This paper is concerned with the design of robust output feedback controllers for a class of linear discrete-time systems with norm-bounded parameter uncertainty. The controller is designed to place the closed-loop poles of the uncertain system in a specified disk and guarantee an upper bound on a quadratic cost function. An existence condition for such controllers is derived. It is shown that the condition is equivalent to the feasibility of a linear matrix inequality (LMI) problem, and the procedure of constructing such a desired controller is presented in terms of the feasible solutions to this LMI. Furthermore, a convex optimization problem is introduced for the selection of a suitable output feedback controller minimizing the upper bound.