## 3 Method Description  
### Placement of the \(N_{p}\) Closed-Loop Poles  
In this paper, the problem of robust pole placement in delayed systems with single-input control is tackled: given a set of desired \(N_{p}<2N\) closed-loop poles, the goal is to compute the control gain vectors \(\mathbf{f}\) and \(\mathbf{g}\) such that the poles are assigned at the prescribed locations and the controlled system satisfy a certain robustness condition. Additionally, it is assumed that the system matrices \(\mathbf{M}\), \(\mathbf{C}\) and \(\mathbf{K}\) are not available and therefore the proposed method should just relies on the measured receptances. Indeed, the knowledge of the system receptances suffices to describe the system properties without the need of knowing the system matrices and therefore allows for designing the controller [16; 22]. The open-loop receptance matrix of the system is defined as [16]  
\[\mathbf{H}(s)=\Big{(}s^{2}\mathbf{M}+s\mathbf{C}+\mathbf{K}\Big{)}^{-1}, \tag{4}\]  
its generic \(pq\)-th entry \(h_{pq}(s)\) is the transfer function from the force applied to the \(q\)-th coordinate to the displacement of the \(p\)-th coordinate. The closed-loop receptance matrix is simply inferred from Equation (3), leading to  
\[\mathbf{\tilde{H}}(s)=\Big{(}s^{2}\mathbf{M}+s\Big{(}\mathbf{C}-\mathbf{B} \mathbf{f}^{T}e^{-s\tau_{f}}\Big{)}+\Big{(}\mathbf{K}-\mathbf{B}\mathbf{g}^{ T}e^{-s\tau_{g}}\Big{)}\Big{)}^{-1}. \tag{5}\]  
The application of the Sherman-Morrison formula [25] to Equation (5) yields [17]  
\[\mathbf{\tilde{H}}(s)=\mathbf{H}(s)+\frac{\mathbf{H}(s)\mathbf{B}\big{(} \mathbf{g}e^{-s\tau_{g}}+s\mathbf{f}e^{-s\tau_{f}}\big{)}^{T}\mathbf{H}(s)}{1 -\big{(}\mathbf{g}e^{-s\tau_{g}}+s\mathbf{f}e^{-s\tau_{f}}\big{)}^{T}\mathbf{ H}(s)\mathbf{B}}. \tag{6}\]  
The closed-loop poles are characterized by those complex values of \(s\) that set to zero the denominator of Equation (6):  
\[1-\big{(}\mathbf{g}e^{-s\tau_{g}}+s\mathbf{f}e^{-s\tau_{f}}\big{)}^{T}\mathbf{ H}(s)\mathbf{B}=0. \tag{7}\]  
The problem of finding the control gains \(\mathbf{f}\) and \(\mathbf{g}\) that assign the desired \(N_{p}\) closed-loop poles \(\beta_{1},...,\beta_{N_{p}}\) can be written as follows [17; 22]:  
\[ \begin{bmatrix}\hat{p}_{1}\mathbf{r_{1}}^{T}e^{-\beta_{1}\tau_{f}}&\mathbf{r_ {1}}^{T}e^{-\beta_{1}\tau_{g}}\\ \hat{p}_{2}\mathbf{r_{2}}^{T}e^{-\beta_{2}\tau_{f}}&\mathbf{r_{2}}^{T}e^{- \beta_{2}\tau_{g}}\\ \vdots&\vdots\\ \hat{p}_{N_{p}}\mathbf{r_{N_{p}}}^{T}e^{-\beta_{N_{p}}\tau_{g}}&\mathbf{r_{N_ {p}}}^{T}e^{-\beta_{N_{p}}\tau_{g}}\end{bmatrix}\begin{bmatrix}\mathbf{f}\\ \mathbf{g}\end{bmatrix} = \begin{bmatrix}1\\ 1\\ \vdots\\ 1\end{bmatrix} , \tag{8}\]where \(\mathbf{r_{i}}=\mathbf{H}(\tilde{p}_{i})\mathbf{B}\), with \(i=1,..,N_{p}\). The system can be formulated, with a more compact notation as the usual form of a linear system, \(\mathbf{G}\mathbf{k}=\mathbf{y}\), with obvious meaning of the notation from Equation (8). In the case of the complete assignment of the closed-loop poles, i.e., \(N_{p}=2N\), if the matrix on the left-hand side of Equation (8) is invertible, the solution of the linear system is: \(\mathbf{k}=\mathbf{G}^{-1}\mathbf{y}\). Therefore, the solution to the complete pole placement problem is unique. Conversely, in the case of partial pole placement \(N_{p}<2N\) desired closed-loop poles are assigned. In this scenario the linear system in Equation (8) has infinite solutions \(\mathbf{k}\). In particular, if \(N_{p}<2N\), the solution of \(\mathbf{G}\mathbf{k}=\mathbf{y}\) is  
\[\mathbf{k}=\mathbf{k_{0}}+\mathbf{k_{h}}, \tag{9}\]  
where \(\mathbf{k_{0}}\) is the particular solution of the non-homogeneous equation. While, \(\mathbf{k_{h}}\) is the solution of the homogeneous equation \(\mathbf{G}\mathbf{k_{h}}=\mathbf{0}\).