Stealthy Control Signal Attacks in Linear Quadratic Gaussian Control Systems: Detectability Reward Tradeoff

The problem of false data injection through compromised cyber links to a physical control system modeled by linear quadratic Gaussian dynamics is studied in this paper. The control input stream is compromised by an attacker who modifies the (cyber) control signals transmitted with the objective of increasing the quadratic cost incurred by the (physical) controller whilst maintaining a degree of stealthiness. The tradeoff between the increase in quadratic cost and the stealthiness (or detectability), are measured by the Kullback-Leibler distance between legitimate and falsified state dynamics is characterized analytically. It is shown that the optimal adversarial strategy is a sequence of independent Gaussian noise signals with carefully chosen variances whose eigenvalues align with those of the legitimate noise covariance with the scaling reflecting the desired quadratic cost increase. As the stealthiness decreases, the optimal tradeoff is shown to be linear with slope inversely proportional to the maximal of maximal eigenvalue of modified reward matrices. Numerical simulations are presented that showcase the optimal tradeoff and the comparison of the legitimate and falsified dynamics under different requirements on detectability.