This paper deals with the robust control of a manipulator robot with n-DOF subject to unknown exogenous disturbance. Our methodology is based chiefly on the use of a high-gain observer for the state estimation and an additional nonlinear observer to estimate the disturbance. Thus, this disturbance observer is injected within the control loop as an auxiliary input that contributes in the robust tracking of a certain desired trajectory. Our design of the observers is achieved via LMI techniques. First, we present an LMI constraint for a general nonlinear system satisfying a Lipschitz condition allowing to determine the parameters of the high-gain observer for which the maximum bound is numerically optimized. Our developed contributions and results are applied on the simple inverted pendulum robot. A new LMI stability condition of the high-gain observer for this simple robotic manipulator is formulated. Simulation results show the effectiveness of the observer-based controller in the tracking of a defined periodic trajectory and therefore in the complete rejection of the disturbance.