This article concerns the design of an observer-based robust control system for a class of nonlinear (NL) systems with uncertainties and disturbances including the input quantization. It addresses nonlinearities by employing simultaneously the quadratically inner-bounded conditions and the one-sided Lipschitz (OSL) technique. Additionally, auxiliary variables that facilitate the derivation of less conservative design conditions in the form of Linear Matrix Inequality (LMI) constraints are added. These LMIs enable the simultaneous computation of observer and controller gains. To demonstrate the effectiveness of this control approach, a numerical case study featuring a nonlinear DC motor system is presented.