## 3 Controllers Design
### Static Output-Feedback \(H_{\infty}\) Centralized Controllers
Following the same design procedure, we obtain the velocity-feedback control gain matrices  
\[\mathbf{K}^{\rm II}= 10^{7}\times\left[ \begin{array}{rrr}-1.4671&0.3369&0.5595\\ 0.0936&-0.6758&-0.4428\\ 0.1090&-0.5970&-0.5382\end{array} \right],\ \mathbf{K}^{\rm IV}= 10^{6}\times\left[ \begin{array}{rrr}-6.8093&-3.7631&-0.3129\\ -1.2553&-8.6084&0.3653\\ 0.5425&0.9675&-2.9665\end{array} \right], \tag{41}\]  
for the control configurations CC3 and CC4, respectively, the control gain matrix  
\[\mathbf{K}^{\rm V}= 10^{7}\times\left[ \begin{array}{rrr}-1.0323&0.4066&0.2143&-0.45 38\\ -0.3656&-0.5823&-0.3065&-0.0591\\ 0.9548&-1.7601&-1.5193&0.2758\\ 0.3918&0.3666&-0.5359&-0.6302\end{array} \right] \tag{42}\]  
for the configuration CC5, and finally, the control gain matrix  
\[\mathbf{K}^{\rm V}= 10^{6}\times\left[ \begin{array}{rrr}-4.7133&-4.9748&-3.9433&1. 9188\\ -4.1747&-5.9723&-2.0281&-1.4118\\ -1.6926&-4.8034&-4.5185&0.1502\\ -4.2366&3.2550&6.0731&-6.8744\end{array} \right] \tag{43}\]  
for the control configuration CC6. The corresponding \(\gamma\)-values and upper bounds \(\widetilde{\gamma}_{t}\) are collected in Table 2. The frequency responses are presented in Figures 6 and 7, using a blue solid line for the unlinked configurations (CC3 and CC5), and a red dash-dotted line for the linked configurations (CC4 and CC6). According to the \(\gamma\)-values in Table 2 and the value \(\gamma_{0}\) in Equation (40), three facts can be noted: (i) all the controllers obtained with the proposed design methodology provide a good level of reduction in the \(H_{\infty}\)-norm value, (ii) an increasing effectiveness is attained with a larger number of actuation devices, and (iii) smaller \(\gamma\)-values are obtained by the unlinked control configurations. All these facts can also be clearly appreciated by observing the peak-values in the frequency plots presented in Figures 5-7. These frequency plots also show that, in all the cases, a significant reduction of the vibrational response is additionally attained in the secondary resonant peaks. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & **CC1** & **CC2** & **CC3** & **CC4** & **CC5** & **CC6** \\ \hline Upper bound \(\widetilde{\gamma}\) & 0.5973 & 0.8851 & 0.3856 & 0.6528 & 0.3396 & 0.4333 \\ \(\gamma\)-value & 0.4967 & 0.7833 & 0.3395 & 0.5037 & 0.2855 & 0.3277 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Active configurations: Actual \(\gamma\)-values and \(\gamma\)-value upper bounds corresponding to the centralized velocity-feedback controllers defined by the gain matrices \(\mathbf{K}^{\mathrm{el}}\). Figure 5: Frequency response corresponding to the centralized velocity-feedback \(H_{\infty}\) controllers with active implementation defined by the control gain matrices \(\mathbf{K}^{\mathrm{el}}\) and \(\mathbf{K}^{\mathrm{el}}\). Maximum singular values of the closed-loop transfer function \(\mathbf{T}_{\mathrm{K}^{\mathrm{el}}}(2\pi fj)\) (thick blue solid line), the closed-loop transfer function \(\mathbf{T}_{\mathrm{K}^{\mathrm{el}}}(2\pi fj)\) (red dash-dotted line) and the open-loop transfer function \(\mathbf{T}(2\pi fj)\) (thin black solid line). Figure 6: Frequency response corresponding to the centralized velocity-feedback \(H_{\infty}\) controllers with active implementation defined by the control gain matrices \(\mathbf{K}^{\mathrm{el}}\) and \(\mathbf{K}^{\mathrm{el}}\). Maximum singular values of the closed-loop transfer function \(\mathbf{T}_{\mathrm{K}^{\mathrm{el}}}(2\pi fj)\) (thick blue solid line), the closed-loop transfer function \(\mathbf{T}_{\mathrm{K}^{\mathrm{el}}}(2\pi fj)\) (red dash-dotted line) and the open-loop transfer function \(\mathbf{T}(2\pi fj)\) (thin black solid line).