Abstract
Bayesian filtering refers to the process of sequentially estimating the current state of a complex dynamic system from noisy partial measurements using Bayes' rule. This thesis considers Bayesian filtering as applied to an important class of state estimation problems, which is describable by a discrete-time nonlinear state-space model with additive Gaussian noise. It is known that the conditional probability density of the state given the measurement history or simply the posterior density contains all information about the state. For nonlinear systems, the posterior density cannot be described by a finite number of sufficient statistics, and an approximation must be made instead. The approximation of the posterior density is a challenging problem that has engaged many researchers for over four decades. Their work has resulted in a variety of approximate Bayesian filters . Unfortunately, the existing filters suffer from possible divergence, or the curse of dimensionality, or both, and it is doubtful that a single filter exists that would be considered effective for applications ranging from low to high dimensions. The challenge ahead of us therefore is to derive an approximate nonlinear Bayesian filter, which is theoretically motivated, reasonably accurate, and easily extendable to a wide range of applications at a minimal computational cost. A journey is easier when we travel together. Interdependence is certainly more valu able than independence. This dissertation is the result of the last three years of work whereby I have been accompanied and supported by many people. • First and foremost , I would like to thank my supervisor Professor Simon Haykin for providing me with an opportunity to undertake a research topic that is truly challenging, intriguing, and important in the field of estimation and control. I am deeply indebted to him for his superb advice 1 Introduction 1 1.1 Problem Statement 1 1.2 Contributions ... 3 1.3 Organization of the Thesis 5 1.4 Related Publications 6 2 Literature Review 8 2.1 Optimal Bayesian Filter 9 2.1.1 Time update .. 11 2.1.2 Measurement Update . 12 2.2 Moment-Matching Algorithms 15 2.3 Innovations-Based Algorithms 22 3 Theory of Cubature Rules 27 3.1 Product Rules ..... . 28 In this thesis, a new approximate Bayesian filter is derived for discrete-time non linear filtering problems, which is named the cubature Kalman fil,ter. To develop this filter, it is assumed that the predictive density of the joint state-measurement random variable is Gaussian. In this way, the optimal Bayesian filter reduces to the problem of how to compute various multi-dimensional Gaussian-weighted moment integrals. To numerically compute these integrals, a third-degree spherical-radial cubature rule is proposed. This cubature rule entails a set of cubature points scaling linearly with the state-vector dimension. The cubature Kalman filter therefore provides an efficient solution even for high-dimensional nonlinear filtering problems. More remarkably, the cubature Kalman filter is the closest known approximate filter in the sense of com pletely preserving second-order information due to the maximum entropy principle. For the purpose of mitigating divergence, and improving numerical accuracy in sys tems where there are apparent computer roundoff difficulties, the cubature Kalman filter is reformulated to propagate the square roots of the error-covariance matrices.
Showing the abstract — retrieve the full paper via the Exa API.