Symmetric Bipartite Containment Tracking of High-Order Networked Agents via Predefined-Time Backstepping Control

Signed networks, which incorporate both cooperative and antagonistic interactions, naturally give rise to symmetric behaviors in multi-agent systems. One such behavior is bipartite containment tracking, where follower agents converge to a symmetric configuration determined by multiple groups of leaders with opposing influence. Moreover, a timely response is critical to ensuring high performance in containment tracking tasks, particularly for high-order multi-agent systems operating in dynamic and uncertain environments. To this end, this paper investigates the predefined-time bipartite containment tracking problem for high-order multi-agent systems affected by external disturbances. A robust tracking control scheme is developed based on the backstepping method to ensure that the tracking errors converge to a predefined residual set within a user-specified time. The convergence time is explicitly adjustable through a design parameter, and the proposed scheme effectively avoids the singularities often encountered in conventional predefined-time control approaches. The stability and robustness of the proposed scheme are rigorously established through Lyapunov-based analysis, and extensive simulation results are provided to validate our theoretical findings.