## 5 Results
### Test-Case 2: Slider-Belt System  
The second test-case exploits the asymmetric model of a slider-belt system with friction, proposed in [2; 6; 19; 26]. The system is sketched in Figure 3. This system has marginally stable, or even unstable, poles depending on the friction coefficient, here denoted as \(\eta\). For this reason, this test-case is here considered to show that the proposed method can handle both the instability and the asymmetry issues. The model independent coordinates are the translations of the three masses: \(q=\left[x_{1},\ y_{3},x_{2},\ y_{2}\right]^{T}\). The asymmetric stiffness matrix \(\mathbf{K}\) is the sum of two parts: the symmetric part \(\mathbf{K}_{s}\) and the asymmetric part \(\mathbf{K}_{a}\). The latter is caused by the pre-compression normal force and the friction force that acts at the slider-belt interface. The resulting system matrices are reported in Equation (29): \begin{table}
\begin{tabular}{c c c c} \hline
**Open-loop poles** & **Desired Closed-Loop Poles** & **Closed-Loop Poles with \(k_{0}\)** & **Closed-Loop Poles with \(k\)** \\ \hline \(-0.006\pm 1.896\)i & \(-0.001\pm 1.50\)i & \(-0.001\pm 1.50\)i & \(-0.001\pm 1.500\)i \\ \(-0.013\pm 2.769\)i & \(-0.001\pm 3.00\)i & \(-0.001\pm 3.00\)i & \(-0.001\pm 3.000\)i \\ \(-0.021\pm 3.569\)i & - & \(0.010\pm 3.520\)i & \(-0.146\pm 3.430\)i \\ \hline \end{tabular}
\end{table} 
Table 2: Open-loop and closed-loop poles (test-case 1). \begin{table}
\begin{tabular}{c c c c} \hline
**Stage 1: \(k_{0}\)** & **Stage 2: \(k\)=\(k_{0}\)+V\(k_{r}\)** \\ \hline \(k_{0}\) & \(k_{0}\) & \(k\) & \(k\) \\ \hline \(-0.0962\) & \(-0.3545\) & \(0.2185\) & \(-0.9705\) \\ \(0.1581\) & \(-1.6452\) & \(-0.5433\) & \(-0.2314\) \\ \(-0.0349\) & \(-3.3434\) & \(0.2670\) & \(-4.0191\) \\ \hline \end{tabular}
\end{table} 
Table 3: Gains (test-case 1). Figure 2: Open-loop and closed-loop poles of the three-mass system (test-case 1). \[\mathbf{M}=\text{diag}(m_{1},m_{3},m_{2},\ m_{2})\ \ \mathbf{C}=\left[ \begin{array}{ c c c}c_{1}&0&-c_{1}&0\\ 0&0&0&0\\ -c_{1}&0&c_{1}&0\\ 0&0&0&c_{0}\end{array} \right]\ \mathbf{b}=\left[1,\ 0,\ 0,\ 0\right]^{T} \tag{29}\] \[\mathbf{K}_{s}=\left[ \begin{array}{ c c c c}k_{1}+k_{2}&0&-k_{2}&0\\ 0&k_{4}+k_{5}&0&-k_{4}\\ -k_{2}&0&k_{2}+0.5k_{3}&-0.5k_{3}\\ 0&-k_{4}&-0.5k_{3}&k_{4}+0.5k_{3}+k_{c}\end{array} \right]\ \mathbf{K}_{a}=\left[ \begin{array}{ c c c c}0&0&0&0\\ 0&0&0&0\\ 0&0&0&\eta k_{c}\\ 0&0&0&0\end{array} \right]\.\]  
The model parameters have been assumed as in [19]: \(m_{1}=m_{2}=m_{3}=1\) kg, \(k_{1}=k_{2}=k_{3}=k_{4}=k_{5}=100\) Nm\({}^{-1}\), \(k_{c}=2\ k_{1}\), \(c_{0}=c_{1}=0.5\) Nsm\({}^{-1}\). The value of the friction coefficient is the one at the flutter instability, i.e., \(\eta=0.3868\). Indeed, a pair of complex conjugate poles with real part appears: \(0.00\pm 8.73\)i [19]. The control is designed to exactly place two pairs of dominant poles into prescribed locations of the complex plane, to ensure stability and the desired dynamic responses, while the remaining two pairs of dominated poles are constrained into an LMI. The results are compared with the standard PPP methods used in vibration control, such as for example those proposed in [6; 7]. Traditional PPP techniques differ from the RPP technique proposed in this paper, since they assign \(p\) poles while the remaining \(2N-p\) poles are kept unchanged, i.e., \(2N-p\) open-loop poles will coincide with the closed-loop ones. To this purpose, the two desired complex conjugate pole pairs in Table 4 are assigned at \(-1\pm 9\)i and \(-1\pm 13.5\)i, while the remaining two complex conjugate pole pairs (\(-0.51\pm 16.75\)i and \(-0.19\pm 19.86\)i) are kept unchanged. The control gains computed by this benchmark, denoted as \(\mathbf{k}_{B}\), are reported in Table 5.