## 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  
The goal of this section is to assign two antiresonances, i.e., two complex conjugate pairs of zeros (\(0\pm 0.5j\) and \(-0.01\pm 1j\)), in the cross-receptance \(h_{3,4}(j\omega)\). For example, let us consider \(\mathbf{b_{C}}=[1,0,0,0,1,0]^{T}\), which ensures the complete assignability of the acceptance zeros. The obtained zeros are reported in Table 8, by solving again Equation (26) with the two methods discussed in Section 4.4.1; the control gains are shown in Table 9.  
Once again, the assignment task is correctly fulfilled. As expected, the solution provided by the pseudoinverse leads to a smaller 2-norm of vector \(\mathbf{k_{0}}\), with the drawback of obtaining a fully populated gain vector which requires more sensors or a state observer. However, it leads to a pair of RHCP complex conjugate poles as well as the solution provided by _mldivide_, as shown through the pole-zero map in Figure 9. This fact highlights that the solution of the linear system in Equation (20) should be done carefully to ensure stability. This issue is tackled in Section 4.4.3 through the method proposed in [51] to cluster the closed-loop poles to ensure the stability of the closed-loop system. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Original System Zeros** & **Method [52]** **Solved Through** & **Method [52]** **Solved Through** \\  & **Pinv** & **MIdivide** \\ \hline - & \(-0.0049\) & - \\ \(-0.005\pm 0.720\) \(j\) & \(-0.005\pm 0.720\) \(j\) & \(-0.005\pm 0.720\) \(j\) \\ \(-0.005\pm 1.176\) \(j\) & \(0\pm 1.000\) \(j\) & \(0\pm 1.000\) \(j\) \\ \(-0.005\pm 1.520\) \(j\) & \(-0.005\pm 1.520\) \(j\) & \(-0.005\pm 1.520\) \(j\) \\ \(-0.005\pm 1.902\) \(j\) & \(-0.005\pm 1.897\) \(j\) & \(-0.004\pm 1.871\) \(j\) \\ \(-0.005\pm 2.042\) \(j\) & \(-0.005\pm 2.042\) \(j\) & \(-0.005\pm 2.042\) \(j\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Zeros of the original and modified receptance \(h_{3,3}(j\omega)\) with the controller with integral action to assign a complex conjugate zero pair. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Gain Vector k** & **Method [52]** **Solved Through Pinv** & **Method [52]** **Solved** \\  & & **Through MIdivide** \\ \hline \(\left\|\mathbf{f}\right\|_{2}\) & \(0.0059\) & \(0.0125\) \\ \(\left\|\mathbf{g}\right\|_{2}\) & \(0.4472\) & \(0.5000\) \\ \(\left\|\mathbf{n}\right\|_{2}\) & \(0.0059\) & \(0\) \\ \hline \(\left\|\mathbf{k}_{0}\right\|_{2}\) & \(0.4473\) & \(0.5002\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Control gain 2-norms in the numerical example of the assignment of zeros through unit-rank active control with integral action. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Original System Zeros** &
\begin{tabular}{c} **State-Feedback Method [50]** \\ **Solved Through Pinv** \\ \end{tabular} & **State-Feedback Method [50]** \\  & **Solved Through Mildivide** \\ \hline \(-0.005\pm 0.765\) \(j\) & \(0\pm 0.500\) \(j\) & \(0\pm 0.500\) \(j\) \\ \(-0.005\pm 1.176\) \(j\) & \(-0.010\pm 1.000\) \(j\) & \(-0.010\pm 1.000\) \(j\) \\ \(-0.005\pm 1.848\) \(j\) & \(-0.001\pm 1.774\) \(j\) & \(-0.105\pm 1.902\) \(j\) \\ \(-0.005\pm 1.902\) \(j\) & \(-0.006\pm 1.938\) \(j\) & \(0.082\pm 1.910\) \(j\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Zeros of the original and modified receptance \(h_{3,4}(j\omega)\) with the different controllers to assign two complex conjugate zero pairs. Figure 9: Open-loop and closed-loop pole-zero maps of receptance \(h_{3,4}(j\omega)\) with different controllers.