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In this paper, we investigate joint sensor-actuator cyber attacks in discrete event systems. We assume that attackers can attack some sensors and actuators at the same time by altering observations and control commands. Because of the nondeterminism in observation and control caused by cyber attacks, the behavior of the supervised system becomes nondeterministic and may deviate from the safety specification. We define the upper-bound on all possible languages that can be generated by the supervised system to investigate the safety supervisory control problem under cyber attacks. After introducing CA-controllability and CA-observability, we prove that the supervisory control problem under cyber attacks is solvable if and only if the given specification language is CA-controllable and CA-observable. Furthermore, we obtain methods to calculate the state estimates under sensor attacks and to synthesize a state-estimate-based supervisor to achieve a given safety specification under cyber attacks. We further show that of all the solutions, the proposed state-estimate-based supervisor is maximally-permissive. |
