A Zonotopic-Based Watermarking Design to Detect Replay Attacks

Abstract

This paper suggests the use of zonotopes for the design of watermark signals. The proposed approach exploits the recent analogy found between stochastic and zonotopic-based estimators to propose a deterministic counterpart to current approaches that study the replay attack in the context of stationary Gaussian processes. In this regard, the zonotopic analogous case where the control loop is closed based on the estimates of a zonotopic Kalman filter (ZKF) is analyzed. This formulation allows to propose a new performance metric that is related to the Frobenius norm of the prediction zonotope. Hence, the steady-state operation of the system can be related with the size of the minimal Robust Positive Invariant set of the estimation error. Furthermore, analogous expressions concerning the impact that a zonotopic/Gaussian watermark signal has on the system operation are derived. Finally, a novel zonotopically bounded water-mark signal that ensures the attack detection by causing the residual vector to exit the healthy residual set during the replay phase of the attack is introduced. The proposed approach is illustrated in simulation using a quadruple-tank process.