In this paper, we present an optimal control approach using Linear Matrix Inequalities (LMIs) for trajectory tracking control of a three-wheeled omnidirectional mobile robot in the presence of external disturbances on the robot's actuators and noise in the robot's sensor measurements. First, a state-space representation of the omnidirectional robot dynamics is derived using a point-mass dynamic model. Then, we propose an LMI-based full-state feedback H∞-optimal controller for the tracking problem. The robot's tracking performance with the H∞-optimal controller is compared to its performance with a classical full-state feedback tracking controller in simulations with circular and bowtie-shaped reference trajectories. In order to evaluate our proposed controller in practice, we also implement the H∞-optimal and classical controllers for these reference trajectories on a three-wheeled omnidirectional robot. The H∞-optimal controller guarantees stabilization of the robot motion and attenuates the effects of frictional disturbances and measurement noise on the robot's tracking performance. Using the H∞-optimal controller, the robot is able to track the reference trajectories with up to a 47.8% and 45.8% decrease in the maximum pose and twist errors, respectively, over a full cycle of the trajectory compared to the classical controller. The simulation and experimental results show that our LMI-based H∞-optimal controller is robust to undesired effects of disturbances and noise on the dynamic behavior of the robot during trajectory tracking and can outperform the classical controller in attenuating their effects.