论文标题
最佳反馈法律通过梯度提高的稀疏多项式回归恢复
Optimal Feedback Law Recovery by Gradient-Augmented Sparse Polynomial Regression
论文作者
论文摘要
提出了用于计算确定性非线性控制中产生的高维最佳反馈定律的稀疏回归方法。该方法利用了汉密尔顿 - 雅各比 - 贝尔曼PDE之间的控制理论链接，该PDE表征了最佳控制问题的价值函数，以及通过Pontryagin的最大原理来表征最佳控制问题的价值函数。后者通过两点边界值问题的解决方案在时空域中的任意点恢复了值函数及其梯度的表示公式。在生成由不同状态值对组成的数据集后，使用套索回归拟合了值函数的双曲线交叉多项式模型。非线性最佳控制中的一组延长的低维数值测试表明，使用梯度信息丰富数据集可减少训练样本的数量，并且稀疏的多项式回归始终产生较低复杂性的反馈定律。
A sparse regression approach for the computation of high-dimensional optimal feedback laws arising in deterministic nonlinear control is proposed. The approach exploits the control-theoretical link between Hamilton-Jacobi-Bellman PDEs characterizing the value function of the optimal control problems, and first-order optimality conditions via Pontryagin's Maximum Principle. The latter is used as a representation formula to recover the value function and its gradient at arbitrary points in the space-time domain through the solution of a two-point boundary value problem. After generating a dataset consisting of different state-value pairs, a hyperbolic cross polynomial model for the value function is fitted using a LASSO regression. An extended set of low and high-dimensional numerical tests in nonlinear optimal control reveal that enriching the dataset with gradient information reduces the number of training samples, and that the sparse polynomial regression consistently yields a feedback law of lower complexity.
