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This article proposes a new discrete-time predictive sliding mode control (DPSMC) for a parallel micropositioning piezostage to improve the motion accuracy in the presence of cross-coupling hysteresis nonlinearities and input constraints. Unlike the traditional linear discrete-time sliding mode control (DSMC), the proposed DPSMC is chattering free and has a faster convergence rate thanks to the design of a nonlinear discrete-time fast integral terminal sliding mode surface. Moreover, by combining with the receding horizon optimization, the sliding mode state is predicted to follow the expected trajectory of a predefined continuous sliding mode reaching law, which also allows the proposed controller to explicitly deal with constraints. The stability of the closed-loop system is analyzed under the model disturbances and constraints, and proves that the proposed DPSMC can offer a smaller quasi-sliding mode bandwidth than the traditional DSMC. The effectiveness of the proposed controller is validated by a series of numerical simulations and experiments. Results demonstrate the advantages of proposed DPSMC over the traditional DSMC method. [ABSTRACT FROM AUTHOR]