## 3 Controllers Design  
### Static Output-Feedback \(H_{\infty}\) Centralized Controllers  
Static output-feedback controllers can be designed to perform a fast and effective computation of the control actions from the available feedback information. Typically, these kind of controllers can be written in the form  
\[\mathbf{u}(t)=\mathbf{K}\mathbf{y}(t), \tag{27}\]  
where \(\mathbf{u}(t)\) is the vector of control actions, \(\mathbf{K}\) is a constant control gain matrix and \(\mathbf{y}(t)\) is a vector of measured outputs. According to the results summarized in Appendix B, a suboptimal static output-feedback \(H_{\infty}\) controller can be computed for the linear system in Equation (22) by solving the LMI optimization \(\mathcal{P}\) given in Equation (A8). The design procedure involves a measured-output vector of the form  
\[\mathbf{y}(t)=\mathbf{C}_{\mathbf{y}}\mathbf{x}(t), \tag{28}\]  
that models the available feedback information, and a vector of controlled outputs  
\[\mathbf{z}(t)=\mathbf{C}_{\mathbf{z}}\,\mathbf{x}(t)+\mathbf{D}_{\mathbf{z}} \,\mathbf{u}(t), \tag{29}\]  
that allows computing the overall cost of the system response and the control action. In this work, we assume that the relative velocities associated to the actuation devices are measurable and consider the vector of measured outputs:  
\[\mathbf{y}(t)=[y_{1}(t),\ldots,y_{n_{d}}(t)]^{T}, \tag{30}\]  
where \(n_{d}\) is the total number of actuation devices and \(y_{j}(t)\) represents the relative velocity corresponding to the actuation device \(d_{j}\). For the control configuration \(\ell\), the vector of measured  
Figure 4: Actuation schemes with four actuation devices. (**a**) Unlinked configuration CC5, with four interstory actuators (\(d_{1}\), \(d_{2}\), \(d_{3}\) and \(d_{4}\)). (**b**) Linked configuration CC6, with three interstory actuators (\(d_{1}\), \(d_{2}\) and \(d_{3}\)) and an interbuilding actuator (\(d_{4}\)). outputs can be written as a linear combination of the state variables defined in Equation (15) using the measured-output matrix  
\[\mathbf{C}_{\mathrm{y}}^{\ell}=\left[ \begin{array}{cc}\left[\mathbf{0}\right]_ {n_{d}\times 9}&\left(\mathbf{T}_{\mathrm{u}}^{\ell}\right)^{T}\end{array} \right], \tag{31}\]  
where \(\mathbf{T}_{\mathrm{u}}^{\ell}\) is the corresponding control location matrix. Thus, for the configuration CC6 displayed in Figure 4b, the number of actuation devices is \(n_{d}=4\) and the measured-output vector has the form \(\mathbf{y}(t)=[y_{1}(t),y_{2}(t),y_{3}(t),y_{4}(t)]^{T}\), where \(y_{1}(t)\) is the interstory velocity at the first-story level in building \(\mathcal{B}^{(1)}\), \(y_{2}(t)\) and \(y_{3}(t)\) denote the interstory velocity at the first and second story levels in building \(\mathcal{B}^{(2)}\), respectively, and \(y_{4}(t)\) is the interbuilding velocity at the four-story level. In this case, by considering the matrix \(\mathbf{T}_{\mathrm{u}}^{\mathrm{u}}\) given in Equation (26), we obtain the measured-output matrix  
\[\mathbf{C}_{\mathrm{y}}^{\mathrm{u}}= \begin{bmatrix}0&0&0&0&0&0&0&0&0&1&0&0&0&0& 0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0&0&-1&1&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0&0&-1&0&0&0&1&0\end{bmatrix} . \tag{32}\]  
To define the controlled-output vector \(\mathbf{z}(t)\), it should be noted that large interstory drifts and interbuilding approaches must both be avoided in order to prevent buildings' structural damage and interbuilding collisions. Additionally, moderate control efforts are also convenient. To this end, we consider the controlled-output matrices  
\[\mathbf{C}_{z}=\left[ \begin{array}{c}\alpha_{\mathrm{r}}\mathbf{C}_{\mathrm{ r}}\\ \alpha_{\mathrm{a}}\mathbf{C}_{\mathrm{a}}\\ \left[\mathbf{0}\right]_{n_{d}\times 18}\end{array} \right],\quad\mathbf{D}_{z} =\left[ \begin{array}{c}\left[\mathbf{0}\right]_{13\times n_{d}}\\ \alpha_{\mathrm{u}}\mathbf{I}_{n_{d}}\end{array}