## 4 Active Antiresonance Assignment with No Additional Masses  
### Motivations  
Along with the usage of ATMDs and SATMDs, other antiresonance assignment methods have been developed through active control paradigms, usually performed through the feedback of a suitable number of displacements, speed, and acceleration signals. These methods perform vibration absorption without adding new masses, and hence new DOFs, in the system. Although they have been less investigated, the recent theoretical enhancement in the theory of feedback control have boosted a quick growth of their effectiveness in the most recent years, leading to new effective solutions.  
Basically, three groups can be defined (see Figure 6) and a summary and comparison of the main methods is provided in Table 3:  
* state-feedback control;
* output-feedback control;
* virtual ATMDs.  
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Paper Reference** & [48, 49] & [50] & [51] & [52] & [53] & [54, 55, 56] & [2] & [57] \\ \hline \hline Damped system & No & Yes & Yes & Yes & Yes & No & Yes \\ \hline Discrete/Continuous system & Discrete & Discrete & Discrete & Discrete & Continuous & Discrete & Discrete/Continuous \\ \hline Symmetric/Asymmetric system & Symm. & Symm. & Both & Symm. & Symm. & Symm. & Symm. & Symm. \\ \hline
1 st order formulation/2 nd order formulation/2 nd order formulation/Receptance method & 2 nd order & Recept. & Recept. and 1 st order & Recept. & 2 nd order & 2 nd order & Recept. & 2 nd order \\ \hline State feedback (SF)/Output & SF. & SF & SF & SF & OF & OF & OF & OF \\ feedback (OF) & & & & & & & & \\ \hline Colocated controller/ & - & - & - & - & Colloc. & Colloc. & Colloc. & Colloc. \\ Non Collocated & & & & & & & & \\ \hline Point acceptance/Cross acceptance & Point & Both & Both & Both & Point & Point & Both & Point \\ \hline Imposed actuation matrix & No & Yes & Yes & Yes & Yes & Yes & Yes & Yes \\ \hline Time delay & No & No & No & No & No & No & No & No \\ \hline Number of assigned zeros & \(N\)-\(J\) complex conjugate pairs & Up to \(N\)-\(I\) complex  
Table 3 shows that antiresonance assignment through active control has been usually solved with ad-hoc methods exploiting either the second-order formulation of the system model or receptance, which are two common representations for vibrating systems. At first glance, even the methods developed in the literature for pole placement through state-space, first-order models seem to be suitable for assigning antiresonance by formulating a first-order model of the adjunct system and then by applying standard techniques, such as the _place_ function of MATLAB (which exploits the algorithm proposed in [46]) or the Ackermann's formula [47]. A deeper investigation reveals that these methods are not suitable for antiresonance assignment for several reasons. First, they require the inversion of the mass matrix of the adjunct system which can be ill-conditioned or even singular (as often happens for cross-receptances). Additionally, the adjunct system is often nearly uncontrollable. A second reason is that standard methods usually need all the poles to be placed it; hence, assigning just a few antiresonances while the other ones are free to assume arbitrary values is not possible. These issues justify the development of methods ad-hoc developed for antiresonance assignment that are discussed in the following paragraphs.