The main contribution of the paper is the modeling approach used to describe a pure threephase shunt active filter, and the application of the $H_{infty}$ control design tool in order to improve the quality of electrical energy. Advanced power electronics devices have widely contributed to the degradation of power quality due to the injection of non-sinuso$"{i}$dal currents into the utility system. Therefore, it is essential to use an active compensator which can attenuate current harmonics to an acceptable level on the line side of the power source. In this work, a three-phase active filter connected in parallel to a supply system feeding a non-linear load is described, in a complex framework, by a linear multivariable state space representation in order to guarantee that the system is mathematically decoupled and therefore to simplify the controller design. This representation includes a sensor to measure perturbations, and allows one to calculate a linear robust control law. The originality of this paper is that a Linear Matrix Inequality based $H_{infty}$ synthesis is performed to design a static state feedback controller with complex-valued parameters. The robustness of this controller with respect to network impedance uncertainties is investigated. Moreover, simulation and experimental results are given to reveal the effectiveness of the synthesized control law.