Sparse Kalman Identification for Partially Observable Systems via Adaptive Bayesian Learning

TL;DR

This work tackles online sparse identification of nonlinear dynamics in partially observable systems. It proposes Sparse Kalman Identification (SKI), which fuses an Augmented Kalman Filter (AKF) with Automatic Relevance Determination (ARD) to learn sparse model structure from sequential data, using an augmented UKF for nonlinearities. A gradient-based ARD update and a KF-like posterior correction enable online relevance learning of basis functions without storing historical data, achieving millisecond-level updates. Extensive simulations and real-world quadrotor experiments demonstrate that SKI yields superior sparsity and identification accuracy (including an $84.21\%$ improvement over the best baseline) while maintaining fast computation, highlighting its practical potential for online control and modeling.

Abstract

Sparse dynamics identification is an essential tool for discovering interpretable physical models and enabling efficient control in engineering systems. However, existing methods rely on batch learning with full historical data, limiting their applicability to real-time scenarios involving sequential and partially observable data. To overcome this limitation, this paper proposes an online Sparse Kalman Identification (SKI) method by integrating the Augmented Kalman Filter (AKF) and Automatic Relevance Determination (ARD). The main contributions are: (1) a theoretically grounded Bayesian sparsification scheme that is seamlessly integrated into the AKF framework and adapted to sequentially collected data in online scenarios; (2) an update mechanism that adapts the Kalman posterior to reflect the updated selection of the basis functions that define the model structure; (3) an explicit gradient-descent formulation that enhances computational efficiency. Consequently, the SKI method achieves accurate model structure selection with millisecond-level efficiency and higher identification accuracy, as demonstrated by extensive simulations and real-world experiments (showing an 84.21\% improvement in accuracy over the baseline AKF).

Figures (12)

• Figure 1: Illustration of polynomial fitting to a quadratic function: (a) underfitting due to an overly simple basis, (b) overfitting resulting from an excessively complex basis, and (c) sparse identification yielding an accurate and parsimonious model.
• Figure 2: Overview of the proposed algorithm: The weight parameters of basis functions are augmented into the state, enabling joint online estimation and adaptive sparsity via ARD within the AKF framework.
• Figure 3: Evolution of the roll angle in the WingRock simulation experiment.
• Figure 4: Estimation results for the coefficients (weight parameters) and their associated 1.96 standard deviation intervals. The red curve denotes the proposed SKI method, the blue curve represents the augmented UKF, and the green curve indicates the augmented EKF. Due to the large estimation error of the SINDy algorithm, its results are omitted for clarity.
• Figure 5: Estimated input gain trajectories for each candidate delay. Blue: Augmented UKF; Red: the proposed SKI method.