Stochastic Watermarking Protection Strategy for CPSs Under Linear Stealthy Attacks

This article investigates the stochastic watermarking protection strategy for cyber-physical systems (CPSs) in the presence of linear stealthy attacks. Unlike deterministic watermarking, stochastic watermarking strategies are developed to counteract scenarios where watermarking signals are deciphered. More specifically, attackers aim to inject attack signals into the vulnerabilities of stochastic watermarking strategies, while defenders intend to intercept attack signals as much as possible. Using zero-sum game theory, we investigate the interactive decision-making process between both sides, proving that the equilibrium strategy depends on maximum attack/defense probabilities and time-varying watermarking parameters. Furthermore, we also present criteria for selecting watermarking parameters to reduce the attacker's payoff. In particular, if the attack probability is not an unknown variable, the proposed defense strategy remains effective. Finally, the validity of the theoretical results is demonstrated through simulation examples.