LQG reference tracking with safety and reachability guarantees under false data injection attacks

Control systems are increasingly targeted by malicious adversaries, who may inject spurious sensor measurements in order to bias the controller behavior and cause suboptimal performance or safety violations. This paper investigates the problem of tracking a reference trajectory while satisfying safety and reachability constraints in the presence of such false data injection attacks. We consider a linear, time-invariant system with additive Gaussian noise in which a subset of sensors can be compromised by an attacker, while the remaining sensors are regarded as secure. We propose a control policy in which two estimates of the system state are maintained, one based on all sensors and one based on only the secure sensors. The optimal control action based on the secure sensors alone is then computed at each time step, and the chosen control action is constrained to lie within a given distance of this value. We show that this policy can be implemented by solving a quadratically-constrained quadratic program at each time step. We develop a barrier function approach to choosing the parameters of our scheme in order to provide provable guarantees on safety and reachability, and derive bounds on the probability that our control policies deviate from the optimal policy when no attacker is present. Our framework is validated through numerical study.