## 5 Results
### Test-Case 2: Slider-Belt System
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Open-Loop** & **Desired** & **Closed-Loop Poles** & **Desired** & **Closed-Loop Poles** & **Closed-Loop Poles with \(\mathbf{k}_{0}\)** & **Closed-Loop Poles with \(\mathbf{k}\)** \\ \hline \(0.00\pm 8.73\)i & \(-1\pm 9\)i & \(-1\pm 9\)i & \(-1\pm 9\)i & \(-1\pm 9\)i & \(-1\pm 9\)i \\ \(-0.05\pm 12.19\)i & \(-1\pm 13.5\)i & \(-1\pm 13.5\)i & \(-1\pm 13.5\)i & \(-1\pm 13.5\)i & \(-1\pm 13.5\)i \\ \(-0.51\pm 16.75\)i & \(-0.51\pm 16.75\)i & \(-0.51\pm 16.75\)i & - & \(-5.55\pm 12.4\)i & \(-5.55\pm 12.4\)i \\ \(-0.19\pm 19.86\)i & \(-0.19\pm 19.86\)i & \(-0.19\pm 19.86\)i & - & \(-0.15\pm 19.98\)i & \(-0.23\pm 20.0\) \\ \hline \end{tabular}
\end{table} 
Table 4: Open-loop and closed-loop poles comparison: traditional technique and the proposed method (test-case 2). Figure 3: Sketch of the slider-belt system (test-case 2). To ensure a fair comparison, the proposed method performs RPP within the LMI \(\text{Re}(\mu_{i})\leq-0.19\) that includes the two open-loop pairs of poles at \(-0.51\pm 16.75\text{i}\) and \(-0.19\pm 19.86\text{i}\). Clearly, \(\mathbf{k}_{B}\) is a possible solution of the assignment problem, although it is expected that the proposed method is able to improve it. The gains after the first and second stage of the proposed method are summarized in Table 5 and compared with \(\mathbf{k}_{B}\). The desired and the obtained poles are shown in Table 4 and in Figure 4, where they are compared with those provided by the traditional partial pole placement as well. The comparison of the control gains in Table 5 reveals that RPP leads to a smaller norm compared to PPP: \(\|\mathbf{k}_{B}\|_{2}=177.9\), while \(\|\mathbf{k}_{0}\|_{2}=26.7\) and \(\|\mathbf{k}\|_{2}=67.6\). To better compare the control effort, the actuator energy in a free response of the controlled system tracking the zero-vibration condition can be evaluated ([19; 27]), by assuming a meaningful set of initial conditions of the state and by simulating a time interval \(T\) covering several oscillation periods:  
\[E=\int\limits_{0}^{T}\left(\mathbf{f}^{T}\dot{\mathbf{q}}(t)+\mathbf{g}^{T}\mathbf{q}(t)\right) ^{2}\mathbf{b}^{T}\mathbf{b}dt. \tag{30}\]  
The following velocity and displacement initial conditions have been set: \(\dot{\mathbf{q}}(0)=1\cdot 10^{-3}\left\{ \begin{array}{ccccc}1&1&1&1&1\end{array}  \right\}^{T}\) and \(\mathbf{q}(0)=1\cdot 10^{-3}\left\{ \begin{array}{ccccc}1&1&1&1&1\end{array}  \right\}^{T}\). The system response has been computed by numerically solving the system ODEs with the ODE45 solver, as shown in Figures 5 and 6. The resulting control effort is drastically lower, since it decreases from 0.0123 J to 0.0053 J, as shown in Figure 7.  
Figure 4: Open-loop and closed-loop poles of the slider-belt system (test-case 2). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{\(k_{B}\)} & \multicolumn{3}{c}{**Stage 1: \(k_{0}\)**} & \multicolumn{3}{c}{**Stage 2: \(\mathbf{k}\)=\(\mathbf{k}_{0}\)+\(\mathbf{V}\mathbf{k}_{\tau}\)**} \\ \hline \(f_{B}\) & \(g_{B}\) & \(f_{0}\) & \(g_{0}\) & \(f\) & \(g\) \\ \hline
3.8949 & 43.7930 & 13.9035 & \(-12.3773\) & 14.0522 & \(-8.9368\) \\ \(-4.2244\) & \(-150.4119\) & \(-13.0355\) & \(-6.5512\) & \(-13.0777\) & \(-33.0146\) \\ \(4.3004\) & \(26.3084\) & \(-1.8911\) & \(-5.1848\) & \(-1.9718\) & \(-16.8434\) \\ \(-2.3322\) & \(-79.6829\) & \(-10.6007\) & \(-3.5384\) & \(-10.6159\) & \(51.3344\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Gains comparison: traditional technique, \(\mathbf{k}_{B}\), and the proposed method, \(\mathbf{k}\) (test-case 2). Figure 5: Velocities of the controlled system comparison: traditional PPP and the proposed method (test-case 2). Figure 6: Displacements of the controlled system comparison traditional PPP and the proposed method (test-case 2). This test also compares the effectiveness of the RPP and of the PPP in vibration control, by analyzing the speed and displacement responses of the controlled system shown in Figures 5 and 6.