Worst-case performance of Kalman filtering against cyber-attacks with true and contaminated information

In this work, we study the worst-case performance of Kalman filtering under the attacks in the framework of multi-sensor with and without safe sensors. The specific setup is that a remote estimator with ${\chi }^2$ detectors performs the state estimation based on data transmitted from a group of sensors via a wireless network. The transmitted data may be modified by an adversary. The previous literature mainly investigates potential impacts of several feasible stealthy integrity attacks on remote estimator in the multi-sensor setup instead of the worst-case estimation performance. It should be pointed out that the analysis of the worst-case performance of remote estimator under the attacks can provide a reference for operators to identify the vulnerability of systems and decide whether corresponding defensive measures need to be executed. Hence, such analyses play an importance role in ensuring system security. In this work, a cyber-attack pattern utilizing true and contaminated information from all sensors is developed. The worst-case estimation performance and corresponding strictly stealthy attack policies in closed-form are provided. Finally, a numerical example is given to verify the effectiveness of the given results.