缓解模糊性的诅咒：最优反馈控制的稀疏网格特征方法和HJB方程
Mitigating the Curse of Dimensionality: Sparse Grid Characteristics Method for Optimal Feedback Control and HJB Equations
摘要：
我们致力于寻找最优反馈控制和Hamilton-Jacobi-Bellman（HJB）方程的半全局解。利用HJB方程的解，反馈最优控制律可以以最小的计算量实时实现。然而，除了具有两个或三个状态变量的系统之外，由于维数灾难，使用传统的技术来数值地找到一般非线性系统的HJB方程的半全局解是不可行的。在这里，我们提出了一种新的计算方法，找到反馈最优控制和解决HJB方程，这是能够减轻维数灾难。我们不直接离散的HJB方程，而是我们在状态空间中引入一个稀疏的网格，并使用庞特里亚金的最大值原理推导出一组必要条件的形式的边值问题，也被称为特征方程，为每个网格点。使用这种方法，该方法是空间因果关系的自由，它享有在稀疏网格上的完美并行的优点。与密集网格相比，稀疏网格具有显著减小的尺寸，这对于具有相对高维度的系统是可行的，诸如示例中所示的$6$ -D系统。一旦在每个网格点处得到解，则使用高精度多项式插值来逼近任意点处的反馈控制。我们证明了一个上限的近似误差和近似数值。用两个利用动量轮的刚体姿态控制算例说明了这种稀疏网格特征方法。
英文摘要：
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 two examples of rigid body attitude control using momentum wheels.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
