Optimal off-line generated ϵ-stealthy attacks under the energy constraint in cyber-physical systems

The malicious attack design helps to accurately assess the vulnerability of cyber-physical systems under attacks. Based on this, an off-line generated attack model with time-varying covariance is proposed under the energy constraint, which aims to maximize the system estimation error while satisfying the ϵ-stealthiness. Subsequently, the problem is equivalently transformed by quantifying the optimization objective based on the derivation of error covariance and deriving the stealthiness condition according to the statistical properties of mutual information and Kullback-Leibler divergence. Due to the coupling relationship between the designed covariance and scheduling, the covariance is derived as a function of the attack scheduling by the Lagrange multiplier method. Then, the optimal attack scheduling is proved to be fixed according to the uniqueness of the optimal parameters. Finally, some numerical simulations are given to validate the effectiveness of results.