# Data-Driven Formation Control for Multi-Vehicle Systems Induced by Leader Motion
## 5 Parameter Synthesis for Finite-Time Transient Behavior  
In this section, we compute the characteristic polynomial as a function of the design parameters. The results that we achieve are the main tools to synthetize the algorithm's parameters through a polynomial approach to achieve finite time consensus. The following technical Lemma 1 establishes the analytic connection between the characteristic polynomial of the original system with protocol (8) and the controlled system when adopting the leader motion mechanism (30) and (31). An analogous result holds when the adopted protocol is (12), in this case, we skip this latter case since the result is analogous by substituting matrix (16) with (17). **Lemma 1** (Characteristic polynomial). : _Let \(p_{\phi}(s)\) be the characteristic polynomial of (33), \(\psi_{a}(s)=a_{0}s^{\mu}+a_{1}s^{\mu-1}+\cdots+a_{\mu}=\sum_{i=0}^{\mu}a_{\mu- i}s^{i}\) and \(\psi_{b}(s)=z^{\mu}+\sum_{i=1}^{\mu}b_{i}z^{i}\). Define:_  
\[p_{\psi}(s)=\sum_{i=1}^{N}\Psi_{ri}\det[(sI-\Phi)_{1i}]+\sum_{i=1}^{N}\Psi_{ri} \det[(sI-\Phi)_{1,N+i}], \tag{51}\]  
_then, it holds:_  
\[p_{\Gamma}(s)=\psi_{b}(s)p_{\phi}(s)-\psi_{a}(s)p_{\psi}(s). \tag{52}\]  
Proof. : System evolution is described by the characteristic polynomial of (33) when using either protocol (8) or (12), as follows:  
\[ \begin{split}& p_{\Gamma}(s)=\det[sI_{\mu}-F+\mathbf{b}\mathbf{e}_{ \mu}]\\ &\cdot\det\Biggl{\{}\begin{bmatrix}sI_{N}-\Phi_{r}-a_{0}\Gamma_{ r}\mathbf{e}_{1}^{\top}&-\Phi_{rv}\\ -\Phi_{vr}-a_{0}\Gamma_{v}\mathbf{e}_{1}^{\top}&sI_{N}-\Phi_{v}\end{bmatrix}- \begin{bmatrix}\Psi_{r}\mathbf{e}_{\mu}^{\top}\\ \Psi_{r}\mathbf{e}_{\mu}^{\top}\end{bmatrix}(sI-F+\mathbf{b}\mathbf{e}_{\mu}^{ \top})^{-1}[(\mathbf{a}-\mathbf{b}a_{0})\mathbf{e}_{1}^{\top}&0]\Biggr{\}}. \end{split}  \tag{53}\]  
Matrix \(F-\mathbf{b}\mathbf{e}_{\mu}\) is a companion form matrix, and it holds:  
\[\mathbf{e}_{\mu}^{\top}(sI-F+\mathbf{b}\mathbf{e}_{\mu}^{\top})^{-1}=\frac{ \mathbf{e}_{\mu}^{\top}\mathrm{adj}(sI-F+\mathbf{b}\mathbf{e}_{\mu}^{\top})}{ \det(sI_{\mu}-F+\mathbf{b}\mathbf{e}_{\mu}^{\top})}=\frac{ \begin{bmatrix}1&s& \ldots&s^{\mu-1}\end{bmatrix} }{s^{\mu}+b_{\mu}z^{\mu-1}+\cdots+b_{2}z+b_{1}}, \tag{54}\]  
so that, considering the definition of \(\psi_{a}(s)\) and \(\psi_{b}(s)\) in the Statement, it is possible to further elaborate Equation (53) as:  
\[p_{\Gamma}(s)=\psi_{b}(s)\det\biggl{\{}sI_{2N}- \begin{bmatrix}\Phi_{r}&\Phi_{ r_{0}}\\ \Phi_{vr}&\Phi_{v}\end{bmatrix} -\frac{\psi_{a}(s)}{\psi_{b}(s)} \begin{bmatrix} \Psi_{r}\\ \Psi_{v}\end{bmatrix}\begin{bmatrix}\mathbf{e}_{1}^{\top}&\Phi_{r}^{\top}\end{bmatrix}  \biggr{\}}. \tag{55}\]  
Recurring to the Laplace expansion of the determinant in (55) along the first column one has:  
\[p_{\Gamma}(s)=\psi_{b}(s)\Biggl{\{}p_{\phi}(s)-\frac{\psi_{a}(s)}{\psi_{b}(s)} \Biggl{(}\sum_{i=1}^{N}\Psi_{ri}\det[(sI-\Phi)_{1i}]+\sum_{i=1}^{N}\Psi_{v_{i}} \det[(sI-\Phi)_{1,N+i}]\Biggr{)}\Biggr{\}}, \tag{56}\]  
and, in view of (51), Equation (56) can be finally put as in (52). **Remark 2**. : _Polynomials \(p_{\phi}(s)\), \(p_{\Gamma}(s)\) are strictly related to the communication graph topology, and indeed, their coefficients can be computed by exploiting the exponential map \(\lambda_{\phi}=e^{\lambda_{\Gamma}}\) (or \(\lambda_{\phi_{2}}=e^{\lambda_{\Gamma_{2}}}\) if protocol (12) is adopted), together with the relations [3]:_  
\[\lambda_{\Gamma}^{i+}=\frac{\gamma\lambda_{\mathcal{L}}^{i}+\sqrt{\gamma^{2} \lambda^{i}\mathcal{L}^{2}+4\lambda^{i}\mathcal{L}}}{2},\qquad\lambda_{ \Gamma}^{i-}=\frac{\gamma\lambda^{i}\mathcal{L}-\sqrt{\gamma^{2}\lambda^{i} \mathcal{L}^{2}+4\lambda^{i}\mathcal{L}}}{2}, \tag{57}\]  
_if (8) is adopted, or:_  
\[\lambda_{\Gamma^{(2)}}^{i+}=\frac{\gamma\lambda^{i}\mathcal{L}-\delta+\sqrt{ \left(\gamma\lambda^{i}\mathcal{L}-\delta\right)^{2}+4\lambda^{i}\mathcal{L}}}{ 2},\qquad\lambda_{\Gamma}^{(2)i-}=\frac{\gamma\lambda^{i}\mathcal{L}-\delta- \sqrt{\left(\gamma\lambda^{i}\mathcal{L}-\delta\right)^{2}+4\lambda^{i} \mathcal{L}}}{2}, \tag{58}\]if the adopted protocol is (12).