## 4 Numerical Results
### Seismic Response  
To illustrate the seismic performance of the designed FVD system, we have computed the interstory-drift, total-acceleration and actuation-force time responses produced by the full-scale 180-component of the El Centro 1940 ground-acceleration seismic record displayed in Figure 5. As a natural reference, we have also computed the time responses corresponding to the uncontrolled building and the optimal SSF \(H_{\infty}\) controller. The obtained absolute peak-values  
\[|r_{i}|_{\max}=\max_{0\leq t\leq 50}|r_{i}(t)|,\quad|a_{i}|_{\max}=\max_{0\leq t \leq 50}|a_{i}(t)|,\quad|u_{i}|_{\max}=\max_{0\leq t\leq 50}|u_{i}(t)|,\quad i=1, \ldots,5 \tag{77}\]  
are displayed in Figure 6, where the red solid lines with asterisks correspond to the FVD system with the damping coefficients in Table 3, the blue dash-dotted lines with circles represent the optimal SSF controller with the control gain matrix \(\widehat{G}\) in Equation (73), and the black solid lines with squares describe the uncontrolled building response. Looking at the plots in Figure 6a, it can be appreciated that the proposed FVD system produces remarkable reductions of the interstory-drift peak-values, which according to the values in Table 4, are superior to 66% in all the building levels. The plots in Figure 6b and the data in Table 5 indicate that the FVD system also produces significant reductions of the total-acceleration peak-values, which are about a 38% at the building bottom level and superior to 64% in the upper levels.  
When compared with the optimal SSF controller, the obtained time-domain results come to confirm the high-performance characteristics of the FVD system pointed out by the frequency-domain responses discussed in Section 4.1. Specifically, despite assuming an ideal active implementation with full-state feedback information, the optimal SSF controller is only able to produce appreciable improved results in the interstory-drift peak-values \(|r_{2}|_{\text{max}}\) and \(|r_{3}|_{\text{max}}\). For the rest of considered peak-values, including the actuation-force peak-values displayed in Figure 6c, the results achieved by the passive FVD system are similar or superior to those attained by the optimal SSF controller.  
To complement the peak-value information and provide a clearer view of the time-domain responses, the interstory-drift, total-acceleration and actuation-force time histories corresponding to the 3 rd building-level are displayed in Figures 7-9, respectively. To facilitate the comparison, Figures 7 and 8 present a close-up view of the FVD and SSF responses, and display in the background the clipped response of the uncontrolled building, which for the 3 rd building-level has an interstory-drift peak-value of 5.11 cm and a total-acceleration peak-value of 9.41 m/s\({}^{2}\) (see Figure 6a,b). Moreover, the corresponding plots of actuation forces versus interstory drifts and interstory velocities are respectively presented in Figures 10 and 11.  
**Remark 4**.: _All the computations have been carried out with Matlab 2020a 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. The optimization LMI problems have been solved with the function \(\mathrm{mincx(}\mathrm{0}\mathrm{)}\) of the Matlab Robust Control Toolbox [45], setting a relative tolerance of \(10^{-5}\)._