# Data-Driven Formation Control for Multi-Vehicle Systems Induced by Leader Motion
## 1 Introduction  
In the last decades, a powerful thrust of research has been oriented to the study of distributed algorithms for their impact on several technological areas, as those of sensor networks and mobile agents [1]. The emerging technologies that were being developed during the same period in the area of communication promoted a significant research thrust in the coordination of mobile sensing agents and cooperative platooning maneuvers for vehicular networks in a distributed fashion [2; 3]. As a result of this impressive research activity, a brand new paradigm emerged for teams of vehicles to perform various fundamental tasks such as exploration, surveillance, environmental monitoring, and distributed coordination based on the so-called _consensus-based_ distributed strategies [4; 5]. Consensus popularity is related to its effectiveness in making all agents reach an agreement on a local variable through a distributed iterative control scheme, which is built on local data [1]. In the earliest stages, consensus protocols were focused on interconnected computing units and computer networks, and algorithms were studied by adopting the simplest first order models [1]. As research moved on, considering the promising potentialities of the results achieved so far, higher-order dynamic agents were largely investigated, and the second-order models were widely considered for describing the motion dynamics of sensing moving devices [2; 3; 4]. Further advancements in vehicle dynamics modeling involve third-order dynamics [4], whose coordination through consensus algorithms is studied in [6]. Consensus techniques have been largely adopted in formation control and vehicle platooning for a long time (see [7] and references therein). However, the drawbacks of slow convergence and poor consensus performances pushed a strong thrust of research toward faster consensus in second-order multi-agent systems and vehicular networks as well. Indeed, in contrast to the powerful features of the consensus condition as a toolfor coordinating teams of agents, it was soon discovered that the convergence toward consensus can be unacceptably low, and may not even be reached in the presence of communication and computation uncertainty or faults [8]. For this reason, convergence to consensus through customized protocols was studied as well [9; 10]. A topic of special interest that has been widely studied and investigated over the years under several facets is the finite-time consensus, and its evolution to fixed-time and prescribed-time consensus. As opposed to the fragility of asymptotic consensus when dealing with an uncertain and erroneous framework, finite-time consensus condition shows a much higher performance and intrinsic disturbance rejection feature [11]. In [12; 13], prescribed-time formation tracking is studied for the networked autonomous surface vehicles with the external disturbances through the combination of a disturbance observer and a sliding mode controller. In [14], the fixed-time formation problem is studied in the presence of external disturbances, model uncertainties, input saturation, and quantization constraints. In [15], model-free cluster formation is sought for a network of marine surface vehicles in an uncertain environment. A notable recent paper is [16], where an additional prescribed-time state observer is introduced to solve the problem regardless of the initial condition. Also in [17], the attainment of finite time consensus for vehicle platooning in the presence of disturbances is studied. Two main approaches to finite time consensus have been studied along the years, namely through the use of discontinuous control and time-varying gain [11]. However, despite the advantages of fixed-time consensus, each of the two main approaches that allow for reaching this condition, namely discontinuous control or time-varying gain is prone to several drawbacks in practical applications.