The da Vinci Research Kit (dVRK) was introduced in 2012 to provide an affordable, open-source platform for research in robotic minimally-invasive surgery. It provides access to all levels of control but, until now, has relied on an analog controller for the motor current, which cannot easily be customized to improve performance. This paper aims to implement the low-level control digitally and to improve the overall control performance. To enable model-based controller design, the system is first identified using measurements provided by the encoders and internal electronics. The digital current controller is then implemented on the existing field programmable gate array (FPGA). Experiments demonstrate that the new digital current controller yields superior performance compared to the original analog design. In addition, the identified system model is used to design an improved position controller that is also implemented on the FPGA and provides better trajectory tracking than the position controller currently implemented on the control PC. The comparison between simulation and measurement, for both the current and position control, verifies the validity of the model based on the system identification, enabling utilization for future adaptations. The improved low-level control enlarges the possibilities for more accurate operation and the achieved digital implementation enables researchers worldwide to easily adapt the low-level control in future versions of the dVRK.