Optimal filter design for cyber-physical systems under stealthy hybrid attacks

This article studies the optimal filtering problem for cyber-physical systems under stealthy hybrid cyber-attacks. Due to the stealthiness of cyber-attacks, the optimal filter (OF) consists of an exponentially increasing number of terms, which makes it computationally infeasible in practice. To address this computational issue, an optimal linear filter (OLF) is developed with its estimation performance fairly close to the optimal one. Furthermore, for a system under the stealthy hybrid cyber-attacks, it is proved that when the system is stable, the exponentially increasing number of terms remain bounded, making the OF stable; and that the OLF is stable if and only if the system is stable. Finally, numerical examples are employed to illustrate the effectiveness of the proposed two filters.