# Laplacian Controllability and Observability of Multi-Agent Systems: Recent Advances in Tree Graphs
## Appendix A Spectral Methods for Controllability and Observability Analysis  
_Appendix A.1. Popov-Belevich-Hautus (PBH) Method_  
The Popov-Belevich-Hautus polynomial approach for controllability and observability analysis is one of the most adopted methods for its effectiveness [4]. Its use in the context of Laplacian-based multi-agent system provides fundamental tools for such analysis. **Proposition A1**. : _System (3) is completely controllable if and only if_  
\[rank[L_{\mathcal{G}}-\lambda I\,|\,B\,]=n\qquad\qquad\forall\lambda\in\mathbb{ C}, \tag{1}\]  
Figure 11: Graphical result of the algorithm applied to the starlike graph in Figure 10.  
and System (4) is completely observable if and only if_  
\[rank \begin{bmatrix}L_{\mathcal{G}}-\lambda\ I\\ C\end{bmatrix} =n\qquad\qquad\forall\lambda\in\mathbb{C}. \tag{10}\]  
We now specialize this criterion considering some peculiarities of (3) and (4). Considering that the Laplacian matrix is symmetric, all its eigenvalues are real and semi-simple, so that \(m_{\alpha}(\lambda)=m_{g}(\lambda)\) for any \(\lambda\in\mathbb{C}\)[30]. Condition (11) is violated if there exist a nonzero vector \(\mathbf{v}\in\mathbb{R}^{N}\) such that  
\[\mathbf{v}^{\top}[L_{\mathcal{G}}-\lambda I|\ B\ ]=\mathbf{0},\]  
and this is equivalent to say that  
\[ \begin{cases}L_{\mathcal{G}}^{\top}\mathbf{v}=\lambda\mathbf{v},\\ \mathbf{e}_{i}^{\top}\mathbf{v}=0\qquad\qquad\forall i\in\mathcal{V}_{i},\end{cases}  \tag{12}\]  
In turn, this means that there should exist an eigenvector with zero value along all the components corresponding to the control nodes. Whenever this condition holds, we call \(\lambda\), \(\mathbf{v}\) an unreachable eigenvalue and eigenvector. Complete system reachability requires that such eigenvectors do not exist. **Remark A1** (Minimal set of control nodes for complete controllability). : _From the above discussion, the number of control/measured nodes must be larger or at least equal to the maximum multiplicity of eigenvectors \(m_{g}(\lambda)\). Moreover, for the second condition of (12), it is fundamental to understand the zero-nonzero pattern of each eigenvector._  
It is possible to derive an analogous condition for observability, requiring that  
\[ \begin{cases}L_{\mathcal{G}}\mathbf{v}=\lambda\mathbf{v},\\ \mathbf{e}_{i}^{\top}\mathbf{v}=0\qquad\qquad\forall i\in\mathcal{V}_{o}\end{cases}  \tag{13}\]  
Namely, there should exist an eigenvector with zero components along the observation nodes. Also, in this case, when this condition holds, \(\lambda\), \(\mathbf{v}\) are called unobservable eigenvalue and eigenvector, and complete system observability requires that such eigenvectors do not exist. Clearly, in the case of observability, analogous considerations hold. However, considering again the symmetry of matrix \(L_{\mathcal{G}}\), a system is reachable through a set of control nodes if and only if it is observable from the same set of measured nodes, and for this reason we deduce that the controllability problem and the observability problem are indeed equivalent for systems evolving with Laplacian dynamics, as (3) and (4). Laplacian Spectrum of Path Graphs, of the Two Class of Matrices \(N_{i}\), \(M_{j}\), and Their Relations  
In the following, we characterize the Laplacian spectrum of path graphs, of the two class of matrices \(N_{i}\), \(M_{j}\), and their relations. Laplacian spectrum and eigenstructure of path graphs have been widely considered and studied. Here, we briefly offer the formulas that characterize it for the sake of completeness. We let \(\mathcal{P}_{n}=(\mathcal{V},\mathcal{E})\) be a _Path graph_, namely we assume that \(\mathcal{V}=\{1,\ldots,n\}\) and \(\mathcal{E}=\{(a,a+1):1\leq a<n\}\).