  The addition of a physical watermarking signal to the control input increases
the detection probability of data deception attacks at the expense of increased
control cost. In this paper, we propose a parsimonious policy to reduce the
average number of watermarking events when the attack is not present, which in
turn reduces the control cost. We model the system as a stochastic optimal
control problem and apply the dynamic programming to minimize the average
detection delay (ADD) for fixed upper bounds on false alarm rate (FAR) and
increased control cost. The optimal solution results in a two threshold policy
on the posterior probability of attack, which is derived from the Shiryaev
statistics for sequential change detection assuming the change point is a
random variable with a geometric distribution. We derive approximate
expressions of ADD and FAR applying the non-linear renewal theory. The
relationship between the average number of watermarking added before the attack
and the increase in control cost is also derived. We design the optimal
watermarking that maximizes the Kullback-Leibler divergence for a fixed
increase in the control cost. Simulation studies are performed to illustrate
and validate the theoretical results.
