Exact stability analysis of second-order leaderless and leader–follower consensus protocols with rationally-independent multiple time delays

An investigation of double-integrator agents with directed asymmetric consensus protocols and multiple rationally independent time delays is presented in this paper from two novel perspectives. First, we complement the group consensus literature on crucial stability analysis, using a recent technique called the Cluster Treatment of Characteristic Roots (CTCR) for the first time on this class of time-delayed systems. The CTCR paradigm is pursued after a block-diagonalization (mode-decoupling) transformation on the system. This treatment produces some unique stability tables for the dynamics in the space of the delays which are non-conservative and exhaustive. Second, a novel concept of spectral delay space is presented, as an overture to the CTCR for the determination of the complete set of stability-crossing (switching) hypersurfaces in the delay space. Examples are provided to display the strengths and efficiency of this new stability analysis mechanism.