Malicious data attacks against dynamic state estimation in the presence of random noise

State estimation in a discrete-time linear dynamical system is considered in the presence of random process noise, in addition to a malicious adversary able to manipulate some measurements of the system. The adversary has access to only a subset of the measurements, but the particular subset is unknown, and it may adjust these measurements in an arbitrary fashion. A specific attack is proposed that gives a lower bound on the mean squared error for any estimator. Two estimators are proposed; one based on a non-convex optimization problem using sparsity constraints, the second a convex relaxation using a mixed ℓ₁/ℓ₂ norm. The performance of both estimators are studied using simulations.