A Framework for Joint Attack Detection and Control Under False Data Injection

In this work, we consider an LTI system with a Kalman filter, detector, and Linear Quadratic Gaussian (LQG) controller under false data injection attack. The interaction between the controller and adversary is captured by a Stackelberg game, in which the controller is the leader and the adversary is the follower. We propose a framework under which the system chooses time-varying detection thresholds to reduce the effectiveness of the attack and enhance the control performance. We model the impact of the detector as a switching signal, resulting in a switched linear system. A closed form solution for the optimal attack is first computed using the proposed framework, as the best response to any detection threshold. We then present a convex program to compute the optimal detection threshold. Our approach is evaluated using a numerical case study.