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In vibration control of compound structures, inter-substructure damper (ISSD) systemsexploit the out-of-phase response of different substructures to dissipate the kinetic vibrationalenergy by means of inter-substructure damping links. For seismic protection of multistory buildings,distributed sets of interstory fluid viscous dampers (FVDs) are ISSD systems of particular interest.The connections between distributed FVD systems and decentralized static output-feedbackcontrol allow using advanced controller-design methodologies to obtain passive ISSD systemswith high-performance characteristics. A major issue of that approach is the computationaldifficulties associated to the numerical solution of optimization problems with structured bilinearmatrix inequality constraints. In this work, we present a novel iterative linear matrix inequalityprocedure that can be applied to obtain enhanced suboptimal solutions for that kind of optimizationproblems. To demonstrate the effectiveness of the proposed methodology, we design a system ofsupplementary interstory FVDs for the seismic protection of a five-story building by synthesizing adecentralized static velocity-feedback H&yen, controller. In the performance assessment, we compare thefrequency-domain and time-domain responses of the designed FVD system with the behavior of theoptimal static state-feedback H&yen, controller. The obtained results indicate that the proposed approachallows designing passive ISSD systems that are capable to match the level of performance attained byoptimal state-feedback active controllers.