## 4 Numerical Results  
### FVD System Design  
To illustrate the effectiveness of the proposed ILMI procedure, in this section we design a system of five interstory FVDs to mitigate the seismic response of a five-story building model with mass matrix (in kg)  
\[\mathbf{M}=10^{3}\times\left[ \begin{array}{ccccc}2152&0&0&0&0\\ 0&209.2&0&0&0\\ 0&0&207.0&0&0\\ 0&0&0&204.8&0\\ 0&0&0&266.1\end{array} \right], \tag{65}\]  
stiffness matrix (in N/m)  
\[\mathbf{K}_{s}=10^{8}\times\left[ \begin{array}{ccccc}2.60&-1.13&0&0&0\\ -1.13&2.12&-0.99&0&0\\ 0&-0.99&1.88&-0.89&0\\ 0&0&-0.89&1.73&-0.84\\ 0&0&0&-0.84&0.84\end{array} \right], \tag{66}\]  
and damping matrix (in Ns/m)  
\[\mathbf{C}_{d}=10^{5}\times\left[ \begin{array}{ccccc}2.602&-0.924&0&0&0\\ -0.924&2.196&-0.810&0&0\\ 0&-0.810&1.995&-0.728&0\\ 0&0&-0.728&1.867&-0.687\\ 0&0&0&-0.687&1.274\end{array} \right]. \tag{67}\]  
The matrices \(\mathbf{M}\) and \(\mathbf{K}_{s}\) correspond to the mass and stiffness coefficients of the five-story building model presented in [43], and \(\mathbf{C}_{d}\) has been computed as a Rayleigh damping matrix with 2% of relative damping in the first and fifth modes [44]. The building natural frequencies (in Hz) are collected in Table 1.  
Following the discussion in Section 2, we assume that the building is equipped with a complete system of interstory force-actuation devices as schematically displayed in Figure 2(a). Our objective is to design a high-performance FVD system by computing a fully decentralized SVF \(H_{\infty}\) controller, which takes the vector of interstory velocities \(\mathbf{v}(t)\) as feedback information. To this end, we consider the system matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{E}\) in Equation (9) corresponding to the value \(n=5\), the matrices \(\mathbf{M}\), \(\mathbf{K}_{s}\) and \(\mathbf{C}_{d}\) in Equations (65)-(67) and the control-input matrix  
\[\mathbf{P}_{5}= \begin{bmatrix}1&-1&0&0&0\\ 0&1&-1&0&0\\ 0&0&1&-1\\ 0&0&0&0&1\end{bmatrix} ; \tag{68}\]  
the measured-output matrix \(\mathbf{C}_{y}=\mathbf{C}_{\mathbf{v}}\) in Equation (17) with \(n=5\); and the controlled-output matrices \(\mathbf{C}_{z}\) and \(\mathbf{D}_{z}\) in Equation (22) corresponding to the values \(n=5\) and \(\alpha=10^{-7.25}\). The matrix \(\mathbf{Q}\), computed by the Matlab command null(), has the following form  
\[\mathbf{Q}= \begin{bmatrix}-\mathbf{I}_{5}\\ [\mathbf{0}]_{5\times 5}\end{bmatrix}  \tag{69}\]  
and, by solving the Lyapunov equation in (64) and applying the formula in (63), we obtain the initial \(L\)-matrix  
\[\mathbf{L}^{(0)}=10^{-2}\times \begin{bmatrix}0.2500&0.1362&0.1097&0.0725&0.0419\\ 0.4246&0.3928&0.2439&0.1735&0.0949\\ 0.5627&0.5400&0.4748&0.2829&0.1612\\ 0.6652&0.6344&0.5756&0.4748&0.2360\\ 0.7089&0.6805&0.6153&0.5134&0.3757\end{bmatrix} , \tag{70}\]  
which produces a feasible LMI problem \(\mathcal{P}(\mathbf{L}^{(0)},\mathbb{D})\) with \(\eta_{0}^{1/2}=0.0933\). As stated in Theorem 1, the subsequent LMI problems \(\mathcal{P}(\mathbf{L}^{(i)},\mathbb{D})\), \(i=1,\ldots,6\) are all feasible and produce the decreasing sequence of values \(\eta_{i}^{1/2}\) collected in Table 2. In the sixth step, we obtain the diagonal gain matrix  
\[\mathbf{K}^{(6)}=10^{6}\times \begin{bmatrix}-8.4144&0&0&0&0\\ 0&-6.65750&0&0&0\\ 0&0&-5.7092&0&0\\ 0&0&0&-5.1799&0\\ 0&0&0&0&-4.9493\end{bmatrix}  \tag{71}\]  
and, by proceeding as indicated in Remark 2, we get the controller \(H_{\infty}\)-norm  
\[\gamma_{K^{(6)}}=0.0890. \tag{72}\]  
According to the discussion in Section 2, as all the diagonal elements in \(\mathbf{K}^{(6)}\) are negative, the decentralized SVF controller \(\mathbf{u}(t)=\mathbf{K}^{(6)}\mathbf{v}(t)\) can be implemented by a system of interbuilding FVDs with the damping coefficients displayed in Table 3. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Mode** & **1** & **2** & **3** & **4** & **5** \\ \hline natural frequency (Hz) & 1.008 & 2.825 & 4.493 & 5.797 & 6.773 \\ \hline \hline \end{tabular}
\end{table}