In this paper a new method for fixed-order output-feedback Linear Parameter Varying (LPV) controller design is presented. First, a set of stabilizing LPV controllers is given as an inner convex approximation of the non-convex set of all stabilizing LPV controllers with the same order. This characterization is based on the decoupling of the state and Lyapunov matrices appearing together in the derivative of the Lyapunov function. As a measure of performance the decay-rate related to exponential stability is considered. The efficiency of the proposed method is illustrated on an appropriate simulation example.