This paper develops an observer-based predictive repetitive control system to track periodic reference signals or reject disturbances with bandlimited frequency content. The new design complements existing approaches to predictive control, where a model of the periodic disturbance is embedded in the controller. In particular, the new design, based on a novel combination of repetitive control and a disturbance observer, results in a significant improvement in design transparency and implementation simplicity. Although the design is undertaken using a state-space-model, frequency response analysis based on the sensitivity and complementary functions is used to demonstrate the characteristics of the repetitive control system for disturbance rejection, reference following and measurement noise attenuation. Moreover, operational constraints can be included in the design for applications where this feature is required. Simulation studies are given to highlight the closed-loop performance achievable in the presence of constraints. Experimental validation results from application to a two-joint robotic arm are also given and discussed.