Prevailing against Adversarial Noncentral Disturbances: Exact Recovery of Linear Systems with the $l_1$-norm Estimator

TL;DR

This paper demonstrates that the $l_1$-norm estimator can exactly recover linear systems affected by adversarial and noncentral disturbances, outperforming traditional methods under certain probabilistic conditions.

Contribution

It introduces conditions under which the $l_1$-norm estimator guarantees exact system recovery despite adversarial and nonzero-mean disturbances, extending robustness in system identification.

Findings

The $l_1$-norm estimator accurately identifies systems with noncentral disturbances when disturbances are sign-balanced.

It achieves exact recovery against adversarial disturbances when attack probability is below 0.5.

The method provides finite-time exact recovery in the presence of arbitrary large noncentral attacks.

Abstract

This paper studies the linear system identification problem in the general case where the disturbance is sub-Gaussian, correlated, and possibly adversarial. First, we consider the case with noncentral (nonzero-mean) disturbances for which the ordinary least-squares (OLS) method fails to correctly identify the system. We prove that the $l_1$-norm estimator accurately identifies the system under the condition that each disturbance has equal probabilities of being positive or negative. This condition restricts the sign of each disturbance but allows its magnitude to be arbitrary. Second, we consider the case where each disturbance is adversarial with the model that the attack times happen occasionally but the distributions of the attack values are arbitrary. We show that when the probability of having an attack at a given time is less than 0.5 and each attack spans the entire space in expectation, the $l_1$-norm estimator prevails against any adversarial noncentral disturbances and the exact recovery is achieved within a finite time. These results pave the way to effectively defend against arbitrarily large noncentral attacks in safety-critical systems.