## 4 Active Antiresonance Assignment with No Additional Masses
### Numerical Examples of Active Control  
4.1 Numerical Examples of Active Control: Assignment of a Single Antiresonance through State or Output Feedback  
The goal of this section is to assign an antiresonance frequency through the techniques proposed in [PERSON] et al. (2018) for output feedback collocated control (Equation (27)), and ([PERSON] et al., 2018) for state feedback (Equation (20)), and then to compare the results. Let us consider the system sketched in Figure 7 that has been taken from the literature as a test case ([PERSON] et al., 2018; [PERSON] et al., 2019; [PERSON] et al., 2020; [PERSON] et al., 2021). The system is made of six lumped masses and springs of unitary value (1 kg for the masses and 1 Nm\({}^{-1}\) for the springs). The system is lightly damped and the Rayleigh damping model is assumed (see e.g., [PERSON] et al. (2018)): the mass and stiffness proportional coefficients are, respectively, \(a\) = 1 10\({}^{-2}\) s\({}^{-1}\), \(b\) = 1 10\({}^{-6}\) s, such that \(\mathbf{C}\) = \(a\mathbf{M}\)+\(b\mathbf{K}\)([PERSON] et al., 2018; [PERSON] et al., 2018). Let us consider the point-receptance \(h_{3,3}(i\omega)\), where an undamped antiresonance is assigned at 1.000 \(\text{rads}^{-1}\) (i.e., the zero locations are 0 \(\pm\) 1\(j\)) exploiting one control force whose force distribution vector is \(\mathbf{b_{C}}=[1,\,0,\,0,\,0,\,0]^{T}\). Such a choice ensures the partial controllability of the adjoint system since only two zeros ensure a rank of the controllability matrix equal to 5. In the case of output-feedback, the output measurement matrix for the collocated control is therefore \(\mathbf{C_{O}}=\mathbf{b_{C}^{T}}\). The zeros obtained by means of the proposed controllers are reported in Table 4, while the control gains are reported in Table 5. The unit-rank receptance-based method [50] leads to an underdetermined linear system whenever antiresonance is assigned. Two solutions are here compared: the first solution is obtained when solving Equation (26) by means of the pseudo-inverse of the coefficient matrix (that is referred to in the tables as _pinv_), that leads to the minimum norm solution; the second one solve the underdetermined system by exploiting the _mldivide_ function of MATLAB, which aims to obtain the fewest possible nonzero components in the control gain vector by exploiting a QR-decomposition algorithm [76] which does not lead necessarily the minimum norm solution. In control system design, minimum norm solution is attractive since it offers a heuristic rule to reduce the actuator effort and noise influence; conversely, the solution with the fewest possible nonzero components enables us to design controllers with fewer coordinates to be measured (i.e., output-feedback controllers). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Original System** & **Output-Feedback** & \begin{tabular}{c} **State-Feedback** \\ **Method [53]** \\ \end{tabular} & \begin{tabular}{c} **State-Feedback** \\ **Method [50]** \\ \end{tabular} & \begin{tabular}{c} **State-Feedback** \\ **Method [50]** \\ \end{tabular} & \
\begin{tabular}{c} **Method [50]** \\ **Through Pinv** \\ \end{tabular} &
\begin{tabular}{c} **Method [50]** \\ **Through MIdivide** \\ \end{tabular} \\ \hline \(-0.005\pm 0.720\,j\) & \(-0.005\pm 0.720\,j\) & \(-0.005\pm 0.720\,j\) & \(-0.005\pm 0.720\,j\) \\ \(-0.005\pm 1.176\,j\) & \(0\pm 1.000\,j\) & \(0\pm 1.000\,j\) & \(0\pm 1.000\,j\) \\ \(-0.005\pm 1.520\,j\) & \(-0.005\pm 1.520\,j\) & \(-0.005\pm 1.520\,j\) & \(-0.005\pm 1.520\,j\) \\ \(-0.005\pm 1.902\,j\) & \(-0.004\pm 1.871\,j\) & \(-0.005\pm 1.897\,j\) & \(-0.004\pm 1.871\,j\) \\ \(-0.005\pm 2.042\,j\) & \(-0.005\pm 2.042\,j\) & \(-0.005\pm 2.042\,j\) & \(-0.005\pm 2.042\,j\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Zeros of the original and modified acceptance \(h_{3,3}(i\omega)\) with the different controllers to assign a complex conjugate zero pair. Figure 7: A sketch of the six-mass system.