Adaptive consensus tracking control of uncertain nonlinear systems: A first-order example

In this paper, we consider the problem of designing distributed adaptive consensus tracking controllers for multiple nonlinear systems with unknown parameters and external disturbances. The desired trajectory is time varying given by the state of a reference system, which is only available to a portion of the group of the systems. Besides, the dynamics of the reference state is bounded but unknown to all of the systems. The communication graph characterizing the interactions among the systems is assumed to have undirected, fixed and connected topology. By introducing distributed estimators for the bound of the reference dynamics, two control schemes are proposed to address the problem. In the first scheme, a sign function is employed and perfect consensus tracking can be achieved. In the second scheme, an alternative control law is developed and the chattering phenomenon caused by the sign function can be reduced. However, new challenge will be triggered which is to compensate for possible destabilizing effects of the coupling elements relating to local parameter estimation errors and the synchronization errors of the neighbors. The overall communication graph is firstly reduced to an undirected spanning tree with single system notified of the reference state. Based on this, new synchronization error for each subsystem is then defined as the weighted distance relative to only one of its neighbors. It is shown that all the synchronization errors will converge to a prescribed bound which can be made as small as desired in this case.