Fast Fixed-Time Output Multi-Formation Tracking of Networked Autonomous Surface Vehicles: A Mathematical Induction Method

In this paper, we aim to exploit an effective way to solve the output multi-formation tracking problem of the networked autonomous surface vehicles (ASVs) in a fast fixed time manner. Specifically, addressing the output multi-formation tracking problem implies that 1) the networked ASVs are divided into multiple interconnected subnetworks with respect to multiple targets; 2) for each subnetwork, the outputs of the networked ASVs form a desired geometric formation with exchanging the local interactions. Besides, solving the fast fixed-time tracking problem in this paper implies that 1) the convergence time is independent of the initial conditions; 2) the system states are forced to reach the employed nonsingular fixed-time sliding surface in a prescribed time, which thus called fast fixed-time control. Then, based on a time-related function and a nonsingular fixed-time sliding surface, a hierarchical fast fixed-time control algorithm is proposed to solve the aforementioned problem within a fast fixed time being independent of the initial conditions. Furthermore, by employing the Lyapunov argument and mathematical induction, we present the sufficient conditions for fast fixed-time convergence of the tracking errors with respect to multiple targets. Finally, numerous simulation examples are presented to demonstrate the effectiveness of the proposed control algorithm.