Huber-based Robust System Identification with Near-Optimal Guarantees Across Independent and Adversarial Regimes

Dynamical systems can confront one of two extreme types of disturbances: persistent zero-mean independent noise, and sparse nonzero-mean adversarial attacks, depending on the specific scenario being modeled. While mean-based estimators like least-squares are well-suited for the former, a median-based approach such as the ℓ1-norm estimator is required for the latter. In this paper, we propose a Huber-based estimator, characterized by a threshold constant μ, to identify the governing matrix of a linearly parameterized nonlinear system from a single trajectory of length T. This formulation bridges the gap between mean- and median-based estimation, achieving provably robust error in both extreme disturbance scenarios under mild assumptions. In particular, for persistent zero-mean noise with a positive probability density around zero, the proposed estimator achieves an O(1/√T) error rate if the disturbance is symmetric or the basis functions are linear. For arbitrary nonzero-mean attacks that occur at each time with probability smaller than 0.5, the error is bounded by O(μ). We validate our theoretical results with experiments illustrating that integrating our approach into frameworks like SINDy yields robust identification of discrete-time systems.