## 4 Active Antiresonance Assignment with No Additional Masses
### State-Feedback Approaches
#### 4.2.1 Assignability of the Zeros through State-Feedback Control  
The assignability of the zeros can be computed exploiting the controllability condition for second-order systems that can be found in several books and papers for the assignment of poles (see e.g., [58]), properly formulated for the adjunct system. Indeed, the zeros of the generic receptance \(h_{pq}\) are completely assignable by means of a control force defined by state-feedback algorithm if and only if:  
\[\text{rank}\Big{[}\begin{array}{c}z_{i}^{2}\,\mathbf{M}_{\mathbf{q}\mathbf{ p}}+z_{i}\,\mathbf{C}_{\mathbf{q}\mathbf{p}}+\mathbf{K}_{\mathbf{q}\mathbf{p}}- \mathbf{B}\mathbf{C}_{\mathbf{q}}^{-}\end{array}\Big{]}=N-1 \tag{15}\]  
for all the zeros \(z_{i}\) of the open-loop system, where \(\mathbf{B}\mathbf{C}_{\mathbf{q}}^{-}\) is obtained by removing the \(q\)-th row from the controller force distribution matrix \(\mathbf{B}_{\mathbf{c}}\). On the other hand, the zeros of \(h_{pq}\) are partially assignable whenever the condition in Equation (15) is satisfied just for some zeros.  
.2.2 Antiresonance Assignment for Undamped Systems and Non-Imposed Control Force Distribution Vector  
One of the first works addressing active antiresonance assignment in vibrating systems was proposed by [PERSON] in [48; 49]. A state feedback method to assign the \(N\)-\(1\) zeros of the \(N\)-th point receptance together with two complex conjugate poles has been developed, while the remaining poles are imposed to match those of the open-loop system. The method computes the position feedback gain, the control force distribution vectors for the case of sparse force distribution vectors are admitted, and the method has been developed with reference to symmetric, undamped systems, with no possibility of modifying the pole or zero damping. Additionally, the method does not allow for imposing the desired force distribution vector, as it is common in practice and requires distributed actuation even if the rank-one control is performed.  
A two-stage procedure is proposed. First, the control gain vector \(\mathbf{g}\) is computed to assign the two prescribed complex conjugated closed-loop poles \(p^{c}\) to a certain value:  
\[\mathbf{g}=-\frac{p_{1}^{o}-p^{c}}{p_{1}^{o}}\mathbf{K}\mathbf{u}_{\mathbf{r},1} \tag{16}\]  
where \(p_{1}^{o}\) and \(\mathbf{u}_{\mathbf{r},1}\) are the eigenvalue-eigenvector pair of the first mode obtained through the open-loop pencil: \(\mathbf{Q}_{\mathbf{o}}(p)=\mathbf{K}-p\mathbf{M}\). In the second stage, the force distribution vector \(\mathbf{b}_{\mathbf{c}}\) is determined such that the \(N\)-\(1\) zeros are assigned. The principal submatrix \(\mathbf{\hat{Q}}_{\mathbf{o}}(z)\in\mathbb{R}^{(N-1)\times(N-1)}\), its eigenvalues \(z_{1}^{o},\ldots,z_{N-1}^{o}\) and eigenvectors \(\mathbf{u}_{\mathbf{z},1},\ldots,\mathbf{u}_{\mathbf{z},N-1}\) are obtained and the system matrices are partitioned as follows:  
\[\mathbf{b}_{\mathbf{C}}=\left\{\begin{array}{c}\mathbf{\hat{b}}_{\mathbf{C} }\\ b_{\mathbf{C},N}\end{array}\right\}\!\!,\quad\mathbf{g}=\left\{\begin{array}{ c}\mathbf{\hat{g}}\\ g_{N}\end{array}\right\}\!\!,\quad\mathbf{K}=\left[\begin{array}{cc}\mathbf{ \hat{K}}&\mathbf{\hat{k}}\\ \mathbf{\hat{k}}^{T}&k_{NN}\end{array}\right] \tag{17}\]  
Finally, the control force distribution vector that enables us to assign the prescribed closed-loop zeros \(z_{1}^{c},\ldots,z_{N-1}^{c}\) is obtained through:  
\[\mathbf{\hat{b}}_{\mathbf{C}}=-\mathbf{\hat{K}}\mathbf{U}_{\mathbf{z}}\mathbf{r} \tag{18}\]  
where \(\mathbf{U}_{\mathbf{z}}=[\mathbf{u}_{\mathbf{z},1},\ldots,\mathbf{u}_{\mathbf{ z},N-1}]\) and \(\mathbf{r}\) is a vector whose coefficients are:  
\[r_{j}=\frac{1}{\mathbf{\hat{g}}^{T}\mathbf{u}_{\mathbf{z},j}}\frac{z_{j}^{c}-z _{j}^{o}}{z_{j}^{o}}\prod_{\begin{array}{c}z_{i}^{o}-z_{j}^{o}\\ z_{i}^{o}-z_{j}^{o}\end{array}}^{N-1}\quad\text{with }j=1,\ldots,N-1 \tag{19}\]while the \(N\)-th entry of \(\mathbf{b_{C}}\) is obtained through a prescribed normalization condition.