## 3 Controllers Design
### Static Output-Feedback \(H_{\infty}\) Centralized Controllers
\right], \tag{33}\]  
where \(\alpha_{\mathrm{r}}\), \(\alpha_{\mathrm{a}}\) and \(\alpha_{\mathrm{u}}\) are scaling coefficients that compensate the different magnitude of interstory drifts, interbuilding approaches and control forces, respectively, \(\mathbf{C}_{\mathrm{r}}\) and \(\mathbf{C}_{\mathrm{a}}\) are the matrices given in Equation (19), and \(n_{d}\) is the total number of actuation devices of the considered control configuration. With this choice, the controlled-output vector satisfies  
\[\|\mathbf{z}(t)\|_{2}^{2}=\alpha_{\mathrm{r}}^{2}\,\|\mathbf{r}(t)\|_{2}^{2}+ \alpha_{\mathrm{a}}^{2}\,\|\mathbf{a}(t)\|_{2}^{2}+\alpha_{\mathrm{u}}^{2}\, \|\mathbf{u}(t)\|_{2}^{2}. \tag{34}\]  
To obtain a velocity-feedback \(H_{\infty}\) controller for the control configuration \(\ell\), we consider the system matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{E}\) corresponding to the control location matrix \(\mathbf{T}_{\mathrm{u}}^{\ell}\), the mass and stiffness values given in Table 1 (see Appendix A) and the damping matrices in Equations (A1) and (A2); the measured-output matrix \(\mathbf{C}_{\mathrm{y}}^{\ell}\) in Equation (31); and the controlled-output matrices \(\mathbf{C}_{\mathrm{z}}\) and \(\mathbf{D}_{\mathrm{z}}\) in Equation (33) defined by the corresponding control dimension \(n_{d}\) and the scaling coefficients  
\[\alpha_{\mathrm{r}}=5,\quad\alpha_{\mathrm{a}}=1,\quad\alpha_{\mathrm{u}}=10^ {-7.4}. \tag{35}\]  
Next, by applying the computational procedure described in Appendix B, we obtain a velocity-feedback control gain matrix \(\mathbf{K}^{\ell}\) and an upper bound \(\widetilde{\gamma}_{\ell}\) of the corresponding \(H_{\infty}\)-norm \(\gamma_{\ell}\), which can be computed by considering the closed-loop transfer function \(\mathbf{T}_{\mathrm{K}^{\ell}}(2\pi fj)\) in Equation (A17) and solving the optimization problem given in Equation (A16). For the control configurations CC1 and CC2, we obtain the following velocity-feedback control gain matrices  
\[\mathbf{K}^{\mathrm{i}}\;=\;10^{7}\times\left[ \begin{array}{cc}-0.7525&0.014 2\\ -0.8003&-1.1890\end{array} \right],\quad\mathbf{K}^{\mathrm{u}}\;=\;10^{7} \times\left[ \begin{array}{cc}-1.0176&0.0748\\ 0.1392&-0.4387\end{array} \right], \tag{36}\]  
and the \(\gamma\)-value upper bounds  
\[\widetilde{\gamma}_{\mathrm{I}}=0.5973,\quad\widetilde{\gamma}_{\mathrm{II}}= 0.8851. \tag{37}\]The actual \(\gamma\)-values are  
\[\gamma_{1}=0.4967,\quad\gamma_{\rm II}=0.7833. \tag{38}\]  
To illustrate the frequency behavior of the velocity-feedback controllers defined by the gain matrices \(\mathbf{K}^{\rm I}\) and \(\mathbf{K}^{\rm II}\), the maximum singular values of the closed-loop transfer functions \(\mathbf{T}_{\rm K^{\rm I}}(2\pi fj)\), \(\mathbf{T}_{\rm K^{\rm II}}(2\pi fj)\) and the open-loop transfer function  
\[\mathbf{T}(2\pi fj)=\mathbf{C}_{x}(2\pi fj\mathbf{I}-\mathbf{A})^{-1}\mathbf{E} \tag{39}\]  
are presented in Figure 5. In this figure, the thin black solid line displays the maximum singular values of the open-loop transfer function and shows the frequency response characteristics of the uncontrolled two-building structure. The peaks in this plot are associated to the natural resonant frequencies of the individual buildings, which are presented in Table 1. The main peak-value is associated to the frequency \(1.0082\,\mathrm{Hz}\) and has a magnitude  
\[\gamma_{0}=2.0479, \tag{40}\]  
which corresponds to the \(H_{\infty}\)-norm of the uncontrolled configuration. The thick blue solid line presents the frequency response of the configuration CC1 with the velocity-feedback control gain matrix \(\mathbf{K}^{\rm I}\), and the red dash-dotted line shows the frequency response of the configuration CC2 with the velocity-feedback control gain matrix \(\mathbf{K}^{\rm II}\). As indicated in Equation (A16), the largest peak-values in these plots correspond to the controllers \(H_{\infty}\)-norms given in Equation (38).