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.

中文翻译：

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EKF–SINDy：通过非线性动力学的稀疏识别增强扩展卡尔曼滤波器


来自动态系统的测量数据可以通过卡尔曼滤波器吸收到预测模型中。需要卡尔曼滤波器的非线性扩展，例如扩展卡尔曼滤波器 (EKF)，才能实现（可能是非线性的）系统动力学和输入参数的联合估计。为了构建 EKF 预测阶段使用的演化模型，我们建议依赖非线性动力学的稀疏辨识（SINDy）。 SINDy 能够直接从初步获取的数据中识别演化模型，从而避免由于错误的假设和不正确的系统动力学建模而可能出现的偏差。此外，与基于有限元等替代策略相比，SINDy 模型的数值积分可节省大量计算量。最后，SINDy 允许立即定义 EKF 所需的雅可比矩阵，以识别系统动力学和属性，这种推导通常与物理模型密切相关。因此，将 EKF 与 SINDy 相结合提供了一种数据驱动的计算高效、易于应用的方法来识别非线性系统，即使在 SINDy 的训练范围之外也能够稳健运行。为了证明该方法的潜力，我们解决了由真实地震图激发的剪切建筑模型组成的线性非自治系统的识别，以及部分观测的非线性系统的识别。当系统状态不可完全访问时使用 SINDy 所带来的挑战已通过延时嵌入得到缓解。 与状态识别相关的高精度和小的不确定性，其中状态已被增强以包括系统属性，强调了所提出的策略的巨大潜力，为不同领域的预测数字孪生的设置铺平了道路。