Computational Optimization and Applications

, Volume 68, Issue 2, pp 289–315 | Cite as

Mitigating the curse of dimensionality: sparse grid characteristics method for optimal feedback control and HJB equations

Article

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.

Keywords

Optimal feedback control Hamilton–Jacobi–Bellman equation Sparse grid Method of characteristics Rigid body attitude control 

Notes

Acknowledgements

This work was supported in part by AFOSR, NRL, DARPA, and CRUSER of Naval Postgraduate School. Thanks to Carlos F. Borges for his time discussing terminology and the naming of the method. Thanks to Arthur J. Krener for his insights on HJB solutions. Thanks to Lars Grüne for his comments on the parallel computation of reachable set [18].

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Copyright information

© Springer Science+Business Media New York (outside the USA) 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsNaval Postgraduate SchoolMontereyUSA

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