Exact Stability Analysis of the Second-Order Affine Formation Maneuver System with Three Independent Time Delays

We investigate the asymptotic stability of a leader-follower type second-order affine formation maneuver system with three independent time delays. These three delays stem from the communication channels of position, velocity, and acceleration. The central idea is to exactly determine the potential stability switching loci in the domain of delays, namely the kernel and offspring hypersurfaces (KOH). For this, we first decompose the system's characteristic equation into a sequence of factors through a factorization procedure. We then adopt the Dixon resultant theory combining the frequency sweeping technique to ascertain KOH of each subsystem exhaustively. Finally, we superpose all KOH and resort to the Cluster Treatment of Characteristic Roots (CTCR) paradigm to create the complete stability map of the whole system. Finally, we provide a case study to demonstrate the effectiveness of these methodologies and point out the positive and negative effects of the delays on the affine formation system.