Mitigating the curse of dimensionality: sparse grid characteristics method for optimal feedback control and HJB equations
Wilcox, Lucas C.
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We address finding the semi-global solutions to optimal feedback control and the Hamilton–Jacobi–Bellman (HJB) equation. Using the solution of an HJB equation, a feedback optimal control law can be implemented 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 approximate 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 three examples of rigid body attitude control using momentum wheels.
The article of record as published may be found at http://dx.doi.org/10.1007/s10589-017-9910-0
RightsThis 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.
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Mitigating the Curse of Dimensionality: Sparse Grid Characteristics Method for Optimal Feeback Control and HJB Equations Kang, Wei; Wilcox, Lucas C. (2015-07);We address finding the semi-global solutions to optimal feedback con- 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 ...
Wilcox, Lucas; Kang, Wei (2015-01);Solving Hamilton-Jacobi-Bellman (HJB) equations is essential in feedback optimal con- trol. Using the solution of HJB equations, feedback optimal control laws can be imple- mented in real-time with minimum computational ...
Wilcox, Lucas; Kang, Wei (2015-12);It is well known that solving the Hamilton-Jacobi- Bellman (HJB) equation in moderate and high dimensions (d > 3) suffers the curse of dimensionality. In this paper, we introduce and demonstrate an example of solving ...