# Data-Driven Formation Control for Multi-Vehicle Systems Induced by Leader Motion
## 2 Notation  
Symbols \(\mathbb{N}\), \(\mathbb{N}_{0}\), and \(\mathbb{R}_{\geq 0}\) represent, respectively, the natural numbers, the non-negative integer numbers, and the non-negative real numbers. Vectors are denoted with bold symbols, and, in particular, \(\mathbf{e}_{i}\) denote the \(i\)-th canonical vector, i.e., \(\mathbf{e}_{1}=\begin{bmatrix}1&0&\ldots&0\end{bmatrix}^{\top}\). Notation \((A)_{ij}\) represents the submatrix of \(A\) obtained by removing the \(i\)-th row and \(j\)-th column from \(A\). Given a vector \(\mathbf{v}\in\mathbb{R}^{d}\), \(v_{i}\) is its \(i\)-th component, so that \(\mathbf{v}=\begin{bmatrix}v_{1}&v_{2}&\ldots&v_{d}\end{bmatrix}^{\top}\).  
For a square matrix \(A\in\mathbb{R}^{n\times n}\), its _eigenvalues_\(\lambda\in\mathbb{C}\) are those values satisfying the _eigenvalue equation_\(A\mathbf{x}=\lambda\mathbf{x}\) for a nonzero \(\mathbf{x}\in\mathbb{C}^{n}\), which is called _eigenvector_. The _characteristic polynomial_ of \(A\) is the polynomial \(p_{A}(s)=\det[sI-A]\), and its zeros correspond to the eigenvalues of \(A\). The _spectrum_ of \(A\in\mathbb{R}^{n\times n}\), denoted by \(\sigma(A)\), is the set of its eigenvalues, and the _spectral radius_ of \(A\), denoted by \(\rho_{A}\), is the maximum modulus of the elements of \(\sigma(A)\). The _algebraic multiplicity_\(m_{a}(\lambda)\) of an eigenvalue \(\lambda\) is its multiplicity as zero of \(p_{A}(s)\), while the _geometric multiplicity_\(m_{g}(\lambda)\) is defined as \(m_{g}(\lambda)=\dim[\ker(A-\lambda I_{n})]\). It is well-known that \(1\leq m_{g}(\lambda)\leq m_{a}(\lambda)\); if \(m_{a}(\lambda)=1\), then \(\lambda\) is called a _simple eigenvalue_ of \(A\), while if \(m_{g}(\lambda)=m_{a}(\lambda)\) for a \(\lambda\in\mathbb{C}\) such that \(m_{a}(\lambda)>1\), then \(\lambda\) is called _semi-simple_. Given a square matrix \(A\), if \(m_{g}(\lambda)=m_{a}(\lambda)\) for each \(\lambda\in\sigma(A)\), then there exists a pair of square matrices \(\Lambda\), \(\Pi\) with \(\Lambda\) diagonal matrix, and \(\Pi\) nonsingular, such that \(A\Pi=\Pi\Lambda\). They are called respectively _spectral matrix_ and _modal matrix_, the diagonal entries of \(\Lambda\) are the eigenvalues of \(A\), and the columns of \(\Pi\) are the corresponding eigenvectors. Conversely, if \(m_{g}(\lambda)<m_{a}(\lambda)\) for at least one \(\lambda\), then it is possible to find a pair of square matrices \(J_{A}\), \(\Pi\) with \(J_{A}\) a matrix in _Jordan form_, namely:  
\[J_{A}=\begin{bmatrix}J_{\lambda_{1}}&0&\ldots\\ 0&J_{\lambda_{2}}&\\ \vdots&&\ddots\\ 0&&J_{\lambda_{s}}\end{bmatrix},\qquad J_{\lambda_{i}}=\begin{bmatrix}J_{1}&0 &\ldots\\ 0&J_{2}&\\ \vdots&&\ddots\\ 0&&J_{m_{a}(\lambda_{i})}\end{bmatrix},\qquad J_{i}=\begin{bmatrix}\lambda_{ i}&1&0&\ldots\\ 0&\lambda_{i}&1&\\ \vdots&&\ddots\end{bmatrix} \tag{1}\]  
and \(\Pi\) nonsingular, such that \(A\Pi=\Pi J_{A}\). In such a case, we call \(J_{A}\) the _generalized spectral matrix_ and \(\Pi\) the _generalized modal matrix_, whose columns are called _generalized eigenvectors_ and they can be built as follows. A _generalized eigenvector of order_\(m\) is a nonzero vector \(\mathbf{v}\
eq 0\) such that:  
\[(A-\lambda I)^{m}\mathbf{v}=0,\qquad(A-\lambda I)^{m-1}\mathbf{v}\
eq 0. \tag{2}\]  
Consider the special case of generalized eigenvector of order two, which is of special interest in this paper. They can be effectively built through the relation:  
\[A\mathbf{v}=\lambda\mathbf{v}+\mathbf{v}_{\lambda}, \tag{3}\]  
where \(\mathbf{v}_{\lambda}\) is an (ordinary) eigenvector of \(\lambda\).  
Finally, given a numbered set of numbers \(\{x_{i}\}\), \(i=1,2,\ldots\), we define the _Hankel matrix of order \(s\)\(\mathcal{H}_{s}(x)\)_ as:\[\mathcal{H}_{s}(x)=\left[\begin{array}{cccc}x_{0}&x_{1}&\ldots&x_{s-1}\\ &\ddots&\ddots&\\ x_{s-1}&x_{s}&\ldots&x_{2s-2}\end{array}\right]. \tag{4}\]