## 1 Introduction  
### State of the Art  
The assignment of the dynamic response of vibrating mechanical systems, such as structures, mechanisms, or multibody systems, is often performed by properly assigning the poles of the controlled systems. Both active feedback control [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19] and passive approaches (i.e., parameter modifications, see e.g., [19; 20; 21]) or are exploited to accomplish this important task. Indeed, the system poles, which are often denoted as the eigenvalues, define the system stability as well as the properties of the transient response such as the damping ratio, the rise time, and the settling time. Therefore, pole placement (also denoted as \"pole assignment\" or \"eigenvalue assignment\") is a broadly adopted vibration control technique, due to the simple definition of the aforementioned specifications. The goal of pole placement through active control is to find the gains, usually exploiting state feedback, to ensure that the closed-loop poles are exactly the desired ones, to feature the desired dynamic performances. The literature proposes several computational methods to compute the gains,assigning the whole set of system poles through active control. A ground-breaking advancement in the field of pole placement in active vibration control is the receptance method proposed by [PERSON] and [PERSON] in [1] that uses only the measured receptances of the system, in lieu of a first-order model. Such a formulation has been later exploited and extended by other researchers, for example to damp friction-induced vibrations [2], in robust pole placement [3], to control of flexible multibody systems [4], or to avoid resonances in planetary gear trains [5]. A practical issue often affects the attempt of assigning all the system poles to prescribed locations of the complex plane. Indeed, usually just few dominant poles define the system response, the so-called \"dominant poles\", which are often the lowest frequency ones or those with higher degree of observability. The remaining ones, in contrast, provide marginal contributions to the system response. In the light of this need, Partial Pole Placement (PPP) has gained more attentions in the last years in the field of vibration control, to assign just the reduced set of dominant poles. The difficulty of PPP relies in the fact that spillover on the unassigned poles might occur, which might result in instability or in the establishment of other low-frequency undamped poles that would negatively dominate the system response. This issue is usually solved by imposing that the remaining open-loop poles, i.e., those which are not dominant, are retained in the closed-loop configuration [6; 7; 8; 9; 10]. Unfortunately, imposing the exact locations to these poles might leads to high control gains since high-frequency poles are often weakly controllable. In the light of all these issues and considering that the exact locations of the dominated poles is not critical as long as they belong to suitable regions of the complex plane, the idea of Regional Pole Placement (RPP) is very attractive. The underlying idea is to place some poles in predefined regions of the complex plane, in lieu of placing them in predefined locations. In practice, it is a \"non-strict\" assignment of some poles. Some solutions have been proposed for some simple electronic power systems or chemistry plants in [11; 12; 13; 14]; however, such methods employ mathematical methods that are not numerically reliable (e.g., the characteristic polynomial, the controllability matrix). Therefore, these approaches are not appropriate for vibrating systems modelled through medium or large dimensional and ill-conditioned matrices. For example, in [14] the polynomial matrix, which defines the stability region in the coefficient space should be approximated by choosing it heuristically, by leading to approximate and sub-optimal results.