# Data-Driven Formation Control for Multi-Vehicle Systems Induced by Leader Motion
## 5 Parameter Synthesis for Finite-Time Transient Behavior
However, Equations (57) and (58) require the knowledge of matrix \(\mathcal{L}\) and, in turn, the topology of graph \(\mathcal{G}\) in advance. This latter condition is out of our Assumptions, indeed one main goal of our algorithm is to retrieve polynomials \(p_{\phi}(s)\), \(p_{\psi}(s)\) from data, as explained in detail in the following._  
As a first step, we deduce a polynomial equation to synthetize the controller's parameters taking into account both stationary and transient conditions. Indeed, based on Equations (51) and (52), it is possible to set a polynomial equation through the approach [31]. However, each of the two protocols, (8) and (12), deserves dedicated investigations. Consider protocol (8) first. In this case, the dynamics of the system under the leader's action are given by Equation (33), and considering the steady state conditions in (34) and (35), one has that \(s=1\) has multiplicity 2, so that we can parametrize the closed loop characteristic polynomial as:  
\[p_{\Gamma}(s)=(s-1)^{2}\hat{p}_{\Gamma}(s),\qquad\qquad\hat{p}_{ \Gamma}(1)\
eq 0, \tag{59}\]  
where \(\hat{p}_{\Gamma}(s)\) describes the transient stage of the controlled system. Considering our goal, namely system convergence in finite time, this part of the characteristic polynomial related to the transient stage should satisfy the following identity:  
\[\hat{p}_{\Gamma}(s)=s^{N+\mu-2}, \tag{60}\]  
so that, plugging (60) into (52) one has:  
\[(s-1)^{2}s^{N+\mu-2}=\psi_{b}(s)p_{\phi}(s)-\psi_{a}(s)p_{\psi}(s). \tag{61}\]  
Consider now that, in view of Equation (27), \(\Phi\) has the eigenvalue \(s=1\) of algebraic multiplicity equal to two, so that \(p_{\phi}(s)\) can be parametrized as \(p_{\phi}(s)=(s-1)^{2}\hat{p}_{\phi}(s)\), \(\hat{p}_{\phi}(1)\
eq 0\). Finally \(\psi_{a}(s)\) can encapsulate conditions (34) and (35), thus allowing us to include consensus conditions. Considering that \(\psi_{a}(s)=a_{0}z^{\mu}+\sum_{i=1}^{\mu}a_{\mu-i}s^{i}\), it holds:  
\[\mathbf{1}^{\top}\mathbf{a}=[\psi_{a}(s)]_{s=1},\qquad\qquad \sum_{i=1}^{\mu}(\mu-i+1)a_{i}=\left[\frac{\mathrm{d}}{\mathrm{d}s}[\psi_{a}(s )]\right]_{s=1}. \tag{62}\]  
In view of equalities (62), conditions (34) and (35) imply that \(\psi_{a}(s)\) should have \(s=1\) as zero of multiplicity 2, so that a choice of \(\psi_{a}(s)=(s-1)^{2}\hat{\psi}_{a}(s)\) allows us to satisfy (34) and (35) for any \(\hat{\psi}_{a}(s)\). Combining all the above conditions, it is possible to set the characteristic polynomial of the controlled system to achieve finite time consensus, as in Equation (60), by solving the following polynomial equation:  
\[s^{N+\mu-2}=y(s)\hat{p}_{\phi}(s)+x(s)p_{\psi}(s), \tag{63}\]  
in the unknown polynomials \(x(s)\), \(y(s)\). Then setting:  
\[\psi_{a}(s)=-(s-1)^{2}x(s),\qquad\qquad\psi_{b}(s)=y(s), \tag{64}\]  
as parameters into (30) and (31) allows us to satisfy (34), (35), together with (60), and hence, the achievement of finite time consensus. On the other hand, when adopting protocol (12), the algebraic properties of \(p_{\phi}(s)\) and \(p_{\psi}(s)\) derive from Equation (28). In view of Equation (28), \(p_{\phi}(s)\) has a simple eigenvalue in \(s=1\), so that, in this case, \(p_{\phi}(s)=(s-1)\hat{p}_{\phi}(s)\), \(\hat{p}_{\phi}(1)\
eq 0\). As for \(\psi_{a}(s)\), condition (28) is equivalent to \(\mathbf{1}^{\top}\mathbf{a}=[\psi_{a}(s)]_{s=1}=0\), meaning also that \(s=1\) must be a simple zero \(\psi_{a}(s)\), so that \(\psi_{a}(s)\) can be written as \(\psi_{a}(s)=(s-1)\hat{\psi}_{a}(s)\), \(\hat{\psi}_{a}(1)\
eq 0\).