In this paper, two robust optimal control strategies: Discrete Model Predictive Control (DMPC) and Linear Quadratic Regulator (LQR) are proposed to solve the problem of backlash nonlinearity present in two mass system and also reducing the sensor noise present at the output of the system. In past, number of attempts has been made to develop the optimum controls for backlash nonlinear system to compress the oscillations in load speed. The (DMPC) and (LQR) are now one of the most successful robust optimal control strategies for highly uncertain nonlinear systems like specially the one we have in industries. The (DMPC) and (LQR) require online information of all the states of the nonlinear system, so role of estimators becomes very prominent in (DMPC) and (LQR). In this paper, Kalman Filter (KF) has been used for the state estimation assuming that sensor noise is also present at the output of the system, so in that case load speed, which is also output of the nonlinear system contains backlash nonlinearity and random sensor noise, so now both (DMPC) and (LQR) have to deal with two problems simultaneously. In simulations, a comparison has been presented between the two control schemes. From simulations, it is quite clear that (DMPC) performance is much better than (LQR), while suppressing oscillations due to presence of backlash and sensor noise at the output of the system. Comparison between two controllers also reveals that (DMPC) is much faster than (LQR), while achieving tracking.