# Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems
## 4 DDS Designs
From a practical point of view, those residual dampers can be removed without any significant loss of performance and, consequently, the DDSs corresponding to the optimal configurations DC2 and DC3 could be implemented with a set of 11 and 10 dampers, respectively. Regarding the optimal \(\gamma\)-values, the data in Table 2 indicate that the considered damping configurations are all able to produce a significant reduction of the system \(H_{\infty}\)-norm when compared with the nonactuated MBS. In particular, an \(H_{\infty}\)-norm decrease around 82% is attained by DC3 and larger reductions of about 88% are achieved by DC1 and DC2. The better results obtained by DC1 and DC2 suggest a superior performance of widely distributed interstory damping schemes. However, it is worth highlighting the potential implementation advantages of DC3, which would only require internal modifications of building \(\mathcal{B}^{(1)}\). As to the computational aspects, the selected GA solver has shown to be very effective in dealing with the mixture of discrete and continuous variables, the variety of optimization constraints and the relatively large number of optimization variables. All the presented damping configurations have been obtained with a common random seed and using the standard parameter setting for large-scale GA optimization problems suggested in the Matlab Global Optimization Toolbox. Considering the dimension and complexity of the problem and the modest computing resources (see Remark 8), the computation  
Figure 7: Damping configuration DC3. Full-linked DDS with a single actuated building, \(\hat{n}=4\) interstory dampers and \(\hat{n}=8\) interbuilding dampers. times are notably short, specially when the GA solver is run in parallel mode. Finally, the large number of objective-function evaluations required to obtain the different optimal configurations indicates that the computational cost of evaluating the objective function can certainly be a critical bottleneck for the overall computational effectiveness of the proposed design methodology. In that sense, the presented matrix formulation for the damped multibuilding model has proved to be a relevant contribution. **Remark 6**. : _As suggested in the GA solver documentation for problems with a large number of optimization variables [31], we have introduced some modifications in the default parameter setting of the solver. Specifically, we have set the values 200 for the_ PopulationSize_, 0.9 for the_ CrossoverFraction_, 20 for the_ EliteCount _and 500 for the_ MaxGenerations _parameters. To take advantage of the CPU multi-core architecture, the GA solver has been enforced to run in parallel mode by enabling the option_ UseParallel_. Also, to improve the relative accuracy in the computation of the H\({}_{\infty}\)-norm_, the tolerance in the function_ hinfnorm _has been decreased to_ \(10^{-3}\)___[_30_]__._  
**Remark 7**. : _As indicated in Remark 5, the stochastic character of the GA solver typically produces distinct suboptimal solutions in different runs of the solver. For simplicity, in this work the Matlab order_ rng(125) _has been used to set a common random seed for all the computed DDS configurations. That random seed has been arbitrarily chosen, which confirms the effectiveness of the proposed design methodology and indicates that improved results could be possibly obtained by exploring a wider set of random seeds [29]._  
**Remark 8**. : _The computation time values presented in Table 2 should only be taken as approximate references, in the sense that small variations can be observed in the computation time of different runs of the GA solver. Moreover, the computation time in parallel mode can be significantly affected by the available number of CPU cores. In this work, all the computations have been carried out with Matlab 2019a on a regular desktop computer equipped with an Intel Core i7-8700 CPU at 3.20 GHz, 16 GB RAM and a 480 GB SSD hard drive._