ADAPTIVE DEEP LEARNING FOR HIGH-DIMENSIONAL HAMILTON-JACOBI-BELLMAN EQUATIONS
ADAPTIVE DEEP LEARNING FOR HIGH-DIMENSIONAL HAMILTON-JACOBI-BELLMAN EQUATIONS
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DOI：
10.1137/19m1288802
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发表时间：
2021-01-01
影响因子：
3.1
通讯作者：
Kang, Wei
中科院分区：
文献类型：
--
作者：
Nakamura-Zimmerer, Tenavi;Gong, Qi;Kang, Wei
计算非线性系统的最优反馈控制通常需要求解Hamilton-Jacobi-Bellman（HJB）方程，当状态维数很大时，这是非常困难的。现有的高维问题的策略往往依赖于特定的，限制性的问题结构，或仅在局部有效的一些名义轨迹。本文提出了一种数据驱动的方法来逼近一般高维非线性系统的HJB方程的半隐式解，并实时计算候选最优反馈控制。为了实现这一点，我们用神经网络（NN）对HJB方程的解进行建模，神经网络是在不离散状态空间的情况下生成的数据上训练的。通过利用问题的已知物理特性并使用部分训练的NN来帮助自适应数据生成，使训练更加有效和数据高效。我们证明了我们的方法的有效性，通过学习的解决方案HJB方程对应的姿态控制的六维非线性刚体和非线性系统的维数高达30所产生的稳定的Burgers '型偏微分方程。然后将训练好的神经网络用于这些系统的实时反馈控制。
Computing optimal feedback controls for nonlinear systems generally requires solving Hamilton-Jacobi-Bellman (HJB) equations, which are notoriously difficult when the state dimension is large. Existing strategies for high-dimensional problems often rely on specific, restrictive problem structures or are valid only locally around some nominal trajectory. In this paper, we propose a data-driven method to approximate semiglobal solutions to HJB equations for general high-dimensional nonlinear systems and compute candidate optimal feedback controls in real-time. To accomplish this, we model solutions to HJB equations with neural networks (NNs) trained on data generated without discretizing the state space. Training is made more effective and data-efficient by leveraging the known physics of the problem and using the partially trained NN to aid in adaptive data generation. We demonstrate the effectiveness of our method by learning solutions to HJB equations corresponding to the attitude control of a six-dimensional nonlinear rigid body and nonlinear systems of dimension up to 30 arising from the stabilization of a Burgers'-type partial differential equation. The trained NNs are then used for real-time feedback control of these systems.