Compensation Strategy-Based Intrusion Tolerant Parameter Estimation for Quantized Nonlinear Hammerstein Systems Under Replay Attacks

In Cyber-Physical Systems security (CPSs), replay attacks disrupt the temporal alignment between inputs and outputs. The dual nonlinearity of Hammerstein system and quantized measurements further increases the difficulty of parameter estimation. This paper investigates the intrusion tolerant (meaning the capable of maintaining performance despite attack interference) identification problem of quantized Hammerstein systems under replay attacks. On the attack side, an optimal attack strategy is formulated by maximizing estimation error under energy constraints. A closed-form solution is derived using quadratic optimization and projection. For the defense design, a sending mechanism based on stochastic intervals and binary detection sequences is proposed. By leveraging the statistical properties of the detection sequence, the mechanism compensates for estimation errors caused by attacks. The system’s nonlinear and quantized structure is reformulated as a solvable nonlinear system, enabling joint estimation of system parameters and attack probabilities. Theoretical analysis demonstrates that the estimators are consistent and asymptotically normal, and an optimal setting for defense factors is presented. Simulation examples validate the effectiveness and robustness of the proposed approach in terms of identification accuracy and attack resistance.