基于Koopman算子与哈密尔顿-雅可比-贝尔曼方程的数据驱动最优控制
Data-Driven optimal control via Koopman operators and Hamilton-Jacobi-Bellman equations
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中文总结 AI 辅助
本文提出数据驱动稳定流形（DD-SM）方法，结合Koopman算子与HJB方程，从原始轨迹数据实现最优反馈控制，经改进范德波尔振子实验验证有效，可1毫秒内输出控制信号。
中文摘要 AI 辅助
本文提出一种数据驱动稳定流形（DD-SM）方法，该方法将Koopman算子表示学习与哈密尔顿-雅可比-贝尔曼（HJB）方程的几何稳定流形方法相结合，可在无需系统动力学先验知识的情况下，从原始轨迹数据实现端到端的最优反馈控制综合。我们在统一对称子空间分解（SSD）与扩展动态模式分解（EDMD）框架下构建增广控制系统，联合近似漂移场、控制矩阵及其空间导数，并推导不变与非不变字典空间的概率有限样本误差界，以得到可证准确的HJB方程近似特征系统。通过李雅普诺夫-佩龙算子与常微分方程扰动分析，我们证明数据驱动稳定流形随训练数据增加实现单调递减的半全局误差；进一步建立闭环指数稳定性并量化最优性间隙，两者均可通过提升模型精度收紧。我们开发了含自适应数据生成与深度神经近似的高效算法管线，可在1毫秒内输出控制信号。对改进范德波尔振子的实验验证了该方法的有效性。
英文摘要
This paper presents a data-driven stable manifold (DD-SM) method, which integrates Koopman operator representation learning with the geometric stable manifold approach to Hamilton-Jacobi-Bellman (HJB) equations, enabling end-to-end optimal feedback control synthesis from raw trajectory data without prior knowledge of system dynamics. We construct an augmented control system under a unified symmetric subspace decomposition (SSD) and extended dynamic mode decomposition (EDMD) framework for joint approximation of the drift field, control matrix and their spatial derivatives, and derive probabilistic finite-sample error bounds for invariant and non-invariant dictionary spaces to yield a provably accurate approximate characteristic system of HJB equation. Via Lyapunov-Perron operator and ODE perturbation analysis, we prove the data-driven stable manifold achieves monotonically decreasing semi-global error with growing training data. We further establish closed-loop exponential stability and quantify the optimality gap, both tightenable by refining model accuracy. An efficient algorithm pipeline with adaptive data generation and deep neural approximation is developed, outputting control signals within 1 millisecond. Experiments on a modified van der Pol oscillator verify the effectiveness of our method.
