## 5 Conclusions  
In this work, we have introduced a novel iterative linear matrix inequality (ILMI) procedure, which can be applied to compute enhanced suboptimal solutions for optimization problems with structured bilinear matrix inequality (BMI) constraints. After discussing the connections between passive damping systems and decentralized static velocity-feedback (SVF) controllers, the design of fluid viscous damper (FVD) systems has been set as a decentralized SVF \(H_{\infty}\) controller-design problem, which admits a BMI formulation and can be solved with the proposed ILMI procedure. The main features of the presented methodology have been demonstrated by designing a system of supplemental interstory FVDs to mitigate the seismic response of a five-story building model. In the performance assessment of the obtained passive damping system, we have studied the frequency-domain and time-domain responses, taking as reference the optimal static state-feedback \(H_{\infty}\) controller. Considering the results achieved in this paper, the following points can be highlighted: (i) Active controller-design strategies can be applied to design FVD systems with high-performance characteristics. (ii) The followed \(H_{\infty}\) controller-design approach does not depend on particular seismic records. (iii) For the considered problem, the proposed ILMI procedure can be initialized by solving a simple matrix Lyapunov equation. (iv) The study of the frequency response functions indicates that the obtained FVD system produces a relevant reduction of the main and secondary resonant peaks. (v) The study of the time-history responses indicates that the obtained FVD system produces a significant and well-balanced reduction of the interstory-drift and absolute-acceleration peak values in all the building levels. (vi) The performance characteristics of the obtained FVD system are similar, and sometimes superior, to those attained by the optimal \(H_{\infty}\) state-feedback active controller.  
Natural extensions of this work are the study of computational effectiveness in large structural problems and the design of ISSD systems with mixed interstory-interbuilding damping links for seismic protection of adjacent buildings. Additional research lines associated to the application of the proposed ILMI procedure to other problems of practical interest include the design of high-performance automotive suspensions [46] and the combination of health monitoring techniques with ISSD system design to define scalable and adaptive retrofitting schemes [47; 48].  
Conceptualization, formal analysis, investigation, methodology and writing--review & editing were conducted collaboratively by all the authors; software, F.P.-Q. and J.R.-M.; visualization, J.R.-M. and J.M.R.; writing--original draft, F.P.-Q., J.R.-M. and H.R.K. All authors have read and agreed to the published version of the manuscript.  
This research was partially supported by the Spanish Ministry of Economy and Competitiveness under Grant DPI2015-64170-R/FEDER and by the Italian Ministry of Education, University and Research under the Project \"Department of Excellence LIS4.0--Lightweight and Smart Structures for Industry 4.0\".  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} BI & base isolation \\ BMI & bilinear matrix inequality \\ BRL & bounded real lemma \\ FRF & frequency response function \\ FVD & fluid viscous damper \\ ILMI & iterative linear matrix inequality \\ ISSD & inter-substructure damper \\ \end{tabular}