This paper presents the fundamentals and the algorithm of a new methodology for the design of robust power system damping controllers. The methodology provides controllers capable of fulfilling various practical requirements of the oscillations damping problem, which could not be simultaneously satisfied by the majority of the proposed robust approaches until now. The design procedure is based on a special formulation of the dynamic output feedback control problem, which is very well suited for damping controller design. With this formulation, the design problem (which is originally stated as a set of bilinear matrix inequalities) can be expressed directly in the form of linear matrix inequalities. Furthermore, the formulation allows the incorporation of decentralization constraints on the controller matrices, which are one of the practical requirements for power system damping controllers. Another practical requirement is satisfied with the use of the polytopic model (to ensure the robustness of the closed-loop system with respect to the variation of operating conditions). Moreover, the inclusion of a regional pole placement criterion, as the design objective, allows the specification of a minimum damping factor for all modes of the controlled system. The results show the controller is able to provide adequate damping for the oscillation modes of interest.