Distributed Fixed-Time Output Regulation for a Class of Nonlinear Multi-Agent Systems with Unity Relative Degree

In this paper, we consider a fixed-time output regulation problem for a class of unknown unity relative degree nonlinear multi-agent systems composed of some subsystems and one exosystem. The goal is to design a smooth distributed adaptive controller such that the fixed-time output regulation problem is solved. Throughout this paper, it is assumed that only a subset of subsystems can obtain information from the exosystem. First, smooth distributed observers are designed to reconstruct the system matrix and state of the exosystem, which can guarantee fixed-time convergence of the estimation errors. Second, an internal model is introduced to design a smooth distributed fixed-time observer to reconstruct the output of the exosystem, which can ensure that the estimation errors approach zero within a fixed time. Next, linearly parameterized models are utilized to model the unknown dynamics of the subsystems. A smooth observer-based adaptive controller is further developed for each subsystem using the output information from each subsystem. At the same time, fixed-time convergence of the closed-loop system is analyzed with the help of a common Lyapunov function method. Finally, as an example, we apply our result to the generalized Lorenz systems.