Feedback particle filter and its applications

The purpose of nonlinear filtering is to extract useful information from noisy sensor data. It finds applications in all disciplines of science and engineering, including tracking and navigation, traffic surveillance, financial engineering, neuroscience, biology, robotics, computer vision, weather forecasting, geophysical survey and oceanology, etc. This thesis is particularly concerned with the nonlinear filtering problem in the continuous-time continuous-valued state-space setting (diffusion). In this setting, the nonlinear filter is described by the Kushner-Stratonovich (K-S) stochastic partial differential equation (SPDE). For the general nonlinear non-Gaussian problem, no analytical expression for the solution of the SPDE is available. For certain special cases, finite-dimensional solution exists and one such case is the Kalman filter. The Kalman filter admits an innovation error-based feedback control structure, which is important on account of robustness, cost efficiency and ease of design, testing and operation. The limitations of Kalman filters in applications arise because of nonlinearities, not only in the signal models but also in the observation models. For such cases, Kalman filters are known to perform poorly. This motivates simulation-based methods to approximate the infinite-dimensional solution of the K-S SPDE. One popular approach is the particle filter, which is a Monte Carlo algorithm based on sequential importance sampling. Although it is potentially applicable to a general class of nonlinear non-Gaussian problems, the particle filter is known to suffer from several well-known drawbacks, such as particle degeneracy, curse of dimensionality, numerical instability and high computational cost. The goal of this dissertation is to propose a new framework for nonlinear filtering, which introduces the innovation error-based feedback control structure to the particle filter. The proposed filter is called the feedback particle filter (FPF). The first part of this dissertation covers the theory of the feedback particle filter. The filter is defined by an ensemble of controlled, stochastic, dynamic systems (the “particles”). Each particle evolves under feedback control based on its own state, and the empirical distribution of the ensemble. The feedback control law is obtained as the solution to a variational problem, in which the optimization criterion is the Kullback-Leibler divergence between the actual posterior, and the common posterior of any particle. The following conclusions are obtained for diffusions with continuous observations: 1) The optimal control solution is exact: The two posteriors match exactly, provided they are initialized with identical priors. 2) The optimal filter admits an innovation error-based gain feedback structure. 3) The optimal feedback gain is obtained via a solution of an Euler-Lagrange boundary value problem; the feedback gain equals the Kalman gain in the linear Gaussian case. The feedback particle filter offers significant variance improvements when compared to the bootstrap particle filter; in particular, the algorithm can be applied to systems that are not stable. No importance sampling or resampling is required, and therefore the filter does not suffer sampling-related issues and incurs low computational burden. The theory of the feedback particle filter is first developed for the continuous-time continuous-valued state-space setting. Its extensions to two other settings are also studied in this dissertation. In particular, we introduce feedback particle filter-based solutions for: i) estimating a continuous-time Markov chain with noisy measurements, and ii) the continuous-discrete time filtering problem. Both algorithms are shown to admit an innovation error-based feedback control structure. The second part of this dissertation concerns the extensions of the feedback particle filter algorithms to address additional uncertainties. In particular, we consider the nonlinear filtering problem with i) model uncertainty, and ii) data association uncertainty. The corresponding feedback particle filter algorithms are referred to as the interacting multiple model-feedback particle filter (IMM-FPF) and the probabilistic data association-feedback particle filter (PDA-FPF). The proposed algorithms are shown to be the nonlinear non-Gaussian generalization of their classic Kalman filter- based counterparts. One remarkable conclusion is that the proposed IMM-FPF and PDA-FPF algorithm retains the innovation error-based feedback structure even for the nonlinear non-Gaussian case. The results are illustrated with the aid of numerical simulations