## 5 Results
### Test-Case 4: Four-Bar Flexible-Link Multibody System
\begin{table}
\begin{tabular}{c c} \hline \hline
**Link Parameters [unit]** & **Value** \\ \hline Young modulus [Pa] & \(210\cdot 10^{9}\) \\ Second area moment [m\({}^{4}\)] & \(11.1\cdot 10^{-10}\) \\ Width [m] & \(6\cdot 10^{-3}\) \\ Thickness [m] & \(6\cdot 10^{-3}\) \\ Linear mass density [kg m\({}^{-1}\)] & \(272\cdot 10^{-3}\) \\ Link 0 length [m] & 0.360 \\ Link 1 length [m] & 0.390 \\ Link 2 length [m] & 0.535 \\ Link 3 length [m] & 0.632 \\ Rayleigh coefficients [s\({}^{-1}\)], [s] & \(\alpha_{\text{R}}=8.5\cdot 10^{-2}\), \(\beta_{\text{R}}=2\cdot 10^{-5}\) \\ \hline \hline \multicolumn{2}{c}{Nodal lumped masses [kg]:} \\ Joint B & \(70\cdot 10^{-3}\) \\ Joint C & \(70\cdot 10^{-3}\) \\ \hline \hline \multicolumn{2}{c}{Joint inertia [kg m\({}^{2}\)]} \\ Joint A & \(5\cdot 10^{-4}\) \\ Joint D & \(12\cdot 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Four-bar flexible-link mechanism parameters ([16]). To tackle the risk of ill numerical conditioning of the SDP programming, due to the different magnitudes of the entries of the mass, damping and stiffness matrices, the system matrices are scaled by defining the scaling parameters \(\gamma\) and \(\beta\):  
\[\widetilde{\mathbf{M}}=\beta^{-2}\gamma^{-1}\mathbf{M},\;\;\widetilde{\mathbf{C} }=\beta^{-2}\gamma^{-1}\mathbf{C}\;,\;\;\widetilde{\mathbf{K}}=\gamma^{-1} \mathbf{K}, \tag{33}\]  
which leads to the scaled receptance matrix (adopted in the first stage):  
\[\widetilde{\mathbf{H}}(\mathrm{i}\widetilde{\omega})=\left(\widetilde{\omega} ^{2}\widetilde{\mathbf{M}}+\mathrm{i}\widetilde{\omega}\widetilde{\mathbf{C} }+\widetilde{\mathbf{K}}\right)^{-1}\text{ with }\widetilde{\omega}=\beta\omega. \tag{34}\]  
The issue of numerical conditioning is exacerbated in this kind of problem due to the medium-large number of DOFs and to the simultaneous presence of low and high frequency vibrational modes. Therefore, as shown in [4], some numerical methods for pole placement often might fail. Accordingly to the scaled receptance matrix, scaled gain vectors \(\widetilde{f}_{0}\) and \(\widetilde{g}_{0}\) are computed in the first stage and then in the unscaled model:  
\[f_{0}=\beta\gamma\widetilde{f}_{0},\;\;g_{0}=\gamma\widetilde{g}_{0}. \tag{35}\]  
In the second stage, the state space system matrices in Equation (12) are scaled accordingly to the definitions provided in Equation (33) The LMI conditions have been scaled too. The following scaling parameters have been here adopted: \(\gamma=1\cdot 10^{5}\) and \(\beta=1\cdot 10^{-4}\). The control specifications prescribe to assign the two real poles, that in practice represent the \"rigid-body\" motion, and the three lowest-frequency complex conjugate pole pairs, as reported in Table 9. The remaining closed-loop poles must belong to the LMI ensuring that \(\xi(\mu_{i})\geq 0.05\). The gain \(\mathbf{k}_{0}\) and \(\mathbf{k}\) are reported in Table 10, while the open-loop and the closed-loop poles are summarized in Table 9 and shown in the complex plane in Figure 11. The application of \(\mathbf{k}_{0}\) alone leads to an unstable pole pair at \(28.6\pm 244.7\)i. The second stage of the method ensures the stabilization of the unstable pole pair, which is placed at \(-74.0\pm 215.7\)i. At the same time, it damps all the remaining unassigned poles accordingly with the requirements, as shown in the low frequency detail of the pole map in Figure 12