Innovation-Based Stealthy Attacks on Remote State Estimators Using the Wasserstein Distance

We study the problem of designing an attacking strategy for a remote state estimator of a linear Gaussian cyber-physical system. We focus on an innovation-based attack on a scenario in which a smart sensor transmits innovations via a wireless communication network. We start by considering the attacking scheme in its most general form, and constrain its design so that it remains stealthy to the attack detector at the receiver side. To this end, we consider two stealthiness constraints. First, the attacker should not alter the autocorrelation of the healthy innovation sequence. Second, the attack needs to ensure that the distance between the healthy and attacked probability distributions is smaller than a certain threshold. To measure this distance we use the Wasserstein metric. We then design the optimal attack, in the sense of maximizing the cumulative estimation error covariance, while maintaining the attack stealthy. We show that this attacking scheme can be chosen to be linear and Gaussian, and give its closed form expression. We present numerical experiments showing that the proposed attacking strategy outperforms other rival alternatives on both stable and unstable systems.