## 4 Numerical Simulations  
In this section, a proper set of numerical simulations is conducted to investigate the vibrational time-response of the two-building system for the considered control configurations. The simulations  
Figure 9: Frequency response corresponding to the decentralized velocity-feedback \(H_{\infty}\) controller with passive implementation defined by the diagonal control gain matrix \(\widehat{\mathbf{K}}^{\mathrm{u}}\). Maximum singular values of the closed-loop transfer function \(\mathbf{T}_{\mathrm{K}^{\mathrm{u}}}(2\pi fj)\) (red dash-dotted line), the closed-loop transfer function \(\mathbf{T}_{\mathrm{K}^{\mathrm{u}}}(2\pi fj)\) (thick blue solid line) and the open-loop transfer function \(\mathbf{T}(2\pi fj)\) (thin black solid line). include the active controllers designed in Section 3.1, the passive controllers computed in Section 3.2 and the response of the uncontrolled buildings, which is taken as a natural reference in the performance assessment. In all the cases, the full-scale _North-South El Centro 1940_ ground acceleration seismic record (see Figure 10) has been used as external disturbance. To describe the vibrational response of the individual buildings and the buildings interactions, the vectors of interstory drifts \(\mathbf{r}(t)\) and interbuilding approaches \(\mathbf{a}(t)\) have been computed. Additionally, the vector of control efforts \(\mathbf{u}(t)\) has also been computed for the controlled configurations. Overall, the simulations include eleven different control configurations plus the uncontrolled case. To provide an intuitive and effective summary of this complex set of numerical results, the control configurations have been grouped by the number of actuation devices. Thus, the peak-values of the absolute interstory drifts and interbuilding approaches corresponding to the control configurations with two actuation devices (CC1 and CC2) are presented in Figure 11, where the interstory drift peak-values corresponding to the four-story building \(\mathcal{B}^{(1)}\) are presented in the left-hand-side graphic, the interbuilding approach peak-values are shown in the central graphic, and the interstory drift peak-values corresponding to the five-story building \(\mathcal{B}^{(2)}\) are displayed in the right-hand-side graphic. In the graphics of this section, the following colors and line styles have been used: blue lines present the values of the unlinked control configurations, red lines represent the linked control configurations, and black lines correspond to the uncontrolled configuration; additionally, active controllers are represented by solid lines, and passive controllers by non-solid lines. Detailed legends and captions have also been included in the figures to facilitate an unambiguous interpretation of the graphics. Specifically, the following colors, line styles and symbols have been used in the plots of Figure 11: black solid line with squares for the uncontrolled configuration, blue solid line with circles for the unlinked control configuration CC1 with the active controller defined by the control gain matrix \(\mathbf{K}^{i}\) given in Equation (36), blue dashed line with asterisks for the unlinked control configuration CC1 with the passive controller defined by the control gain matrix \(\widehat{\mathbf{K}}^{i}\) given in Equation (45), red solid line with triangles for the linked control configuration CC2 with the active controller defined by the control gain matrix \(\mathbf{K}^{i}\) given in Equation (36), and red dotted line with hexagrams for the linked control configuration CC2 with the passive controller defined by the control gain matrix \(\widehat{\mathbf{K}}^{i}\) given in Table 3. The peak-values of the corresponding absolute control efforts are presented in Figure 12a.