Stealthy bias injection attack detection based on Kullback-Leibler divergence in stochastic linear systems

Abstract

This paper studies the design of detection observers against stealthy bias injection attacks in stochastic linear systems under Gaussian noise, considering adversaries that exploit noise and inject crafted bias signals into a subset of sensors in a slow and coordinated manner, thereby achieving malicious objectives while remaining stealthy. To address such attacks, we formulate the observer design as a max-min optimization problem to enhance the detectability of worst-case BIAs, which attain a prescribed attack impact with the least detectability evaluated via Kullback-Leibler divergence. To reduce the computational complexity of the derived non-convex design problem, we consider the detectability of worst-case BIAs at three specific time instants: attack onset, one step after attack occurrence, and the steady state. We prove that the Kalman filter is optimal for maximizing the BIA detectability at the attack onset, regardless of the subset of attacked sensors. For the one-step and steady-state cases, the observer design problems are approximated by bi-convex optimization problems, which can be efficiently solved using alternating optimization and alternating direction method of multipliers. Moreover, more tractable linear matrix inequality relaxations are developed. Finally, the effectiveness of the proposed stealth-aware detection framework is demonstrated through an application to a thermal system.