# Data-Driven Formation Control for Multi-Vehicle Systems Induced by Leader Motion
## 3 Problem Framework
In the following, considering [5], we say that system (15) reaches a _consensus_ condition if for any initial conditions \(\mathbf{r}_{i}(0),\mathbf{v}_{i}(0),i=1,2,\ldots,N\), there exist \(\boldsymbol{\alpha}=\boldsymbol{\alpha}(\mathbf{r}_{i}(0),\mathbf{v}_{i}(0)) \in\mathbb{R}^{2}\) and \(\boldsymbol{\beta}=\boldsymbol{\beta}(\mathbf{r}_{i}(0),\mathbf{v}_{i}(0))\in \mathbb{R}^{2}\) so that:  
\[\lim_{t\to+\infty}\mathbf{r}_{i}(t)=\boldsymbol{\alpha}, (i=1,\ldots,N),\] \[\lim_{t\to+\infty}\mathbf{v}_{i}(t)=\boldsymbol{\beta}, (i=1,\ldots,N), \tag{9}\]  
or equivalently, \(\lim_{t\to+\infty}\mathbf{r}(t)=\boldsymbol{\alpha}(\mathbf{r}_{i}(0),\mathbf{ v}_{i}(0))\mathbf{1}_{N}\) together with \(\lim_{t\to+\infty}\mathbf{v}(t)=\boldsymbol{\alpha}(\mathbf{r}_{i}(0), \mathbf{v}_{i}(0))\mathbf{1}_{N}\). The values of \(\alpha\) and \(\beta\), that are functions of the initial values \((\mathbf{r}_{i}(0),\mathbf{v}_{i}(0))\), are called the _consensus values_ (or _collective decision_) [1] for system (15). However, adopting protocol (8), the consensus value on the positions is a linear function in \(t\), and the team diverges linearly, keeping the selected formation. In technical details, for system (5) under protocol (8), consensus is reached if graph \(\mathcal{G}\) is strongly connected and \(\gamma\) is sufficiently large [5], and it is equal to:  
\[\alpha(\mathbf{r}_{i}(0),\mathbf{v}_{i}(0))=\frac{\mathbf{w}_{\mathcal{L}}^{ \top}\mathbf{r}(0)}{\mathbf{w}_{\mathcal{L}}^{\top}\mathbf{1}_{N}}+t\frac{ \mathbf{w}_{\mathcal{L}}^{\top}\mathbf{v}(0)}{\mathbf{w}_{\mathcal{L}}^{ \top}\mathbf{1}_{N}}, \tag{10}\]  
where \(\mathbf{w}_{\mathcal{L}}\) is a vector in the left null space of the Laplacian matrix \(\mathcal{L}\) of the communication graph. The second term in (10) expresses a drift term caused by the asymptotic velocity of the teammates, indeed under (8), velocity itself shows a consensus condition:  
\[\beta=\lim_{t\to+\infty}\mathbf{v}_{i}(t)=\frac{\mathbf{w}_{\mathcal{L}}^{ \top}\mathbf{v}(0)}{\mathbf{w}_{\mathcal{L}}^{\top}\mathbf{1}_{N}},\qquad \quad\forall\;i\in\{1,\ldots,N\}. \tag{11}\]  
However, it is often preferable to reach a static consensus condition [20], thus seeking \(\beta=0\) for all initial conditions. To reach this goal, a slightly different protocol has been studied in [20]:  
\[\mathbf{u}_{i}(t)=-\delta\mathbf{v}_{i}(t)+M_{i}\sum_{j=1}^{N}[\mathcal{A}]_{ ij}k_{ij}[(\mathbf{r}_{j}(t)-\mathbf{r}_{i}(t))+\gamma(\mathbf{v}_{j}-\mathbf{v}_{i })], \tag{12}\]  
with \(\delta>0\) (the so called _absolute damping_). Under the same conditions as in (10), System (5) under protocol (12) reaches the consensus condition:  
\[\lim_{t\to+\infty}\mathbf{r}_{i}(t)=\alpha(\mathbf{r}_{i}(0),\mathbf{v}_{i}(0 ))=\frac{\mathbf{w}_{\mathcal{L}}^{\top}\mathbf{r}(0)}{\mathbf{w}_{\mathcal{L }}^{\top}\mathbf{1}_{N}}+\frac{1}{\delta}\frac{\mathbf{w}_{\mathcal{L}}^{\top }\mathbf{v}(0)}{\mathbf{w}_{\mathcal{L}}^{\top}\mathbf{1}_{N}}, \tag{13}\]  
and \(\lim_{t\to+\infty}\mathbf{v}_{i}(t)=0\)[20]. Protocol (12) is the special case of a general reference tracking protocol [3]:  
\[\mathbf{u}_{i}(t)=\dot{\mathbf{v}}^{\prime}(t)-\delta(\mathbf{v}_{i}(t)- \mathbf{v}^{\prime}(t))+M_{i}\sum_{j=1}^{N}[\mathcal{A}]_{ij}k_{ij}[(\mathbf{r }_{j}(t)-\mathbf{r}_{i}(t))+\gamma(\mathbf{v}_{j}-\mathbf{v}_{i})], \tag{14}\]  
through which consensus is reached on a reference velocity \(\mathbf{v}^{\prime}(t)\) (namely \(\mathbf{r}_{i}(t)\to\mathbf{r}_{j}(t)\) and \(\mathbf{v}_{i}(t)\to\mathbf{v}^{\prime}(t)\)[24]). In this paper, we study the influence of leader motion when all participants adopt either protocol (8) or (12). We now briefly introduce the matrix description of system (5) under either (8) or (12), that is useful for the asymptotic analysis of the team evolution. In the following of the paper, since Equation (21) are decoupled, we consider \(m=1\) for the sake of simplicity [24]. If we collect the agents' states into a single \(2N\) state vector, the overall second-order multi-agent system under protocol (8) can be written as:  
\[