EKF-SINDy：以非线性动力学的稀疏辨识增强扩展卡尔曼滤波器
EKF-SINDy: Empowering the extended Kalman filter with sparse identification of nonlinear dynamics
AI总结：
本文提出将EKF与SINDy结合，用数据驱动方式高效辨识非线性系统动力学，并通过时间延迟嵌入处理部分观测问题，为预测性数字孪生奠定基础。
摘要：
来自动力系统的测量数据可以通过卡尔曼滤波器同化到预测模型中。卡尔曼滤波器的非线性扩展，例如扩展卡尔曼滤波器（EKF），需要能够联合估计（可能非线性的）系统动力学和输入参数。为了构建EKF预测阶段所使用的演化模型，我们提出依赖非线性动力学的稀疏辨识（SINDy）。SINDy能够直接从预先获取的数据中辨识演化模型，从而避免由于错误假设和对系统动力学的不正确建模而可能产生的偏差。此外，与基于例如有限元的替代策略相比，SINDy模型的数值积分可以带来巨大的计算节省。最后，SINDy允许立即定义EKF辨识系统动力学和特性所需的雅可比矩阵，而这种推导对于物理模型通常极其复杂。因此，将EKF与SINDy相结合，为非线性系统的辨识提供了一种数据驱动、计算高效且易于应用的方法，即使在SINDy训练范围之外也能稳健运行。为了证明该方法的潜力，我们处理了一个线性非自治系统的辨识，该系统由受真实地震图激励的剪切建筑模型组成，以及一个部分观测的非线性系统的辨识。当系统状态不完全可访问时，使用SINDy所带来的挑战已通过时间延迟嵌入得到缓解。状态辨识（其中状态已被增广以包含系统特性）所伴随的高精度和小不确定性凸显了所提出策略的巨大潜力，为在不同领域建立预测性数字孪生铺平了道路。
英文摘要：
Measured data from a dynamical system can be assimilated into a predictive model by means of Kalman filters. Nonlinear extensions of the Kalman filter, such as the Extended Kalman Filter (EKF), are required to enable the joint estimation of (possibly nonlinear) system dynamics and of input parameters. To construct the evolution model used in the prediction phase of the EKF, we propose to rely on the Sparse Identification of Nonlinear Dynamics (SINDy). SINDy enables to identify the evolution model directly from preliminary acquired data, thus avoiding possible bias due to wrong assumptions and incorrect modelling of the system dynamics. Moreover, the numerical integration of a SINDy model leads to great computational savings compared to alternate strategies based on, e.g., finite elements. Last, SINDy allows an immediate definition of the Jacobian matrices required by the EKF to identify system dynamics and properties, a derivation that is usually extremely involved with physical models. As a result, combining the EKF with SINDy provides a data-driven computationally efficient, easy-to-apply approach for the identification of nonlinear systems, capable of robust operation even outside the range of training of SINDy. To demonstrate the potential of the approach, we address the identification of a linear non-autonomous system consisting of a shear building model excited by real seismograms, and the identification of a partially observed nonlinear system. The challenge arising from the use of SINDy when the system state is not entirely accessible has been relieved by means of time-delay embedding. The great accuracy and the small uncertainty associated with the state identification, where the state has been augmented to include system properties, underscores the great potential of the proposed strategy, paving the way for the setting of predictive digital twins in different fields.
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