Sequential Detection of Replay Attacks with a Parsimonious Watermarking Policy

In this paper, we have proposed a technique for Bayesian sequential detection of replay attacks on networked control systems with a constraint on the average number of watermarking (ANW) events used during normal system operations. Such a constraint limits the increase in the control cost due to watermarking. To determine the optimal sequence regarding the addition or otherwise of watermarking signals, first, we formulate an infinite horizon stochastic optimal control problem with a termination state. Then applying the value iteration approach, we find an optional policy that minimizes the average detection delay (ADD) for fixed upper bounds on the false alarm rate (FAR) and ANW. The optimal policy turns out to be a two thresholds policy on the posterior probability of attack. We derive approximate expressions of ADD and FAR as functions of the two derived thresholds and a few other parameters. A simulation study on a single-input single-output system illustrates that the proposed method improves the control cost considerably at the expense of small increases in ADD. We also perform simulation studies to validate the derived theoretical results.