A fast-steering mirror （FSM） is a key instrument in high-precision optical systems. However， the flexible-support FSM driven by a voice coil actuator （VCA） possesses complex coupling characteristics； this leads to a complex system model and severely impacts the control performance. To address this issue， this study proposes a dual-axis integral augmented sliding-mode control method， based on system identification and model reduction. Firstly， an accurate coupled VCA-FSM model is established using the Hankel-matrix system-identification method based on pulse response. Subsequently， based on balanced realization and truncation， the high-order model is reduced， while ensuring model accuracy. Next， an integral augmented sliding-mode controller is designed， adopting a modern control-theory method and using the reduced-order model. A state observer is constructed to create the sliding-mode switching function and control law， and the sign function is improved in the control design to eliminate sliding-mode chattering. Finally， frequency-domain and time-domain performance-testing experiments are conducted on the VCA-FSM servo control-system experimental platform. The experimental results demonstrate that the proposed control method significantly improves the control performance of the VCA-FSM servo system， compared to the single-axis sliding-mode and PID control methods. The closed-loop tracking bandwidth is increased by approximately 50.3% and 251.3%， respectively， the disturbance-suppression bandwidth is increased by approximately 39.9% and 451.9%， respectively， the settling time of the step response is reduced by approximately 29.7% and 97.7%， respectively， and the tracking accuracy of the spiral line is improved by approximately 48.5% and 97.8%， respectively. Moreover， the proposed control method controls the decoupling of the VCA-FSM with strong coupling characteristics. The control method proposed in this study greatly improves the control performance of the VCA-FSM.