Traditional underwater vehicles are limited in speed due to dramatic friction drag on the hull. A supercavitating vehicle exploits supercavitation to induce a gaseous cavity that contains most part of the vehicle and separates the vehicle hull from its surrounding water. Thus friction drag is substantially reduced. A supercavitating underwater vehicle can achieve very high speed, but also poses technical challenges in stability, control, and maneuvering due to various characteristics such as instability in open-loop dynamics, nonlinearity, cavity memory effect, etc. Among the existing literature on the control design for supercavitating vehicles, the cavity memory effect is often neglected to simplify system dynamics. In this paper, we take into account the cavity memory effect and model the supercavitating vehicle as a time-delay Quasi-Linear-Parameter-Varying system. Then a robust controller is designed to handle the switched, time-delay dependent behavior of the vehicle. The uncertainties considered in the presented control design include both parameter and planing force modeling uncertainties.