# Data-Driven Formation Control for Multi-Vehicle Systems Induced by Leader Motion
## 4 Sampled-Data Leader Motion Mechanism Design
### Conditions to Reach Consensus  
As a first step, we derive conditions for \(a_{i}\) and \(b_{i}\) under which the sampled multi-agent system (25) with input (31) built upon (33) reaches a consensus, either adopting protocol (8), or (12). We consider protocol (8) first; this case is trickier as a consequence of the spectral properties of matrix \(\Gamma\) (namely, \(\lambda=1\) is a semi-simple eigenvalue with \(m_{a}(1)=2\)). The case of (12) follows, which is more in line with [19] for the spectral properties of \(\Gamma_{2}\). Proposition 2: _Consider the sampled multi-agent system Equation (25) with input Equation (30). Under the hypothesis that it has a stable transient, then the equilibrium of the multi-agent system is a consensus condition, as in (10), if and only if parameters \(\mathbf{a}\) satisfy:_  
\[\mathbf{1}^{\top}\mathbf{a}=0, \tag{34}\]  
\[\sum_{i=1}^{\mu}(\mu-i+1)a_{i}=0. \tag{35}\]  
Proof: Consider the overall system evolution (33) under the action of the leader motion, generated by (30) and (31):\[ \begin{bmatrix}\mathbf{r}(k+1)\\ \mathbf{v}(k+1)\\ \boldsymbol{\varphi}(k+1)\end{bmatrix} = \begin{bmatrix}\[PERSON]_{r}+a_{0}\[PERSON]_{r} \mathbf{e}_{1}^{\top}&\[PERSON]_{r0}&\[PERSON]_{r}\mathbf{e}_{h}^{\top}\\ \[PERSON]_{vr}+a_{0}\[PERSON]_{v}\mathbf{e}_{1}^{\top}&\[PERSON]_{v}&\[PERSON]_{v}\mathbf{e}_{h}^{ \top}\\ (\mathbf{a}-\mathbf{b}a_{0})\mathbf{e}_{1}^{\top}&0&F-\mathbf{b}\mathbf{e}_{h} ^{\top}\end{bmatrix}\begin{bmatrix}\mathbf{r}(k)\\ \mathbf{v}(k)\\ \boldsymbol{\varphi}(k)\end{bmatrix} , \tag{36}\] \[\mathbf{r}(0)=\mathbf{r}_{0},\qquad\mathbf{v}(0)=\mathbf{v}_{0}, \qquad\boldsymbol{\varphi}(0)=\boldsymbol{0}.\]  
In order to study the steady state behavior of (36), we study the solutions of the stationary equations [30]:  
\[ \begin{bmatrix}\mathbf{r}\\ \mathbf{v}\\ \boldsymbol{\varphi}\end{bmatrix} = \begin{bmatrix}\[PERSON]_{r}+a_{0}\Psi_{r} \mathbf{e}_{1}^{\top}&\[PERSON]_{rv}&\Psi_{r}\mathbf{e}_{h}^{\top}\\ \Phi_{vr}+a_{0}\Psi_{v}\mathbf{e}_{1}^{\top}&\[PERSON]_{v}&\[PERSON]_{v}\mathbf{e}_{h}^{ \top}\\ (\mathbf{a}-\mathbf{b}a_{0})\mathbf{e}_{1}^{\top}&0&F-\mathbf{b}\mathbf{e}_{h} ^{\top}\end{bmatrix}\begin{bmatrix}\mathbf{r}\\ \boldsymbol{\varphi}\end{bmatrix} . \tag{37}\]  
Considering the spectral properties of matrix \(\Phi\) when adopting protocol (8) derived in (27), the equilibrium condition corresponding to consensus analogously to (9) requires that the two vectors \(\mathbf{v}_{\lambda},\mathbf{v}_{\lambda}^{(2)}\in\mathbb{R}^{2N+\mu}\):  
\[\mathbf{v}_{\lambda}= \begin{bmatrix}\mathbf{1}_{N}\\ \boldsymbol{0}_{N}\\ \bar{\mathbf{v}}\end{bmatrix} ,\qquad\mathbf{v}_{\lambda}^{(2)}= \begin{bmatrix} \boldsymbol{0}_{N}\\ \mathbf{1}_{N}\\ \mathbf{v}_{1}\end{bmatrix} , \tag{38}\]  
where \(\bar{\mathbf{v}},\mathbf{v}_{1}\in\mathbb{R}^{\mu}\), must be, respectively, an eigenvector and a generalized eigenvector of order two relative to \(\lambda=1\) of matrix (37). We first impose that \(\mathbf{v}_{\lambda}\) in (38) is an eigenvector relative to \(\lambda=1\) by exploiting the eigenvalue equation as follows:  
\[ \begin{bmatrix}\Phi_{r}+a_{0}\bar{\mathbf{v}}_{r}\mathbf{e}_{1}^{\top}&\Phi_{ rv}&\Psi_{r}\mathbf{e}_{h}^{\top}\\ \Phi_{vr}+a_{0}\bar{\mathbf{v}}_{r}\mathbf{e}_{1}^{\top}&\[PERSON]_{v}&\[PERSON]_{v} \mathbf{e}_{h}^{\top}\\ (\mathbf{a}-\mathbf{b}a_{0})\mathbf{e}_{1}^{\top}&0&F-\mathbf{b}\mathbf{e}_{h} ^{\top}\end{bmatrix}\begin{bmatrix}\mathbf{1}_{N}\\ \mathbf{0}_{N}\\ \bar{\mathbf{v}}\end{bmatrix} = \begin{bmatrix}\[PERSON]_{r}\mathbf{1}_{N}+a_{0}\bar {\mathbf{v}}_{r}+\[PERSON]_{r}(\bar{\mathbf{v}})_{h}\\ \[PERSON]_{vr}\mathbf{1}_{N}+a_{0}\bar{\mathbf{v}}_{v}+\Psi_{v}\bar{\mathbf{v}}_{h} \\ (\mathbf{a}-\mathbf{b}a_{0})+(F-\mathbf{b}\mathbf{e}_{h}^{\top})\bar{\mathbf{v}} \end{bmatrix} = \begin{bmatrix}\mathbf{1}_{N}\\ \mathbf{0}_{N}\\ \bar{\mathbf{v}}\end{bmatrix}  \tag{39}\]  
so that, considering (27), we obtain the condition:  
\[\left\{