## 3 Controllers Design
### Static Velocity-Feedback \(H_{\infty}\) Decentralized Controllers  
From a practical point of view, the controllers defined by the velocity-feedback gain matrices \(\mathbf{K}^{\ell}\) computed in the previous section have the important advantage of using a reduced system of sensors which are naturally associated to the actuation devices. However, they also present two serious drawbacks. Firstly, the complete vector of measured outputs is used to compute the control actions and, consequently, a wide communication system would be necessary in the controller implementation. Secondly, producing the corresponding actuation forces would require active devices with a large power consumption and potential reliability issues. These two disadvantages, typically present in active vibration control of large structures, can be properly overcome by considering fully decentralized velocity-feedback controllers with control gain matrices of the form  
\[\mathbf{\hat{K}}^{\ell}=\text{diag}(\xi_{1}^{\ell},\dots,\xi_{n_{d}}^{\ell}), \tag{44}\]  
which can be obtained by solving the LMI optimization problem \(\mathcal{P}\) in Equation (A8) with the same matrices used in the previous controller designs and constraining the LMI variable matrices \(\mathbf{X}_{\text{R}}\) and \(\mathbf{Y}_{\text{R}}\) to a diagonal form. As indicated in [17; 40], if the gain matrix elements \(\xi_{i}^{\ell}\) are all negative, then this kind of controllers admit a passive implementation using linear viscous dampers. Thus, for instance, by applying this design methodology to the control configuration CC1, we obtain the following diagonal gain matrix:  
\[\mathbf{\hat{K}}^{\text{t}}=10^{7}\times\left[ \begin{array}{cc}-0.8543&0\\ 0&-1.2968\end{array} \right]. \tag{45}\]  
Hence, the decentralized velocity-feedback controller defined by the diagonal gain matrix \(\mathbf{\hat{K}}^{\text{t}}\) can be implemented using two interstory linear damping devices \(d_{1}\) and \(d_{2}\) with respective damping constants \(0.8543\times 10^{7}\,\text{Ns/m}\) and \(1.2968\times 10^{7}\,\text{Ns/m}\). The frequency response characteristics of this passive control system are displayed in Figure 8 using a red dash-dotted line. The frequency response corresponding to the uncontrolled configuration (thin black solid line) and the centralized controller defined by the gain matrix \(\mathbf{K}^{\text{t}}\) (thick blue solid line) are also included as a reference. The plots in the figure clearly show the good behavior of the obtained passive controller, which practically matches the performance of the active centralized controller over most of the frequency range, and produces a small increment (of about 10%) in the peak-value corresponding to the main resonant frequency. Figure 7: Frequency response corresponding to the centralized velocity-feedback \(H_{\infty}\) controllers with active implementation defined by the control gain matrices \(\mathbf{K}^{\text{v}}\) and \(\mathbf{K}^{\text{v}}\). Maximum singular values of the closed-loop transfer function \(\mathbf{T}_{\text{K}^{\text{v}}}(2\pi fj)\) (thick blue solid line), the closed-loop transfer function \(\mathbf{T}_{\text{K}^{\text{v}}}(2\pi fj)\) (red dash-dotted line) and the open-loop transfer function \(\mathbf{T}(2\pi fj)\) (thin black solid line).