Retraction Notice: Multiple Transformation Matrix-Based Adaptive Projective Vortex Formation Tracking for Multiagent...

This article systematically studies projective vortex formation tracking (PVFT) of linear multiagent systems (MASs) on directed graphs through multiple transformation matrices, in which the input of leader and its upper bound information are not available to any follower. First, an innovative class of distributed adaptive observer is designed using the projection matrices to capture multiple desired virtual signals of the leader. Next, a novel kind of distributed PVFT protocol based on distributed observer, local observer and coordinates coupling matrices are proposed. Two different adaptive update mechanisms and nonlinear functions are introduced in the observer and controller to override the unknown input of the leader. Then, an algorithm is given, and the protocol under the algorithm is proved from three processes of estimation, aggregation, and rotation to enable linear systems to achieve PVFT. Finally, the effects of parameters on aggregation and rotation are analyzed systematically, and several simulation examples are given to illustrate the reliability of the results.