# Data-Driven Formation Control for Multi-Vehicle Systems Induced by Leader Motion
## 4 Sampled-Data Leader Motion Mechanism Design
### Conditions to Reach Consensus
\begin{array}{rcl}(\bar{\mathbf{v}})_{\mu}&=&-a_{0},\\ (\mathbf{a}-\mathbf{b}a_{0})+\left(F-\mathbf{b}\mathbf{e}_{\mu}^{\top}\right) \bar{\mathbf{v}}-\bar{\mathbf{v}}&=&(\mathbf{a}-\mathbf{b}a_{0})+\left(F-I_{ \mu}\right)\bar{\mathbf{v}}+\mathbf{b}\bar{\mathbf{v}}_{\mu}=0.\end{array} \right. \tag{40}\]  
The second relation provides the structure of \(\bar{\mathbf{v}}\):  
\[\bar{\mathbf{v}}= \begin{bmatrix}a_{1}\\ a_{1}+a_{2}\\ \vdots\\ a_{1}+\cdots+a_{\mu}\end{bmatrix} , \tag{41}\]  
and plugging (41) into the first equation, one obtains the following condition on the coefficients \(a_{i}\):  
\[a_{0}+\sum_{i=1}^{\mu}a_{i}=0. \tag{42}\]  
We now exploit the condition that \(\mathbf{v}_{\lambda}^{(2)}\) is a second-order generalized eigenvector through (3), so that:  
\[ \begin{bmatrix}\[PERSON]_{r}+a_{0}\bar{\mathbf{v}}_{r}\mathbf{e}_{1}^{\top}&\Phi_{ rv}&\Psi_{r}\mathbf{e}_{h}^{\top}\\ \Phi_{vr}+a_{0}\bar{\mathbf{v}}_{r}\mathbf{e}_{1}^{\top}&\Phi_{v}&\Psi_{v} \mathbf{e}_{h}^{\top}\\ (\mathbf{a}-\mathbf{b}a_{0})\mathbf{e}_{1}^{\top}&0&F-\mathbf{b}\mathbf{e}_{h} ^{\top}\end{bmatrix}\begin{bmatrix}\boldsymbol{0}_{N}\\ \mathbf{1}_{N}\\ \mathbf{v}_{1}\end{bmatrix} = \begin{bmatrix}\boldsymbol{0}_{N}\\ \mathbf{1}_{N}\\ \mathbf{v}_{1}\end{bmatrix} + \begin{bmatrix}\boldsymbol{1}_{N}\\ \boldsymbol{0}_{N}\\ \bar{\mathbf{v}}\end{bmatrix} , \tag{43}\]  
where \(\bar{\mathbf{v}}\) must satisfy (40), and \(\alpha\in\mathbb{R}-\{0\}\) is a nonzero scaling coefficient. By computation of the left-hand side and exploiting relations (27), one obtains:\[ \begin{bmatrix}\mathbf{1}_{N}\Delta+\Psi_{r}v_{1_{H}}\\ \mathbf{1}_{N}+\Psi_{v}v_{1_{H}}\\ (F-\mathbf{b}\mathbf{e}_{\mu}^{\top})\mathbf{v}_{1}\end{bmatrix} = \begin{bmatrix} \mathbf{0}_{N}\\ \mathbf{1}_{N}\\ \mathbf{v}_{1}\end{bmatrix} +\alpha \begin{bmatrix}\mathbf{1}_{N}\\ \mathbf{0}_{N}\\ \bar{\mathbf{v}}\end{bmatrix} . \tag{44}\]  
From the second of (44), we get that \(v_{1_{H}}=0\), and this choice solves the first line of (44) upon setting \(\alpha=\Delta\). Finally, exploiting the last line, one obtains:  
\[\big{(}I_{\mu}-F\big{)}\mathbf{v}_{1}=-\Delta\bar{\mathbf{v}}, \tag{45}\]  
and in view of Equation (41), it is possible to express \(\mathbf{v}_{1}\) as a function of parameters \(a_{i}\) as follows:  
\[\mathbf{v}_{1}= \begin{bmatrix}-\Delta a_{1}\\ -\Delta(2a_{1}+a_{2})\\ \vdots\\ -\Delta\big{(}\mu a_{1}+(\mu-1)a_{2}+\cdots+2a_{\mu-1}+a_{\mu}\big{)}\end{bmatrix} , \tag{46}\]  
thus, considering the previous condition \(v_{1_{H}}=0\), we finally achieve the condition for consensus (10) in terms of coefficients \(a_{i}\) as follows:  
\[\sum_{i=1}^{\mu}(\mu-i+1)a_{i}=0, \tag{47}\]  
and hence, the Statement holds. When dealing with protocol (12), eigenvalue \(\lambda=1\) is simple, and it is straight to obtain the consensus condition, as follows. The proof follows the lines of first-order systems, so here, we give a brief sketch. Proposition 3: _Consider the sampled multi-agent system Equation (25) under protocol (12) with input Equation (30). Under the hypothesis that it has a stable transient, the equilibrium of the multi-agent system is a consensus condition (13) if and only if parameters \(\mathbf{a}\) satisfy:_  
\[\mathbf{1}^{\top}\mathbf{a}=0. \tag{48}\]  
Proof: Consider the overall system evolution (33) when adopting protocol (12), the steady state behavior satisfies the stationary equations (37) using protocol (12). In order to reach the consensus condition, we impose the stationarity of the vector:  
\[\mathbf{v}_{\lambda}= \begin{bmatrix}\mathbf{1}_{N}\\ \mathbf{0}_{N}\\ \bar{\mathbf{v}}\end{bmatrix} , \tag{49}\]  
so that we seek values of \(\mathbf{a}\), \(\mathbf{b}\) so as to satisfy:  
\[\left\{ \begin{array}{rcl}\Phi_{r}^{(2)}\mathbf{1}+a_{0}\Psi_{r}^{(2)}+\Psi_ {r}^{(2)}\varphi_{\mu}&=&\mathbf{1},\\ \Phi_{vr}^{(2)}\mathbf{1}+a_{0}\Psi_{r}^{(2)}+\Psi_{r}^{(2)}\varphi_{\mu}&=& \mathbf{0},\\ \boldsymbol{\varphi}=(\mathbf{a}-\mathbf{b}a_{0})+\Big{(}F-\mathbf{b}\mathbf{e }_{\mu}^{\top}\Big{)}\boldsymbol{\varphi}.\end{array} \right.