Data-driven discovery of linear dynamical systems from noisy data

In modern science and engineering disciplines, data-driven discovery methods play a fundamental role in system modeling, as data serve as the external representations of the intrinsic mechanisms within systems. However, empirical data contaminated by process and measurement noise remain a significant obstacle for this type of modeling. In this study, we have developed a data-driven method capable of directly uncovering linear dynamical systems from noisy data. This method combines the Kalman smoothing and sparse Bayesian learning to decouple process and measurement noise under the expectation-maximization framework, presenting an analytical method for alternate state estimation and system identification. Furthermore, the discovered model explicitly characterizes the probability distribution of process and measurement noise, as they are essential for filtering, smoothing, and stochastic control. We have successfully applied the proposed algorithm to several simulation systems. Experimental results demonstrate its potential to enable linear dynamical system discovery in practical applications where noise-free data are intractable to capture.