A game theoretic approach to analyze false data injection and detection in LQG system

The interplay between an intrusion detection mechanism and a false data injection mechanism is investigated in this work in the context of an infinite horizon Linear Quadratic Gaussian Control System (LQG). A game theoretic framework is proposed between the attacker, who wishes to impair the operation of the control system while remaining stealthy, and the detector, who wishes to detect the presence of an attacker using the control system observations. In an LQG control system, the control input stream is compromised by an attacker who modifies the control signals with the objective of increasing the quadratic cost, and the controller is equipped with the intrusion detection mechanism that monitors the sequence of system states. The problem is formulated as a two player zero-sum game with the false alarm probability as the payoff, wherein the attacker aims to achieve a target increase in controller's cost while keep stealth, and a detector who wishes to minimize the payoff while remaining consistent. It is shown that in such a game an e-equilibrium exists. The equilibrium attacker strategy is the one that minimize the KL-divergence between legitimate and falsified state dynamics, and the equilibrium detector strategy is the corresponding likelihood test. Numerical simulations are presented that showcase the equilibrium strategy pair and the comparison of the equilibrium and naive strategy pairs.