The consensus problem in networks with transmission delays

Abstract

We study discrete- and continuous-time consensus problems on networks in the presence of distributed time delays. We focus on information transmission delays, as opposed to information processing delays, so that each node of the network compares its current state with the past states of its neighbours. We consider directed and weighted networks where the connection structure is described by a normalized Laplacian matrix and show that consensus is achieved if and only if the underlying graph contains a directed spanning tree. This statement holds independently of the transmission delays, which is in contrast to the case of processing delays. Furthermore, we calculate the consensus value explicitly, and show that it is determined by the history of the system over an interval of time, unlike the case of processing delays where the consensus value depends only on the initial state of the system at time zero. This provides the consensus algorithm with improved robustness against noise.

On the Delayed Scaled Consensus Problems

“…An important implication is that, in the event of additive measurement noise with zero temporal mean, the delayed system may have better performance by choosing a suitable τ to reduce the noise effect by averaging over [−τ, 0] via (14). In [18], a similar procedure for delayed standard consensus problems has been discussed.…”

“…Signal transmission delay often appears in networks of oscillators and has been investigated in, e.g., [14,18] for standard consensus problems. In fact, (6) can be generalized to accommodate a history dependence over the interval [t − τ, t] instead of a discrete past time instant t − τ:…”

“…Therefore, the eigenvalue condition in Theorem 3 can be replaced by a more geometric (sufficient) condition 0 ≤ τ < π 4 max 1≤i≤n |α i |d i . We mention that a deep investigation regarding the conditions guaranteeing Re(λ i ) < 0 (i ≥ 2) would be critical for tackling the general system (18).…”

Asymptotic Behaviors of an Opinion Dynamical System without or with a Leader

“…The consensus problems as one of the forms of self-organized systems have been widely observed from Physics [1], Chemistry [2], Biology [3]- [5], Computer science [6], [7], Human society, Psychology and Education [8]- [11] (in the emergence of language and spectator violence), Management science [12] and so on. Opinion consensus systems as one kind of consensus problems have attracted more and more researchers to study.…”

Dynamics, control and information in delay-coupled systems: an overview

“…The article by Atay [4] derives stability conditions for synchronization in networks with discrete or distributed delay. Much progress has been made recently in the understanding of zero-lag and cluster synchronization in networks with large delay.…”