Drilling is one of the most critical mechanical process in oil and gas industry in which its operational parameters should properly be tuned to reduce drilling time and consequently enhance efficiency of the drilling process. The main objective in this paper is to present a new method to regulate and optimize the Rate of Penetration (ROP) of the system with top drive rotary motor torque in drill string. The paper presents a formulation of a robust receding horizon controller to track piecewise constant references. To achieve this, a tube-based Robust Model Predictive Control (RMPC) is introduced in which the tubes are based on reachable sets. A drilling system is assumed as a test bed for evaluating the performance of the proposed control scheme. The assumed drilling system is modeled as a linear system with additive bounded uncertainties by using Bourgoyne and Young model which is known as a complete mathematical drilling model. The most important novelty part of this manuscript corresponds to integration of both tracking and regulatory objectives in one control framework. Simulations demonstrate the effectiveness of the stability and robust characteristics the proposed RMPC scheme in terms of its stability and robust characteristics with respect to the usual control approaches.