For an air-breathing hypersonic vehicle model subject to input constraints, this paper is concerned with the application of linear control theory to design its flight control system. In the framework of integral quadratic constraints (IQCs), the standard linear parameter-varying (LPV) control method is generalized to deal with the parametric, dynamic uncertainties and saturation nonlinearities associated with the model. Based on open-loop analysis, an inner/outer loop structure is proposed to design the flight control system that is gain-scheduled over dynamic pressure and Mach number. The inner-loop consists of robust LPV controller and LPV anti-windup compensator to enhance attitude stability and overcome actuator saturation. The outer-loop is designed as a gain-scheduled proportional-integral (PI) controller to achieve precise trajectory tracking performance. The effectiveness of the control strategy is demonstrated on the full nonlinear model.