# Data-Driven Formation Control for Multi-Vehicle Systems Induced by Leader Motion
## 5 Parameter Synthesis for Finite-Time Transient Behavior
Plugging these relations into Equation (61), one has to solve the following polynomial equation:\[s^{N+\mu-1}-s^{N+\mu-2}=y(s)\hat{p}_{\phi}(s)+x(s)p_{\psi}(s), \tag{65}\]  
and then, based on the solution polynomials \(x(s),y(s)\), set:  
\[\lambda_{\Gamma}^{i+}=\frac{\gamma\lambda_{\mathcal{L}}^{i}+\sqrt{\gamma^{2} \lambda^{i}\mathcal{L}^{2}+4\lambda^{i}\mathcal{L}}}{2},\qquad\lambda_{ \Gamma}^{i-}=\frac{\gamma\lambda^{i}\mathcal{L}-\sqrt{\gamma^{2}\lambda^{i} \mathcal{L}^{2}+4\lambda^{i}\mathcal{L}}}{2}, \tag{57}\]0  
as parameters into (30) and (31), thus satisfying (34), (35), together with (60), and hence, the achievement of finite time consensus. Retrieving Polynomial Coefficients from Data  
In view of the results achieved so far, the polynomial equations, Equations (63)-(65), are the fundamental tool for the synthesis of the leader motion mechanism (30) and (31) to obtain finite time consensus. However, as stated in Remark 1, they require the knowledge of \(p_{\phi}(s)\), \(p_{\psi}(s)\), which can be either deduced from the eigenstructure of the graph Laplacian matrix, or directly retrieved from a suitable set of experimental data. Regarding the former method, it is possible to build \(p_{\phi}(s)\) and \(p_{\psi}(s)\) by their zeros (e.g., exploiting Equations (57) and (58)). However, this method requires the full a priori knowledge of the graph topology, and this is out of our Assumptions. As regards the latter method, we describe in the following a strategy to retrieve such polynomials from experimental data, which requires Assumptions A1-A2 only, as follows. As a first step, polynomial \(p_{\phi}(s)\) is the characteristic polynomial of (25), so that we can retrieve its coefficients from the response of (25) setting \(u_{k}=0\), as in [21], as follows. Let \(u_{k}=0\) for \(k=0,\ldots,2N-1\), and collect \(r_{1}(k)\) in the same interval \(k=0,\ldots,2N-1\). Build the Hankel matrix based on the sequence \(\{r_{1}(k)\}\), namely:  
\[\mathcal{H}(r_{1})= \begin{bmatrix}r_{1}(0)&r_{1}(1)&\ldots&r_{1}(N-1)\\ &\ddots&\ddots\\ r_{1}(N-1)&r_{1}(N)&\ldots&r_{1}(2N-2)\end{bmatrix} . \tag{67}\]  
Then, if we parametrize \(p_{\phi}(s)\) as \(p_{\phi}(s)=s^{N}+\phi_{N-1}s^{N-1}+\cdots+\phi_{1}s+\phi_{0}\), then it holds:  
\[ \begin{bmatrix}\phi_{N-1}\\ \phi_{N-2}\\ \vdots\\ \phi_{0}\end{bmatrix} =\mathcal{H}(r_{1})^{-1} \begin{bmatrix}r_{1}(N)\\ r_{1}(N+1)\\ \vdots\\ r_{1}(2N-1)\end{bmatrix} . \tag{68}\]  
For any detail of the above procedure, the reader is referred to [21]. It is worth mentioning that the above method strongly relies on Assumptions A1-A2. Indeed, based on A1, we know \(N\) in advance, and we can build \(\mathcal{H}(r_{1})\) as in (67). Moreover, Equation (68) directly provides the coefficients of the minimal annihilator polynomial, which necessarily coincides with the characteristic polynomial under A2. For a broader discussion of the above algorithm, even when the above hypotheses are not met, the interested reader is referred to [21]. As regards polynomial \(p_{\phi}(s)\), the following inspection provides an effective strategy for an experimental setup useful to derive it. Consider Equation (25) and set \(u_{k}=\pi r_{1}(k)\), thus applying a static output feedback to the network from node 1. The multi-agent evolution under this condition takes the form:  
\[ \begin{cases}\quad\mathbf{r}(k+1)&=&\Phi_{r}\mathbf{r}(k)+\Phi_{r\mathbf{v}} \mathbf{v}(k)+\alpha\Psi_{r}r_{1}(k),\\ \mathbf{v}(k+1)&=&\Phi_{vr}\mathbf{r}(k)+\Phi_{v}\mathbf{v}(k)+\alpha\Psi_{v} r_{1}(k),\\ \mathbf{r}(0)=\mathbf{r}_{0},&\mathbf{v}(0)=\mathbf{v}_{0},\end{cases}  \tag{69}\]which is a special special case of Equation (33) for the static output feedback \(u_{k}=\alpha r_{1}(k)\). Equation (69) can be rewritten also as:  
\[ \begin{bmatrix}\mathbf{r}(k+1)\\ \mathbf{v}(k+1)\end{bmatrix} = \begin{bmatrix}\Phi_{r}+\alpha\[PERSON]_{r}\mathbf{e}_{ 1}^{\top}&\[PERSON]_{rv}\\ \[PERSON]_{vr}+\alpha\[PERSON]_{r}\mathbf{e}_{1}^{\top}&\[PERSON]_{v}\end{bmatrix}\begin{bmatrix} \mathbf{r}(k)\\ \mathbf{v}(k)\end{bmatrix}