Sparse Collocation Methods for Solving High Dimension PDES in Estimation and Control of Dynamical Systems
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The development of stable optimal feedback control laws and uncertainty propagation through nonlinear systems poses significant challenges in the study of closed loop dynamical systems. This is due to the fact that both problems require the solution of Partial Differential Equations (PDEs) that are multivariate functions of states and time. The optimal feedback solutions are derived from the nonlinear PDE known as Hamilton Jacobi Bellman (HJB) equation whereas the evolution of state Probability Density Function (PDF) is governed by Fokker-Planck-Kolmogorov (FPK) equation. The development of a systematic solution process for the governing PDEs is directly impeded by the curse of dimensionality, whereby the number of spatial variables for both the PDEs is equal to state space dimension. The main focus of this dissertation is to develop a unified approach to solve both HJB and FPK equations in a computationally attractive manner, while exploiting recent advances in sparse approximation and non-product quadrature rules. To accomplish this, the recently developed Conjugate Unscented Transformation (CUT) technique is used in conjunction with a sparse approximation method to construct a framework to drive the uncertainty propagation and optimal feedback control realization processes. The suite of developed algorithms is tested on several benchmark engineering problems to demonstrate the efficacy of the proposed research.