Watermarking-Based Protection Strategy Against Stealthy Sensor Attacks on Zonotopic Set-Membership State Estimation

In this article, the zonotopic set-membership state estimation (ZSMSE) for cyber-physical systems (CPSs) is investigated in the presence of stealthy false data injection (FDI) sensor attacks. A set-membership state estimation approach is established in which the true state is enclosed by a parallelotope (a special class of zonotope) at each sampling instant. Due to the openness of the network transmission process, malicious attackers have the ability to modify sensor measurement information by injecting false data. Therefore, from the perspective of the defender, the condition for an attack to bypass the detector and destroy the ZSMSE is discussed. Specifically, the stealthiness definition for the FDI sensor attacks and the vulnerability definition for ZSMSE are given. Moreover, the necessary and sufficient condition for the ZSMSE to be vulnerable is derived. Subsequently, a watermarking-based protection strategy is proposed for vulnerable ZSMSE to ensure the validity of state estimation results under stealthy attacks. The watermarking matrix is designed to break the stealthiness of the attack while ensuring the convergence of the ZSMSE. Finally, a series of illustrative examples are provided to demonstrate the effectiveness of the proposed strategy.