Nanopositioning systems with piezoelectric actuation have found extensively applications in micro-/nanomanipulation domain. This paper reports on the design, realisation and testing of a new output-based discrete-time terminal sliding mode control (DTSMC) scheme, which is applied to achieve precision motion control of a piezoelectric nanopositioning system by overcoming the unmolded non-linearity and disturbance. The controller is easy to implement because it does not require a hysteresis model and a state observer. It is able to deliver a precise tracking without chattering phenomenon. The stability of the control system is proved and the effectiveness of the control scheme is demonstrated through experimental studies. Results show that the non-linear DTSMC robust control is superior to the conventional linear discrete-time sliding-mode control in motion tracking application. The DTSMC enables the generation of a larger bandwidth than the conventional approaches. Moreover, the DTSMC endows the system a well robustness feature in the presence of model uncertainty and external disturbances.