. Simulation results show that SIR, GSPF, UPF and NSUTAPF achieved better estimation performance when using the optimized sensor parameter.