Using the physics-based model for the vibration isolation system, the model uncertainties are described. With the model including parameter perturbations, the robust controller to meet the robust performance and stability is designed through $mu$-synthesis by DK-iteration. The order of controller is reduced by virtue of Hankel norm approximation technique to allow the efficient implementation in the real-time experimental environment without any performance degradation. The performance of the reduced $mu$-controller is accessed in comparison with the original one. The experiments validate the superiority of the proposed control scheme against the model uncertainties and its applicability with varying payload.