A computational optimization-based approach for designing fixed-order controllers through robust pole placement for uncertain (interval) linear time-invariant SISO plants is presented. The design objective is the minimization of the over-all deviation from the desired performance for the closed-loop system, as specified by a polytope of characteristic polinomials. The robust pole placement design problem is associated with the solution of an interval Diophantine equation, whose basic properties are analysed. Robust pole placement controllers are viewed as inner solutions of the interval Diophantine equation. Simple and computationally efficient characterizations of the set of all robust pole placement controllers are then obtained and some of its geometric properties discused. Several aspects of the design of robust controllers by Interval Analysis are integrated into a linear goal programming formulation, which can incorporate addicional constraints on the controller parameters. 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