Stability margin of undirected homogeneous relative sensing networks: A geometric perspective

In this paper, we study the stability margin (SM) of undirected homogeneous relative sensing networks (UH-RSNs) from a geometric point of view. SM is an important robustness measure indicating the amount of simultaneous gain and phase perturbations in the feedback channels before the instability occurs. A UH-RSN is characterized by the identical local dynamics (a single-input-single-output (SISO) open-loop transfer function T_loc(s)) of individual agent and the graph Laplacian L_G representing how the agents are connected. It is shown in this paper that UH-RSNs having multiple inputs and outputs in general may be represented as a unity feedback system including the SISO T_loc(s) and one of the real eigenvalues of L_G. This representation then helps to identify a class of cooperative T_loc(s) for which (1) SM becomes maximized or equal to 1 when the network's connectivity (the second smallest eigenvalue of L_G) is greater than or equal to the curvature of the Nyquist plot of T_loc(s) at the origin; and (2) two bounds on SM are obtained for the SM estimation based on the geometric shape of Nyquist plot. Also, the representation of unity feedback system implies that UH-RSNs with non-cooperative T_loc(s) become unstable when the agents are joined with high connectivity. Numerical examples are provided to demonstrate these findings.