On Kalman filtering in the presence of a compromised sensor: Fundamental performance bounds

Consider a scalar linear time-invariant system whose state is being estimated by an estimator using measurements from a single sensor. The sensor may be compromised by an attacker. The attacker is allowed to replace the measurement sequence by an arbitrary sequence. When the estimator uses this sequence, its estimate is degraded in the sense that the mean square error of this estimate is higher. The estimator monitors the received data to detect if an attack is in progress. The aim of the attacker is to degrade the estimate to the maximal possible amount while remaining undetected. By defining a suitable notion of stealthiness of the attacker, we characterize the trade-off between the fundamental limits of performance degradation that an attacker can induce versus its level of stealthiness. For various information patterns that characterize the information available at every time step to the attacker, we provide information theoretic bounds on the worst mean squared error of the state estimate that is possible and provide attacks that can achieve these bounds while allowing the attacker to remain stealthy even if the estimator uses arbitrary statistical ergodicity based tests on the received data.