Observability of linear systems under adversarial attacks

We address the problem of state estimation for multi-output continuous-time linear systems, for which an attacker may have control over some of the sensors and inject (potentially unbounded) additive noise into some of the measured outputs. To characterize the resilience of a system against such sensor attacks, we introduce a new notion of observability - termed “observability under attacks” - that addresses the question of whether or not it is possible to uniquely reconstruct the state of the system by observing its inputs and outputs over a period of time, with the understanding that some of the available system´s outputs may have been corrupted by the opponent. We provide computationally efficient tests for observability under attacks that amount to testing the (standard) observability for an appropriate finite set of systems. In addition, we propose two state estimation algorithms that permit the state reconstruction in spite of the attacks. One of these algorithms uses observability Gramians and a finite window of measurements to reconstruct the initial state. The second algorithm takes the form of a switched observer that asymptotically converges to the correct state estimate in the absence of additive noise and disturbances, or to a neighborhood of the correct state estimate in the presence of bounded noise and disturbances.