Mitigating the curse of dimensionality: sparse grid characteristics method for optimal feedback control and HJB equations
Abstract
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.
Description
The article of record as published may be found at http://dx.doi.org/10.1007/s10589-017-9910-0
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.Collections
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