## 4 Active Antiresonance Assignment with No Additional Masses
### Numerical Examples of Active Control
#### 4.4.2 Numerical Examples of Active Control: Assignment of More Antiresonances
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Gain Vector \(\mathbf{k_{0}}\)** & **State-Feedback Method**[50] & **State-Feedback Method**[50] \\  & **Solved Through Pinv** & **Solved Through MIdivide** \\ \hline
**Velocity gains f** & & \\ \hline f\({}_{1}\) & \(0.0129\) & \(0.0263\) \\ f\({}_{2}\) & \(0.0093\) & \(0\) \\ f\({}_{3}\) & \(0\) & \(0\) \\ f\({}_{4}\) & \(-0.0093\) & \(0\) \\ f\({}_{5}\) & \(-0.0201\) & \(0\) \\ f\({}_{6}\) & \(-0.0324\) & \(-0.0580\) \\ \hline
**Position gains g** & & \\ \hline g\({}_{1}\) & \(-0.391\) & \(-0.500\) \\ g\({}_{2}\) & \(-0.109\) & \(0\) \\ g\({}_{3}\) & \(0\) & \(0\) \\ g\({}_{4}\) & \(0.109\) & \(0\) \\ g\({}_{5}\) & \(-0.457\) & \(0\) \\ g\({}_{6}\) & \(-0.609\) & \(-0.920\) \\ \hline \hline \(\|\mathbf{k_{0}}\|_{2}\) & \(0.8707\) & \(1.0495\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Control gains in the numerical example of the assignment of zeros through the unit-rank active control for receptance \(h_{3,4}(j\omega)\). If the integral action is included too, as in [52], both the solutions of Equation (26), obtained through _mldivide_ and _pinv_ lead to RHCP poles, as shown in Figure 10.  
4.3 Numerical Examples of Active Control: Assignment of More Antiresonances with Embedded an a-Priori Stability Condition  
The test case of Section 4.4.2 is solved in this section through the two-stage approach proposed in [51] to ensure that the closed-loop poles must belong to the left half of the complex plane for asymptotic stability, i.e., introducing a LMI with \(\mathbf{R}=0\) and \(\mathbf{Z}=1\) in Equation (25). The first stage is solved, again, through the _pinv_ and _mldivide_ functions leading to the same control gains \(\mathbf{k_{0}}\) reported in Table 9 and whose pole-zero map are shown in Figure 10. In the second stage, the semi-definite programming in Equation (24) is solved through the YALMIP toolbox for MATLAB [77] adopting [PERSON] as the solver. Figure 11 highlights that in both cases, the control gains \(\mathbf{k}=\mathbf{k_{0}}+\mathbf{Vk_{r}}\) (reported in Table 10) ensure the asymptotic stability of the closed-loop system, since the closed-loop poles belong to the LHCP (whose shape is highlighted by the dashed line in Figure 11). Figure 10: Open-loop and closed-loop pole-zero maps of receptance \(h_{3,4}(j\omega)\) with different controllers featuring integral action. The steady-state responses of the open-loop and closed-loop systems, controlled with the stabilizing gains computed through _mldivide_, are compared in Figure 12a to corroborate the capability of absorbing vibrations when a harmonic disturbance with a frequency equal to 0.500 \(\text{rads}^{-1}\) excites coordinate 4. The control force is reported in Figure 12b. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Gain Vectork = k0+Vk\({}_{\text{r}}\)**} & **State-Feedback Method** & **State-Feedback Method** \\  & [51](**k0 Obtained Through Pinv**) & [51](**k0 Obtained Through MIdivide**) \\ \hline \multicolumn{3}{c}{**Velocity gains f**} \\ \hline f\({}_{1}\) & 0.0134 & 0.0274 \\ f\({}_{2}\) & 0.0227 & 0.0119 \\ f\({}_{3}\) & 0.0141 & 0.0131 \\ f\({}_{4}\) & 0.0052 & 0.014 \\ f\({}_{5}\) & \(-\)0.0309 & \(-\)0.0192 \\ f\({}_{6}\) & \(-\)0.0244 & \(-\)0.0439 \\ \hline \multicolumn{3}{c}{**Position gains g**} \\ \hline g\({}_{1}\) & \(-\)0.391 & \(-\)0.500 \\ g\({}_{2}\) & \(-\)0.109 & 0 \\ g\({}_{3}\) & 0 & 0 \\ g\({}_{4}\) & 0.109 & 0.001 \\ g\({}_{5}\) & \(-\)0.458 & 0 \\ g\({}_{6}\) & \(-\)0.608 & \(-\)0.921 \\ \hline \multicolumn{3}{c}{\(\|\mathbf{k}\|_{2}\)} \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Stabilizing control gains obtained in the second stage of the method in the numerical example of the assignment of zeros through the unit-rank active control for acceptance \(h_{3,4}(i\omega)\) with LMI constraints. Figure 11: Open-loop and closed-loop pole-zero maps of receptance \(h_{3,4}(i\omega)\): (**a**) with the _mldivide_ gains, (**b**) with the _pirv_ gains.