False Data-Injection Attack Detection in Cyber-Physical Systems: A Wasserstein Distributionally Robust Reachability Optimization Approach

Cyber-physical system (CPS) is the foundational backbone of modern critical infrastructures, so ensuring its security and resilience against cyber-attacks is of pivotal importance. This paper addresses the challenge of designing anomaly detectors for CPS under false-data injection (FDI) attacks and stochastic disturbances governed by unknown probability distribution. By using the Wasserstein ambiguity set, a prevalent data-driven tool in distributionally robust optimization (DRO), we first propose a new security metric to deal with the absence of disturbance distribution. This metric is designed by asymptotic reachability analysis of state deviations caused by stealthy FDI attacks and disturbance in a distributionally robust confidence set. We then formulate the detector design as a DRO problem that optimizes this security metric while controlling the false alarm rate robustly under a set of distributions. This yields a trade-off between robustness to disturbance and performance degradation under stealthy attacks. The resulting design problem turns out to be a challenging semi-infinite program due to the existence of distributionally robust chance constraints. We derive its exact albeit non-convex reformulation and develop an effective solution algorithm based on sequential minimization. Finally, a case study on a simulated three-tank is illustrated to demonstrate the efficiency of our design in robustifying against unknown disturbance distribution.