Input-Output Pseudospectral Bounds for Transient Analysis of Networked and High-Order Systems

Abstract

This paper studies the problem of coordinating a group of n th -order integrator systems. As for the wellstudied conventional consensus problem, we consider linear and distributed control with only local and relative measurements. We propose a closed-loop dynamic that we call serial consensus and prove it achieves n th order consensus regardless of model order and underlying network graph. This alleviates an important scalability limitation in conventional consensus dynamics of order n ≥ 2, whereby they may lose stability if the underlying network grows. The distributed control law which achieves the desired closed loop dynamics is shown to be localized and obey the limitation to relative state measurements. Furthermore, through use of the small-gain theorem, the serial consensus system is shown to be robust to both model and feedback uncertainties. We illustrate the theoretical results through examples.

Fundamental limitations of case isolation

“…In general, if the adjacency matrix W captures a physical network, then a controller u = Ax with A ∈ A q (W, c) means u i only requires signals from Agent i's q-hop neighborhood. This is readily proven; we refer the reader to [26].…”

“…The proof is found in Appendix B, which is a version of what appeared in [26]. The implementation of the serial consensus will be further discussed after our main results.…”

“…For the conventional consensus a theorem like this does not exist, since using, e.g., a Laplacian corresponding to a directed cycle can result in an unstable closed loop as noted in [11], [14]. Serial consensus, on the other hand, can also be shown to be robustly stable, see [26] for the robustness criteria.…”

“…The proof, a version of which appeared in [26], is presented in Appendix A. Per Theorem 1, the serial consensus system has a stable consensus equilibrium and, since this holds independently of the number of agents, it is also scalably stable.…”