用深度学习将最优控制问题转为分类，实现快速实时控制。
Time-optimal neural feedback control of nilpotent systems as a binary classification problem
- 将最优控制转化为多项式根求解，再通过神经网络学习开关序列。
- 在高维积分器上验证，控制精度高且可实时运行。
- 适合需要快速响应的控制系统设计者参考。
本文提出一种计算方法，用于合成线性幂零系统的时最优反馈控制律。基于饱合控制定理，将时最优轨迹表征为控制切换序列的参数依赖多项式系统。采用消去型牛顿法遍历求解该多项式系统的实根，根数估计由赫尔米特二次型提供精确上界。第二部分中，对多项式系统采样求解，构建合成数据集，通过监督学习训练深度神经网络——作为二分类器——生成近似控制律。在维度递增的积分器上进行数值测试，评估了近似控制律的精度、鲁棒性及实时控制能力。
原文摘要 · Abstract (English)
A computational method for the synthesis of time-optimal feedback control laws for linear nilpotent systems is proposed. The method is based on the use of the bang-bang theorem, which leads to a characterization of the time-optimal trajectory as a parameter-dependent polynomial system for the control switching sequence. A deflated Newton's method is then applied to exhaust all the real roots of the polynomial system. The root-finding procedure is informed by the Hermite quadratic form, which provides a sharp estimate on the number of real roots to be found. In the second part of the paper, the polynomial systems are sampled and solved to generate a synthetic dataset for the construction of a time-optimal deep neural network -- interpreted as a binary classifier -- via supervised learning. Numerical tests in integrators of increasing dimension assess the accuracy, robustness, and real-time-control capabilities of the approximate control law.
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