## 4 Active Antiresonance Assignment with No Additional Masses
### State-Feedback Approaches
#### 4.2.4 The Rank-One Acceptance Method with Regional Pole Placement  
The main limitation of solving Equation (20) alone is that it does not account for spillover on the unassigned \(2N\)-\(n_{z}\) poles, and hence it does not ensure the asymptotic stability of the controlled system. To tackle this issue, a two-stage approach has been recently proposed in [51] by the authors of this paper, handling asymmetric and ill-conditioned systems too. Let us first recast the linear system in Equation (20) in a more compact form:  
\[\mathbf{G}\mathbf{k}=\mathbf{y} \tag{22}\]  
with the obvious meaning of \(\mathbf{G}\) and \(\mathbf{y}\). Since the linear system in Equation (22) is under-determined, as a matter of fact, \(\mathbf{G}\in\mathbb{C}^{n_{z}\times 2N}\) with \(n_{z}\leq 2(N-1)\), its solution \(\mathbf{k}\) can be expressed as the sum of the particular solution of the non-homogeneous equation, denoted \(\mathbf{k_{0}}\), and a solution of the homogeneous system \(\mathbf{G}\mathbf{V}\mathbf{k_{r}}=0\), hence:  
\[\mathbf{k}=\mathbf{k}_{0}+\mathbf{V}\mathbf{k_{r}} \tag{23}\]  
where the columns of matrix \(\mathbf{V}\) span the null space of \(\mathbf{G}\), i.e., \(\mathbf{V}\in\text{span}(\text{ker}(\mathbf{G}))\). A similar approach exploiting the null space of the linear system has been successfully applied to pole assignment, too [60; 61]. In the first stage of the method proposed in [51], the particular solution \(\mathbf{k_{0}}\) is computed to assign the prescribed \(n_{z}\) zeros for receptance \(h_{pq}\). In the second stage, the set of solutions of the homogeneous system related to the zero-assignment problem is exploited to cluster the closed-loop system poles into \(D\)-stable regions, i.e., portions of the complex plane that ensure stability and the prescribed dynamic properties for the closed-loop system. In such a way, in the second-stage of the method, the powerful LMI theory [60; 61; 62; 63] is adopted to compute the (\(2N\)-\(n_{z}\))-dimensional vector \(\mathbf{k_{r}}\  
and the velocity feedback gain \(c_{c}\) for assigning antiresonance \(\omega_{z}^{d}\) through the scalar control force \(u(t)=-k_{c}x_{N-1}(t)-c_{c}\dot{x}_{N-1}(t)\) are computed through the following relation:  
\[k_{c}+j\omega_{z}^{d}c_{c}=-\frac{\det\biggl{(}-\left(\omega_{z}^{d}\right)^{2} \hat{\mathbf{M}}+j\omega_{z}^{d}\hat{\mathbf{C}}+\hat{\mathbf{K}}\biggr{)}}{ \det\left(-\left(\omega_{z}^{d}\right)^{2}\hat{\mathbf{M}}+j\omega_{z}\tilde{ \mathbf{C}}+\hat{\mathbf{K}}\right)}, \tag{27}\]  
where \(\hat{\mathbf{M}}\), \(\hat{\mathbf{C}}\), \(\hat{\mathbf{K}}\) are obtained by removing the \(N\)-th row and column from \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) (i.e., the point receptance index). \(\hat{\mathbf{M}}\), \(\tilde{\mathbf{C}}\), \(\tilde{\mathbf{K}}\) are, in turn, computed from \(\hat{\mathbf{M}}\), \(\hat{\mathbf{C}}\), \(\hat{\mathbf{K}}\), removing the row and columns corresponding to the (\(N\)-1)-th row and column of \(\mathbf{M}\), \(\mathbf{C}\), \(\mathbf{K}\) (i.e., the index of the collocated actuator-sensor pair). The method is validated by means of numerical example on systems with few coordinates. Few other approaches have been proposed in the literature to solve some specific problems. In [54; 55; 56], the problem of absorbing the harmonic response at the end of an axially vibrating rod is considered. The authors proposed different strategies, e.g., formulations based on the deformation or on the spectral data are carried out to compute the collocated displacement feedback control gain at a particular location that ensures the assignment of a prescribed antiresonance frequency. .3.2 Assignment for Point-Receptances with Collocated Controllers through Multi-Rank Output Feedback  
The receptance-based approach has been further exploited by [PERSON] and his co-workers in [2] to assign zeros to a point-receptance through more collocated pairs of sensors and actuators.