  We examine the merit of Bernoulli packet drops in actively detecting
integrity attacks on control systems. The aim is to detect an adversary who
delivers fake sensor measurements to a system operator in order to conceal
their effect on the plant. Physical watermarks, or noisy additive Gaussian
inputs, have been previously used to detect several classes of integrity
attacks in control systems. In this paper, we consider the analysis and design
of Gaussian physical watermarks in the presence of packet drops at the control
input. On one hand, this enables analysis in a more general network setting. On
the other hand, we observe that in certain cases, Bernoulli packet drops can
improve detection performance relative to a purely Gaussian watermark. This
motivates the joint design of a Bernoulli-Gaussian watermark which incorporates
both an additive Gaussian input and a Bernoulli drop process. We characterize
the effect of such a watermark on system performance as well as attack
detectability in two separate design scenarios. Here, we consider a correlation
detector for attack recognition. We then propose efficiently solvable
optimization problems to intelligently select parameters of the Gaussian input
and the Bernoulli drop process while addressing security and performance
trade-offs. Finally, we provide numerical results which illustrate that a
watermark with packet drops can indeed outperform a Gaussian watermark.
