Abstract: For discrete-time systems with unknown but bounded noise, the multiplicative fault detection based on zonotopic set-membership estimation is studied. Firstly, a full-dimensional state observer is established to generate residual signals, and the iterative relationship of the center vector and the generated matrix of the residual zonotopes is obtained through the analysis of the filtering error system with the state joint estimation. Secondly, the optimal observer gain with robustness to disturbance and measurement noise which can ensure that the P-radius of the residual zonotope converges to the specified upper bound is obtained by solving the convex optimization problem. Finally, the simulation example of the low-frequency linear model of the permanent magnet DC motor is utilized to verify the effectiveness of the proposed method.