  This paper considers the problem to design physical watermark signals to
protect a control system against replay attacks. We first define the replay
attack model, where an adversary replays the previous sensory data in order to
fool the system. The physical watermarking scheme, which leverages a random
control input as a watermark, to detect the replay attack is introduced. The
optimal watermark signal design problem is then proposed as an optimization
problem, which achieves the optimal trade-off between the control performance
and attack detection performance. For the system with unknown parameters, we
provide a procedure to asymptotically derive the optimal watermarking signal.
Numerical examples are provided to illustrate the effectiveness of the proposed
strategy.
