Improved frequency sweeping technique and stability analysis of the second-order consensus protocol with distributed delays

The consensus problem of a general class of second-order multi-agent system with distributed delays is investigated from a new perspective. We propose a consensus protocol with distributed delays to enhance the delay robustness of the system. First, we establish a spectral equivalence of such a system and the one with multiple discrete delays from the stability viewpoint. We then employ a factorization procedure to reduce the computational complexity. Next, we uniquely use the Dixon resultant theory to determine the exact upper bound of the imaginary spectra, leading to the kernel and offspring hypersurfaces (KOH) in the delay domain. Furthermore, the associated standing roots of the system are analyzed to determine the precise stationary root boundaries, which constitute the potential stability changing curves together with the KOH. Finally, we resort to the Cluster Treatment of Characteristic Roots (CTCR) paradigm to examine the exact and complete stability of the system.