Coherence and convergence rate in networked dynamical systems

Abstract

We study two metrics in stochastic consensus dynamics with leaders or stubborn agents: network coherence (defined in terms of the system H2 and H∞ norms), and convergence rate. We allow each agent to maintain an individual level of stubbornness in deviating from its initial values. We give bounds on the convergence rate and present sufficient conditions under which the bounds become tight. Moreover we study the effect of the level of stubbornness of the agents on network coherence and convergence rate. We then characterize these two metrics in random regular graphs and Erdos-Renyi random graphs. From a leader selection point of view, we show that maximizing H∞ coherence is equivalent to maximizing convergence rate. Moreover we study conditions under which the optimal leader for maximizing H2 coherence differs from the optimal leader for maximizing convergence rate, and conversely, provide sufficient conditions on the network for a single leader to maximize both metrics simultaneously.

Robustness of Leader–Follower Networked Dynamical Systems

“…Moreover, based on Weyl's inequality (26) and according to the fact that |S| = o(np) we have λ n−|S| (L g ) = Θ(np), which yields the result.…”

“…Proof: Similar to the proof of Theorem 6, we write the grounded Laplacian matrix L g as L g =L + E. Using Weyl's inequality (26) for constant p we have (1 − o(1))(n − |S|)p ≤ λ n−|S| (L) ≤ (1 + o(1))(n − |S|)p asymptotically almost surely [43]. By considering |S| = o(np)…”

“…As we describe in the paper, the robustness metrics of interest are a function of the spectrum of the grounded Laplacian matrix (obtained by removing certain rows and columns from the Laplacian matrix [23]- [25]). Hence, one of the main contributions of this paper, which is an extended version of the conference paper [26], is to propose graph-theoretic bounds on the extreme eigenvalues of this matrix. These eigenvalue bounds consequently provide bounds on the robustness metrics in general graphs, and tight bounds on such metrics in random graphs.…”

New methods to realize the cluster consensus for multi‐agent networks

“…Much attention will be paid to relax the assumption of strong connectivity for the considered system. Moreover, convergence rate and H ∞ coherence proposed/discussed in [11] will also be explored for directed multi‐agent networks.…”

“…Over the past decades, the consensus problem of multi-agent networks has attracted much attention due to its wide applications in various fields, such as unmanned air vehicles (UAVs) [1], flocking [2,3] and attitude alignment of satellites [4]. Many different types of consensus phenomena have been investigated in the previous literature [5], which include consensus [6][7][8][9][10][11][12], practical consensus [13], stochastic consensus [14], double-integrator consensus [15], finite-time consensus [16][17][18], fixed-time consensus [19], scaled consensus [20], cluster consensus [21][22][23] and so on. The main purpose of this paper is to study the cluster consensus of linearly coupled multi-agent systems.…”