Conditions for parameter unidentifiability of linear ARX systems for enhancing security

For an adversarial observer of parametric systems, the identifiability of parameters reflects the possibility of inferring the system dynamics and then affects the performance of attacks against the systems. Hence, achieving unidentifiability of the parameters, which makes the adversary unable to get identification with low variance, is an attractive way to enhance security. In this paper, we propose a quantitative definition to measure the unidentifiability based on the lower bound of identification variance. The lower bound is given via the analysis of the Fisher Information Matrix (FIM). Then, we propose the necessary and sufficient condition for unidentifiability and derive the explicit form of the unidentifiability condition for linear autoregressive systems with exogenous inputs (ARX systems). It is proved that the unidentifiability of linear ARX systems can be achieved through quadratic constraints on inputs and outputs. Finally, considering an optimal control problem with security concerns, we apply the unidentifiability constraint and obtain the optimal controller. Simulations demonstrate the effectiveness of our method.