By analyzing the mechanism relationship between the input angular velocity and the output position of the engraving machine system in speed control mode, the system is simulated as a model connected by an integrating factor and a stable transfer function. By introducing a differential filter to process the sampling data, the critical unstable integrating model is transformed into a stable model which is easy to identify. This paper proposes a recursive least squares algorithm which can accurately estimate the model with integrating factor. Besides, the instrumental variables method is applied to eliminate the influence of colored noise caused by filtering. The proposed algorithm can ensure the accurate estimation of the integrating system in open loop state. A fault-tolerant tracking control algorithm based on the intermediate observer is proposed for the multi-axis engraving machine motion control system. Aiming at the process fault matching with the input channel, a fault-tolerant control feedback rate based on fault estimation value is designed to effectively compensate. For the mismatched fault, it can be fully suppressed by adjusting intermediate observer gain. Finally, compared with the existing algorithm, the feasibility and superiority of the proposed algorithm are verified by experiment tests.