This paper aims at developing a robust gain-scheduled proportional-integral-derivative (PID) control design method for a linear-parameter-varying (LPV) system. It is recognized in the literature that robust fixed-order controller design can be formulated as a feasibility problem of a bilinear matrix inequality (BMI) constraint. Unfortunately, the search for a feasible solution of a BMI constraint is a NP hard problem in general. A common way to solve this dilemma is to apply a linearization method, such as variable change method or congruence transformation, to transform the BMI into LMI. The applicability of the linearization method depends on the specific structure of the problem at hand and cannot be generalized. This paper formulates the gain-scheduled PID controller design as a feasibility problem of a quadratic matrix inequality (QMI) constraint, which covers the BMI constraint as a special case. An augmentation of the newly developed sequential LMI optimization method is proposed to search for a feasible solution of a QMI constraint iteratively. In the application part, a vehicle lateral control problem is presented to demonstrate the applicability of the proposed algorithm to a real-world output feedback control design.