Controlled particle systems for nonlinear filtering and global optimization

This thesis is concerned with the development and applications of controlled interacting particle systems for nonlinear filtering and global optimization problems. These problems are important in a number of engineering domains. In nonlinear filtering, there is a growing interest to develop geometric approaches for systems that evolve on matrix Lie groups. Examples include the problem of attitude estimation and motion tracking in aerospace engineering, robotics and computer vision. In global optimization, the challenges typically arise from the presence of a large number of local minimizers as well as the computational scalability of the solution. Gradient-free algorithms are attractive because in many practical situations, evaluating the gradient of the objective function may be computationally prohibitive. The thesis comprises two parts that are devoted to theory and applications, respectively. The theoretical part consists of three chapters that describe methods and algorithms for nonlinear filtering, global optimization, and numerical solutions of the Poisson equation that arise in both filtering and optimization. For the nonlinear filtering problem, the main contribution is to extend the feedback particle filter (FPF) algorithm to connected matrix Lie groups. In its general form, the FPF is shown to provide an intrinsic coordinate-free description of the filter that automatically satisfies the manifold constraint. The properties of the original (Euclidean) FPF, especially the gain-times-error feedback structure, are preserved in the generalization. For the global optimization problem, a controlled particle filter algorithm is introduced to numerically approximate a solution of the global optimization problem. The theoretical significance of this work comes from its variational aspects: (i) the proposed particle filter is a controlled interacting particle system where the control input represents the solution of a mean-field type optimal control problem; and (ii) the associated density transport is shown to be a gradient flow (steepest descent) for the optimal value function, with respect to the Kullback--Leibler divergence. For both the nonlinear filtering and optimization problems, the numerical implementation of the proposed algorithms require a solution of a Poisson equation. Two numerical algorithms are described for this purpose. In the Galerkin scheme, the gain function is approximated using a set of pre-defined basis functions; In the kernel-based scheme, a numerical solution is obtained by solving a certain fixed-point equation. Well-posedness results for the Poisson equation are also discussed. The second part of the thesis contains applications of the proposed algorithms to specific nonlinear filtering and optimization problems. The FPF is applied to the problem of attitude estimation - a nonlinear filtering problem on the Lie group SO(3). The formulae of the filter are described using both the rotation matrix and the quaternion coordinates. A comparison is provided between FPF and the several popular attitude filters including the multiplicative EKF, the invariant EKF, the unscented Kalman filter, the invariant ensemble Kalman filter and the bootstrap particle filter. Numerical simulations are presented to illustrate the comparison. As a practical application, experimental results for a motion tracking problem are presented. The objective is to estimate the attitude of a wrist-worn motion sensor based on the motion of the arm. In the presence of motion, considered here as the swinging motion of the arm, the observability of the sensor attitude is shown to improve. The estimation problem is mathematically formulated as a nonlinear filtering problem on the product Lie group SO(3)XSO(2), and experimental results are described using data from the gyroscope and the accelerometer installed on the sensor. For the global optimization problem, the proposed controlled particle filter is compared with several model-based algorithms that also employ probabilistic models to inform the search of the global minimizer. Examples of the model-based algorithms include the model reference adaptive search, the cross entropy, the model-based evolutionary optimization, and two algorithms based on bootstrap particle filtering. Performance comparisons are provided between the control-based and the sampling-based implementation. Results of Monte-Carlo simulations are described for several benchmark optimization problems