This paper studies how a group of different linear agents—think robots or vehicles with different dynamics—can be guided to stay inside the area defined by some leaders’ motions, even when some actuators are under cyber-attack and the followers do not know the leaders’ internal plans. The leaders produce bounded, continuous trajectories, but their dynamics, speed limits, and motion envelopes are undisclosed to the followers. The goal is resilient output containment: make the followers’ outputs end up inside the convex hull of the leaders’ outputs despite attacks and limited information.

The authors model a fairly broad class of actuator attacks. The false data injected into actuators can depend on the system state and on control inputs, and it can include bounded external disturbance terms. The network linking agents is directed, so information can flow one way along some links. The paper assumes only partial state measurements are available to each follower.

To handle these challenges the paper proposes a continuous two-layer adaptive control architecture. The first layer is a virtual-actuator reconfiguration layer. It uses the available partial state measurements to compensate for actuator attacks in the local tracking-error dynamics. The second layer is a network interface that produces task-space commands through an adaptive interaction protocol. That protocol relies only on neighbor-exchanged interface states whose size matches the plant output. Importantly, it does not require knowledge of the full network graph for choosing tuning parameters.

On the theory side, under a graph condition called a leader-rooted united spanning tree, the authors use a nonsmooth Lyapunov analysis (an energy-like proof method that handles non-differentiable terms) to show asymptotic containment at the command level. In other words, the command signals generated by the network interface converge as desired. The actual physical outputs of the agents then converge into the leaders’ convex hull up to a residual error that depends on the performance of the local command-tracking controllers.