dc.contributor.author Kang, Wei dc.contributor.author Wilcox, Lucas C. dc.date.accessioned 2015-07-29T18:26:28Z dc.date.available 2015-07-29T18:26:28Z dc.date.issued 2015-07 dc.identifier.uri http://hdl.handle.net/10945/45647 dc.description.abstract We address finding the semi-global solutions to optimal feedback con- en_US trol and the Hamilton–Jacobi–Bellman (HJB) equation. Using the solu- tion of an HJB equation, a feedback optimal control law can be imple- mented in real-time with minimum computational load. However, except for systems with two or three state variables, using traditional techniques for numerically finding a semi-global solution to an HJB equation for general nonlinear systems is infeasible due to the curse of dimensionality. Here we present a new computational method for finding feedback optimal control and solving HJB equations which is able to mitigate the curse of dimensionality. We do not discretize the HJB equation directly, instead we introduce a sparse grid in the state space and use the Pontryagin’s maximum principle to derive a set of necessary conditions in the form of a boundary value problem, also known as the characteristic equations, for each grid point. Using this approach, the method is spatially causality free, which enjoys the advantage of perfect parallelism on a sparse grid. Compared with dense grids, a sparse grid has a significantly reduced size which is feasible for systems with relatively high dimensions, such as the 6-D system shown in the examples. Once the solution obtained at each grid point, high-order accurate polynomial interpolation is used to approx- imate the feedback control at arbitrary points. We prove an upper bound for the approximation error and approximate it numerically. This sparse grid characteristics method is demonstrated with two examples of rigid body attitude control using momentum wheels. dc.description.sponsorship This work was supported in part by AFOSR, NRL, and CRUSER of Naval Postgraduate School. en_US dc.rights This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States. en_US dc.title Mitigating the Curse of Dimensionality: Sparse Grid Characteristics Method for Optimal Feeback Control and HJB Equations en_US dc.type Article en_US dc.contributor.department Applied Mathematics en_US
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