Kumar, Mrinal (2009-12). Design and Analysis of Stochastic Dynamical Systems with Fokker-Planck Equation. Doctoral Dissertation.
This dissertation addresses design and analysis aspects of stochastic dynamical systems using Fokker-Planck equation (FPE). A new numerical methodology based on the partition of unity meshless paradigm is developed to tackle the greatest hurdle in successful numerical solution of FPE, namely the curse of dimensionality. A local variational form of the Fokker-Planck operator is developed with provision for h- and p- refinement. The resulting high dimensional weak form integrals are evaluated using quasi Monte-Carlo techniques. Spectral analysis of the discretized Fokker- Planck operator, followed by spurious mode rejection is employed to construct a new semi-analytical algorithm to obtain near real-time approximations of transient FPE response of high dimensional nonlinear dynamical systems in terms of a reduced subset of admissible modes. Numerical evidence is provided showing that the curse of dimensionality associated with FPE is broken by the proposed technique, while providing problem size reduction of several orders of magnitude. In addition, a simple modification of norm in the variational formulation is shown to improve quality of approximation significantly while keeping the problem size fixed. Norm modification is also employed as part of a recursive methodology for tracking the optimal finite domain to solve FPE numerically. The basic tools developed to solve FPE are applied to solving problems in nonlinear stochastic optimal control and nonlinear filtering. A policy iteration algorithm for stochastic dynamical systems is implemented in which successive approximations of a forced backward Kolmogorov equation (BKE) is shown to converge to the solution of the corresponding Hamilton Jacobi Bellman (HJB) equation. Several examples, including a four-state missile autopilot design for pitch control, are considered. Application of the FPE solver to nonlinear filtering is considered with special emphasis on situations involving long durations of propagation in between measurement updates, which is implemented as a weak form of the Bayes rule. A nonlinear filter is formulated that provides complete probabilistic state information conditioned on measurements. Examples with long propagation times are considered to demonstrate benefits of using the FPE based approach to filtering.