## 5 Simulations  
To evaluate the effectiveness of the proposed controller, simulations of system response in the closed loop were performed with the above derived transfer function \(G(s)\). The excitation signal was a summation of the two sinusoidal signals of modal frequencies 1.5 Hz and 9 Hz for a period of 10 s at 0.5 V. A saturation block with limits of \(\pm\)5 V was incorporated to limit the control effort, as it would be in implementation. The simulated time responses for the open and closed loop system in a total period of 40 s are shown in Figure 12.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Open Loop Poles** & **Open Loop Damping Ratio** & **Closed Loop Poles** & **Close Loop Damping Ratio** \\ \hline \(-\)0.036 \(\pm\) 9.33i & 0.004 & \(-\)6.59 \(\pm\) 9.33i & 0.707 \\ \(-\)0.21 \(\pm\) 56.9i & 0.004 & \(-\)40.23 \(\pm\) 56.9i & 0.707 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Comparisons between the closed loop and open loop poles.  
Figure 11: Optimal locations of the closed loop poles.  
Figure 12: Simulated time responses for open and closed loop system. PP: Poles Placement.  
As plotted in Figure 12, the excited vibrations for the open loop system take more than 30 s to eliminate, while with the poles placement controller, the system vibrations die down in less than 15 s. Obviously, system vibrations are effectively suppressed. Furthermore, two sets of optional locations for the closed loop poles, as given in Table 8, are chosen to verify the effectiveness of the proposed controller.  
After the locations of the desired poles are determined, Equation (15) can be rewritten as  
\[|s\mathbf{I}-(A-BK)|=(s-p_{d})\prod_{i=1}^{2}(s-p_{i})(s-\hat{p}_{i}) \tag{16}\]  
where \(p_{i}\) and \(\hat{p}_{i}\) are the complex-conjugate poles, and \(p_{d}\) is the pole corresponding to the first-order inertia element. Then, the feedback gain matrix can be solved using Equation (16).  
Simulation results with different locations of closed loop poles are illustrated in Figure 13a-d. It can be seen that a further location of the closed loop poles leads to a better control performance, but a severer spillover phenomenon is caused as shown in Figure 13b. On the other hand, the spillover phenomenon is avoided with a closer location of the closed loop poles as plotted in Figure 13c. However, the effectiveness of the controller decreases unavoidably. It can be seen more clearly from the power spectral density plot Figure 13d, there are excitation energy peaks at 50 Hz for the cases \(\zeta\) = 0.707 and 0.85. Therefore, the spillover phenomenon appears as the peaks are outside the desired frequency bandwidth.  
\begin{table}
\begin{tabular}{l l c} \hline \hline
**Optional Location** & **Closed-Loop Poles** & **Damping Ratio** \\ \hline \multirow{2}{*}{Closer location} & \(-3.73\pm 9.33\)i & \multirow{2}{*}{0.4} \\  & \(-22.76\pm 56.9\)i & \\ \hline \multirow{2}{*}{Optimal location} & \(-6.59\pm 9.33\)i & \multirow{2}{*}{0.707} \\  & \(-40.23\pm 56.9\)i & \\ \hline \multirow{2}{*}{Further location} & \(-7.93\pm 9.33\)i & \multirow{2}{*}{0.85} \\ \hline \hline \end{tabular}
\end{table}
Table 8: Comparisons of optional locations for closed loop poles.  
Figure 13: _Cont._