Abstract: To address the accuracy degradation caused by hysteresis nonlinearity in high-frequency trajectory tracking of a piezoelectric-driven fast tool servo system, a composite control strategy based on neural-network Koopman operator feedforward compensation and nonsingular fast terminal sliding mode control is proposed. In the feedback loop, NFTSMC is adopted to achieve finite-time convergence and suppress chattering. In the feedforward loop, an autoencoder is introduced to learn the optimal Koopman observables, through which a linearized predictive model of the system is constructed to estimate and compensate for the dynamic hysteresis in real time. Simulation results demonstrate that, for 50 Hz sinusoidal and 100 Hz sawtooth trajectories, the prediction accuracy of the model is superior to that of a back-propagation neural network and Gaussian process regression; the single-step computation time is only 0.0118 ms, which accounts for 23.6% of the sampling period and satisfies the requirement for online real-time compensation. For 300 Hz and 800 Hz sinusoidal trajectory tracking, the steady-state errors of the composite control strategy are reduced by 35.4% and 18.8%, respectively, compared with those of pure NFTSMC. Under both periodic disturbances and sudden step disturbances, the best tracking accuracy and robustness are exhibited by the composite strategy.