This paper addresses the design of output controllers of chosen order in a given set for ensuring a number of desired properties, such as robust D-stability and robust performance. It is shown that a sufficient condition for determining such controllers can be obtained by solving convex optimization problems with linear matrix inequality constraints through the introduction of a Lyapunov function depending polynomially on both uncertainty and controller. The proposed condition is also necessary for sufficiently large degrees of the polynomials involved under some assumptions.