On the confidentiality of the reference signal under sensor attacks

In this paper, we investigate the confidentiality of the reference signal of a feedback system under sensor attacks. In particular, we analyze the conditions for when an attacker with perfect model knowledge and access to the sensor measurements can obtain an unbiased estimate of the reference signal in a feedback control system such that the estimate’s error covariance converges to zero. We call such an estimate a perfect estimate. Under the assumption of linear dynamics for the plant, the controller, and the reference signal, we show that the attacker can perfectly estimate the reference signal if and only if the reference dynamics do not have eigenvalues outside the unit circle. This implies that an attacker is able to perfectly estimate common reference signals such as step functions and sinusoidal signals from noisy measurements. The convergence rate of the estimate is, however, not exponentially fast for common reference signals and depends on the reference signal to be estimated as well as the controller used. We verify our results numerically with a simulation of a three-tank system.