Delay-Induced Stability-Switching Properties of Linear MAS with Directed Topologies

In this paper, the consensus properties of a linear multi -agent system are studied in the presence of a uniform delay and a directed interaction topology. The stability-switching behavior is considered, such that stability and instability of the system's modes alternate several times as the delay increases. First, the delay values causing imaginary roots of the characteristic equation are computed. The main result shows that the crossing direction of the imaginary axis is strictly related to the magnitude slope of the system's frequency response function. This result allows the depiction of a graph that contains explicitly the crossing direction information. A numerical example illustrates the effectiveness of our method applied to a consensus protocol found in the literature.