# Data-Driven Formation Control for Multi-Vehicle Systems Induced by Leader Motion
## 3 Problem Framework
\begin{bmatrix}\dot{\mathbf{r}}(t)\\ \dot{\mathbf{v}}(t)\end{bmatrix} =\boldsymbol{\Gamma} \begin{bmatrix}\mathbf{r}(t )\\ \mathbf{v}(t)\end{bmatrix} , \tag{15}\]  
where \(\mathbf{r}(t)\) is the \(N\)-dimensional vector of the stacked \(\mathbf{r}_{i}(t)\) and \(\mathbf{v}(t)\) analogously for the velocities, and matrix \(\boldsymbol{\Gamma}\) is equal to:  
\[\boldsymbol{\Gamma}= \begin{bmatrix}\mathbf{0}_{n\times n}&I_{n}\\ -\mathcal{L}&-\gamma\mathcal{L}\end{bmatrix} , \tag{16}\]  
with \(\mathcal{L}:=\operatorname{diag}\{\sum_{j\
eq 1}[\mathcal{A}]_{1j},\ldots, \sum_{j\
eq N}[\mathcal{A}]_{Nj}\}-\mathcal{A}=[\ell_{ij}]\in\mathbb{R}^{N \times N}\) is the _Laplacian matrix_[25; 26] associated with the adjacency matrix \(\mathcal{A}\) (equivalently, with the communication graph \(\mathcal{G}\)). When dealing with protocol (12), the same logic leads to a set of matrix equations as in (15), with matrix \(\boldsymbol{\Gamma}\) equal to:  
\[\boldsymbol{\Gamma}_{2}= \begin{bmatrix}\mathbf{0}_{n\times n}&I_{n}\\ -\mathcal{L}&-\delta I_{n}-\gamma\mathcal{L}\end{bmatrix} . \tag{17}\]  
One main reason for studying separately both protocols (8) and (12), is that both approaches are largely adopted, but the spectral properties of \(\boldsymbol{\Gamma}\) and \(\boldsymbol{\Gamma}_{2}\) in terms of \(\lambda=0\) are noticeably different (multiplicity and eigenstructure), so each protocol deserves a separate discussion. One main successful application of the above setting is _formation control_[5]. Indeed, by properly redefining the position variables in order to take account of the relative displacements \(\delta_{ij}\in\mathbb{R}\) related to a desired formation \(\mathbf{r}^{d}\in\mathbb{R}^{N}\), the equilibrium condition can be expressed in geometric terms as:  
\[(\mathbf{r}^{d})_{i}=(\mathbf{r}^{d})_{j}-\delta_{ij}, \tag{18}\]  
and it is possible to design cooperative distributed formation control strategies by applying either protocol (8) or (12) to \(\dot{\mathbf{r}}(t)=\mathbf{r}^{d}-\mathbf{r}(t)\). In particular, in this paper, we adopt the above strategy as applied in (Section 5 of [24]), where a time-varying desired geometric configuration is achieved through protocol (14) for a team of four multiple rotary-wing micro air vehicles. One main peculiarity of this approach is that the consensus value among nodes provides the formation center (see Figure 3 of [3]). We discuss this feature in detail later in Section 6. We are now ready to state the problem posed in this paper. Problem StatementConsider the set-up described so far as a current running protocol among agents, and assume that one agent, which we call the _leader_, has the intention to improve the overall performance through its motion. More precisely, we adopt the approach of sampled-data cooperative control as surveyed in [27; 28], and we assume that the leader agent, which we label without loss of generality as node 1, takes samples of its motion:  
\[r_{1_{\ell}}=r_{1}(\ell\Delta),\qquad\ell\in\mathbb{N}, \tag{19}\]  
and it computes a supplementary input, in addition, to (8) based on the elaboration of local data of its past motion:  
\[\bar{a}_{k}=\mathbf{u}(r_{1_{\ell}},\ell\leq\kappa), \tag{20}\]  
with the goal of improving the overall team performance. We assume that the input is kept constant between two samples, and it is set as a continuous piecewise constant function, as in ([27], Section 3). In general, the additional input of the leader node has the goal of improving the collective performances (e.g., to achieve a more precise consensus value to counteract against faults and uncertainties, to verify data consistency and detect network anomalies) [21]. In this paper, we design an internal elaboration mechanism for the leader past motion data to produce a future motion of the leader node \(r_{1}(t)\) so as to let the agents reach a consensus in finite time. Finally, we seek a procedure for the algorithm synthesis based on local data only, with a leader having poor a priori knowledge of graph topology.