## 6 Conclusions  
In this work a novel, two-stage, pole placement method is proposed and an embedded a-priori stabilizing condition is introduced. In the first stage the receptance method for symmetric or asymmetric systems is applied to compute the gains that exactly assign a subset of the system poles. These poles are usually the dominant ones, such as those with the lowest frequency or with higher controllability and observability. In the second stage, by leverage of a first-order model formulated with a reduced state and by exploiting the Inclusion Principle, the unassigned poles are clustered into some region of the complex plane exploiting the LMI theory. An SDP programming is solved to reduce the control effort, by exploiting the additional degrees of freedom introduced by the RPP, in lieu of the \"exact\" pole placement. A novel formulation has been proposed by exploiting the aforementioned mathematical tools and formulations and some methods to improve the numerical reliability and solvability of the problem.  
Four meaningful and challenging test-cases, also taken from common benchmark in the literature on pole placement in vibrating systems, are proposed. In the first test-case, a simple three-mass system and some features of the method are introduced. In the second test-case, it is provided that the method handles unstable (or marginally stable) and asymmetric systems as well. Moreover, the comparison with traditional partial pole placement techniques, under comparable requirements, shows the benefits of relaxing the requirements on the less relevant poles. Not only can the control effort be reduced and the robustness increased, but the dynamic response of the system can also be improved in term of reducing the settling time. The third test case tackles the control of an aircraft wing under an air stream by stabilizing its unstable open-loop behavior and by improving its closed-loop dynamic response. The fourth test case is the challenging linearized model of a nonlinear flexible multibody system perturbed by gravity, which makes it unstable. Due to the medium-large number of degrees of freedom, to the presence of an unstable pole, to the presence of real poles and to a wide spectrum that includes high-frequency poles, this test case is numerically ill-conditioned and hence very challenging  
Figure 12: Pole map of the four-bar linkage: low frequency detail (test-case 4).  
Additionally, if PPP is not performed correctly, spillover on the unassigned poled might lead to unstable poles. Again, the proposed method has been shown to be effective.  
The results in the four test-cases, and the comparison with the existing literature too, confirm the benefits of the proposed method.  
Conceptualization, R.B., D.R.; methodology, R.B., D.R., I.T.; software, R.B., I.T.: validation R.B., D.R., I.T.; formal analysis R.B., D.R., I.T.; investigation R.B., D.R., I.T.; data curation, R.B., D.R., I.T.; writing, R.B., D.R., I.T., A.T.; visualization, R.B., D.R., I.T., A.T.; supervision, D.R., A.T. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The third author acknowledges the financial support of the Cariparo foundation (\"Fondazione Cassa di Risparmio di Padova e Rovigo\") through a Ph.D. scholarship.  
The authors declare no conflict of interest.