Kullback-Leibler divergence-based filter design against bias injection attacks

Cyber-physical systems (CPS) are increasingly deployed in safety-critical applications, making them prime targets for adversarial attacks. Timely detection and mitigation of such attacks are imperative for the safe operation of CPS. This paper proposes a novel residual generator design method for enhanced detection of bias injection attacks (BIAs) in linear CPS driven by white Gaussian noise. Specifically, we define a flexible attack impact metric based on the weighted norm of the injected bias and a detectability metric based on the Kullback-Leibler divergence. Using these two metrics, we characterize the worst-case BIAs as those that minimize detectability while maintaining a specified minimum impact. For residual generation filter synthesis, we formulate two optimization problems: one for maximizing the detectability of worst-case BIAs at the attack onset and the other at steady state. Since these two problems are inherently conflicting, we employ the ϵ-constraint method to obtain Pareto-optimal solutions that balance transient and steady-state detectability. The effectiveness of the proposed filter design method is demonstrated through numerical simulations, with a comparison against two state-of-the-art benchmarks: the Kalman filter and the H_ / H2 filter.