## 4 Numerical Results
### FVD System Design

Table 1: Natural frequencies of the five-story building model. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Step** & **0** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Upper bound \(\eta_{i}^{1/2}\) & 0.0933 & 0.0911 & 0.0902 & 0.0900 & 0.0899 & 0.0898 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of the upper bounds \(\eta_{i}^{1/2}\) obtained in the steps \(i=0,\ldots,6\). To assess the performance of the obtained FVD system, we compute a SSF controller by solving the LMI optimization problem \(\hat{\mathcal{P}}_{\text{ssf}}\) in Equation (38) with the same system matrices \(\mathbf{A}\), \(\mathbf{B}\), \(\mathbf{E}\) and controlled-output matrices \(\mathbf{C}_{z}\) and \(\mathbf{D}_{z}\) used in the SVF controller design. That LMI optimization problem produces the SSF gain matrix  
\[\widehat{G}=10^{6}\times\left[ \begin{smallmatrix}-1.3624&1.1294&0.0298&-0.13 21&-0.4593&-0.3984&-0.4896&-0.5908&-0.6849&-0.9904\\ 2.1548&-2.0831&1.1686&-0.2736&-0.4753&-0.1066&-0.6238&-0.7467&-0.8551&-1. 2284\\ 1.4879&2.2908&-2.5344&0.7248&-0.5361&-0.1069&-0.2389&-0.8118&-0.8980&-1. 2754\\ 0.4214&1.2851&2.3282&-2.8258&0.0721&-0.0117&-0.2123&-0.3096&-0.8843&-1. 2178\\ 0.3549&0.2391&0.7108&2.4487&-2.8454&-0.0629&-0.1324&-0.1903&-0.2222&-1. 0714\\ \end{smallmatrix} \right] \tag{73}\]  
and the optimal \(H_{\infty}\)-norm  
\[\gamma_{\text{ssf}}=0.0860. \tag{74}\]  
The frequency response produced by the designed FVD system is displayed in Figure 4, where the red solid line corresponds to the closed-loop FRF defined by the diagonal SVF gain-matrix \(\mathbf{K}^{(6)}\), the blue dash-dotted line represents the closed-loop FRF associated to the optimal SSF gain-matrix \(\widehat{\mathbf{G}}\) and the black dotted line aptertains to the open-loop FRF  
\[\mathbf{T}_{\text{o}}(\omega)=\mathbf{C}_{z}\left(2\pi\omega j\mathbf{I}_{n_{x}}-\mathbf{A} \right)^{-1}\mathbf{E}, \tag{75}\]  
which describes the frequency response of the uncontrolled building. The overall view in Figure 4a shows the building resonant peaks, which are located at the natural frequencies given in Table 1. The magnitude of the main resonant peak is  
\[\gamma_{\text{o}}=0.3619 \tag{76}\]  
and corresponds to the \(H_{\infty}\)-norm of the uncontrolled building. Looking at the plots in that figure, it can be appreciated that the proposed FVD system produces a remarkable reduction of the main resonant peak and, quite surprisingly, a slightly better overall response than the optimal SSF controller. The frequency response of the FVD system and the SSF active controller can be more effectively compared from the close-up view in Figure 4b, where it can be observed that the FVD system produces a uniformly smaller frequency-response with the exception of a tiny interval around the main resonant frequency, which is consistent with the \(\gamma\)-values in Equations (72) and (74) and the \(H_{\infty}\) optimality of the SSF controller. Figure 4: Maximum singular values of the FRFs corresponding to the uncontrolled building (black dotted line), the optimal SSF controller (blue dash-dotted line) and the computed FVD system (red solid line). (**a**) Overall view. (**b**) Close-up view of the main resonant peaks. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Building Level** & **1** & **2** & **3** & **4** & **5** \\ \hline \hline Damping coefficient \(\hat{\epsilon_{i}}\) (\(\times 10^{6}\) Ns/m) & 8.4144 & 6.6570 & 5.7092 & 5.1799 & 4.9493 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Damping coefficients of the interstory FVD system.