Convex parameterization of fixed-order robust, stabilizing controllers for systems with polytopic uncertainty is represented as a linear matrix inequality (LMI) using the Kalman-Yakubovich-Popov (KYP) lemma. This parameterization is a convex inner approximation of the whole nonconvex set of stabilizing controllers, and depends on the choice of a central polynomial. It is shown that, with an appropriate choice of the central polynomial, the set of all stabilizing fixed-order controllers that place the closed-loop poles of a polytopic system in a disk centered on the real axis can be outbounded with some LMIs. These LMIs can be used for robust pole placement of polytopic systems.