## 3 Controllers Design
### Static Velocity-Feedback \(H_{\infty}\) Decentralized Controllers
Using the same design strategy to compute decentralized velocity-feedback controllers for the other control configurations considered in this paper, we have obtained diagonal gain matrices with the values presented in Table 3. Comparing the corresponding \(\gamma\)-values collected in Table 4 and the value \(\gamma_{0}\) in Equation (40), it can be appreciated that the same three facts observed in the centralized controller designs also apply to the decentralized controllers: (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. However, two new elements appear in the decentralized design: (iv) the risk of failure of the controller design procedure is higher in the decentralized designs, and (v) the \(\gamma\)-value attained by some decentralized controllers is lower than the one obtained by the associated centralized controller. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & **CC1** & **CC2** & **CC3** & **CC4** & **CC5** & **CC6** \\ \hline Upper bound \(\widetilde{\gamma}\) & 0.6537 & 0.8865 & 0.5745 & 0.7657 & – & 0.6079 \\ \(\gamma\)-value & 0.5500 & 0.7646 & 0.3241 & 0.5987 & – & 0.3058 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Passive configurations: Actual \(\gamma\)-values and \(\gamma\)-value upper bounds corresponding to the decentralized velocity-feedback controllers defined by the diagonal gain matrices \(\mathbf{\hat{K}}^{\ell}\). The values of the control configuration CC5 are missing due to the feasibility issues encountered in the associated linear matrix inequality (LMI) optimization problem. Figure 8: Frequency response corresponding to the decentralized velocity-feedback \(H_{\infty}\) controller with passive implementation defined by the diagonal control gain matrix \(\mathbf{\hat{K}}^{\ell}\). Maximum singular values of the closed-loop transfer function \(\mathbf{T}_{\mathbf{\hat{K}}^{\ell}}(2\pi fj)\) (red dash-dotted line), the closed-loop transfer function \(\mathbf{T}_{\mathbf{\hat{K}}}(2\pi fj)\) (thick blue solid line) and the open-loop transfer function \(\mathbf{T}(2\pi fj)\) (thin black solid line). The increased risk of failure in the decentralized design procedure can be explained by the additional structure constraints introduced in the LMI optimization problem, which can produce feasibility issues. This kind of numerical problems are poorly understood and sometimes depend on the particular numerical solver and the options used in the LMI optimization procedure. In our case, the feasibility issues have only been encountered in the design of a diagonal gain matrix for the control configuration CC5. Regarding the \(\gamma\)-value produced by some decentralized controllers, by comparing the \(\gamma\)-values in Tables 2 and 4, it can be seen that a lower \(\gamma\)-value is attained by the decentralized controllers for the control configurations CC2, CC3 and CC6. A clear view of this situation can be obtained in Figure 9, which presents the frequency response corresponding to the passive control system defined by the diagonal gain matrix \(\widehat{\mathbf{K}}^{\mathrm{u}}\) (red dash-dotted line) and the corresponding centralized controller defined by the gain matrix \(\mathbf{K}^{\mathrm{u}}\) given in Equation (43) (thick blue solid line). To explain this unexpected fact, it should be noted that the LMI optimization procedure is based on the upper bound \(\widetilde{\gamma}\) and produces a suboptimal \(H_{\infty}\) controller.