Fractional-Order PID Cooperative Pose Control of Multi-Agent Rigid-Body Systems on TSE(3)^N

Fractional-order PID control extends conventional PID control by allowing the integral and derivative orders to take non-integer values, thereby introducing α and β as additional tuning parameters beyond the conventional PID gains. Two strategies for fractional-order PIβDα cooperative pose control of a multi-agent rigid body system are considered assuming both undirected and directed communication networks, together with Caputo fractional derivative and integral operators. The rigid-body pose configurations are described using exponential coordinates associated with the Lie group SE(3), while the state space is TSE(3)N. Analytical stability bounds are obtained using Matignon's theorem in terms of the proportional, derivative, and integral gains for the attitude and translational controllers, as well as the fractional orders, that guarantee global Mittag-Leffler stability. The fractional control strategy provides additional flexibility in tuning the closed-loop response beyond that of conventional integer-order PID control. Since the fractional derivative states are not directly measurable, a Luenberger observer is employed to estimate these states from the measurable pose exponential coordinates, and the resulting estimates are used in the implementation of the feedback protocols. Simulation examples of formation establishment further incorporate cohesion and collision-avoidance strategies based on artificial potential functions, illustrating the effectiveness and practical implementation of the proposed fractional PIβDα control protocols.