将稀疏回归与卡尔曼滤波结合，实现实时动态系统建模。
On-line learning of dynamic systems: sparse regression meets Kalman filtering
- 把未知参数当状态变量，用卡尔曼滤波在线估计。
- 在洛伦兹系统和真实飞行数据上成功识别时变非线性模型。
- 无需手动调参，自动估计稀疏度和切换点，适合实时系统监控。
从数据中学习控制方程是理解物理系统行为的核心，涵盖物理、生物和工程等多个领域。SINDy 算法利用稀疏性有效识别非线性动力系统的简洁模型。本文通过融合控制理论中的基石算法——卡尔曼滤波（KF），将稀疏驱动方法扩展至实时学习。提出的 SINDy 卡尔曼滤波器（SKF）将两者统一：将未知系统参数视为状态变量，实现对复杂时变非线性模型的在线推断，这是单一方法无法达到的。此外，SKF改进了卡尔曼滤波的参数辨识策略，特别是通过前瞻误差显著简化了稀疏度、方差参数及切换时刻的估计。我们在参数漂移或切换的混沌洛伦兹系统上验证了 SKF 的有效性，并展示了其在真实飞行数据构建的稀疏非线性飞机模型上的实时识别能力。
原文摘要 · Abstract (English)
Learning governing equations from data is central to understanding the behavior of physical systems across diverse scientific disciplines, including physics, biology, and engineering. The Sindy algorithm has proven effective in leveraging sparsity to identify concise models of nonlinear dynamical systems. In this paper, we extend sparsity-driven approaches to real-time learning by integrating a cornerstone algorithm from control theory -- the Kalman filter (KF). The resulting Sindy Kalman Filter (SKF) unifies both frameworks by treating unknown system parameters as state variables, enabling real-time inference of complex, time-varying nonlinear models unattainable by either method alone. Furthermore, SKF enhances KF parameter identification strategies, particularly via look-ahead error, significantly simplifying the estimation of sparsity levels, variance parameters, and switching instants. We validate SKF on a chaotic Lorenz system with drifting or switching parameters and demonstrate its effectiveness in the real-time identification of a sparse nonlinear aircraft model built from real flight data.
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