Fast steering mirror (FSM) typically operates in harsh environments, susceptible to vibrations, temperature fluctuations, and other factors, which can lead to malfunctions. Focusing on the most prevalent constant bias fault, this paper proposes an LMI-based fault observer design method, aiming to enhance the reliability of fault detection and strengthen the stability and anti-interference capabilities of the FSM. Firstly, the model identification method based on Hankel matrix is employed to identify the two-axis fast steering mirror model including the coupling effect. Then, the fault model of the fast steering mirror system is established, and the fault observer of the fast steering mirror is designed by using the LMI-based method. Finally, the proposed method is verified through simulations and experiments. The results indicate that when both axes of the fast steering mirror have constant bias faults in the actuators and sensors, the Riccati-based fault observer can only detect the fault in one axis, while the LMI-based fault observer can detect faults within 0.1 seconds after the fault of the X-axis occurs, and detect faults within 0.06 seconds after the fault of the Y-axis occurs. Therefore, the fault observer designed by the LMI method proposed in this paper can improve the fault detection performance of the fast steering mirror.