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This paper presents an LMI based robust decentralized dynamic output feedback controller design for power systems. The problem of designing fixed-order robust decentralized output dynamic feedback controllers is formulated as a minimization problem of linear objective function under linear matrix inequality (LMI) constraints in tandem with bilinear matrix inequality (BMI) constraint. In the design, the robust connective stability of the overall system is guaranteed while the upper bounds of the uncertainties arising from the interconnected subsystems as well as the nonlinearities within each subsystem are maximized. The paper also proposes a new approach to solve such optimization problems using an iterative LMI programming method to determine the (sub)-optimal decentralized controllers for the system. Moreover, the approach is flexible enough to allow the inclusion of additional design constraints such as the stability degree of the whole system and different controller orders for each subsystem while at the same time maximizing the tolerable upper bounds on the class of perturbations. The approach is demonstrated by designing power system stabilizers (PSSs) for a test system.