Network coordination and synchronization in a noisy environment with time delays

Fast consensus in a large-scale multi-agent system with directed graphs using time-delayed measurements

This article is on fast-consensus reaching in a class of multi-agent systems (MAS). We present an analytical approach to tune controllers for the agents based on the premise that delayed measurements in the controller can be preferable to standard controllers relying only on current measurements. Controller tuning in this setting is however challenging due to the presence of delays. To tackle this problem, we propose an analytic geometry approach. The key contribution is that the tuning can be implemented for complex eigenvalues of the arising graph Laplacian of the network, complementing the current state of the art, which is limited to real eigenvalues. Results, therefore, extend our knowledge beyond symmetric graphs and enable the study of the MAS under directed graphs. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Multiple Intentional Delays Can Facilitate Fast Consensus and Noise Reduction in a Multiagent System

We investigate the use of a derivative-free control scheme called the multiple-delay proportional-retarded (PR) protocol to achieve fast consensus in a large-scale multiagent system. The delays are intentionally introduced in the PR with the purpose of creating derivative-like controllers that rely only on position measurements, thus mitigating undesirable noise effects. The main result places the spectral abscissa of the consensus dynamics at a desired locus and also achieves a user-defined spectral "separation," both of which directly influence convergence to consensus. Design rules and limitations arising from analytical derivations to achieve these results are laid out. Case studies are provided to demonstrate the concepts.

A consensus dynamics with delay-induced instability can self-regulate for stability via agent regrouping

Dynamics of many multi-agent systems is influenced by communication/activation delays τ. In the presence of delays, there exists a certain margin called the delay margin τ^*, less than which system stability holds. This margin depends strongly on agents' dynamics and the agent network. In this article, three key elements, namely, the delay margin, network graph, and a distance threshold conditioning two agents' connectivity are considered in a multi-agent consensus dynamics under delay τ. We report that when the dynamics is unstable under this delay, its states can be naturally bounded, even for arbitrarily large threshold values, preventing agents to disperse indefinitely. This mechanism can also make the system recover stability in a self-regulating manner, mainly induced by network separation and enhanced delay margin. Under certain conditions, unstable consensus dynamics can keep separating into smaller stable subnetwork dynamics until all agents stabilize in their respective subnetworks. Results are then demonstrated on a previously validated robot coordination model, where specifically robustness of τ^* is studied against the delay τ_inh inherently present in the orientation measurements of the robots. To this end, a mathematical framework to compute τ^* with respect to τ_inh in quasi-state is developed, demonstrating that τ^* can be sensitive to τ_inh, yet robot regrouping and stabilization of subnetworks is still possible.

Extreme fluctuations in stochastic network coordination with time delays

We study the effects of uniform time delays on the extreme fluctuations in stochastic synchronization and coordination problems with linear couplings in complex networks. We obtain the average size of the fluctuations at the nodes from the behavior of the underlying modes of the network. We then obtain the scaling behavior of the extreme fluctuations with system size, as well as the distribution of the extremes on complex networks, and compare them to those on regular one-dimensional lattices. For large complex networks, when the delay is not too close to the critical one, fluctuations at the nodes effectively decouple, and the limit distributions converge to the Fisher-Tippett-Gumbel density. In contrast, fluctuations in low-dimensional spatial graphs are strongly correlated, and the limit distribution of the extremes is the Airy density. Finally, we also explore the effects of nonlinear couplings on the stability and on the extremes of the synchronization landscapes.

Synchronization of networks of oscillators with distributed delay coupling

This paper studies the stability of synchronized states in networks, where couplings between nodes are characterized by some distributed time delay, and develops a generalized master stability function approach. Using a generic example of Stuart-Landau oscillators, it is shown how the stability of synchronized solutions in networks with distributed delay coupling can be determined through a semi-analytic computation of Floquet exponents. The analysis of stability of fully synchronized and of cluster or splay states is illustrated for several practically important choices of delay distributions and network topologies.

Exact dynamics of stochastic linear delayed systems: application to spatiotemporal coordination of comoving agents

Delayed dynamics result from finite transmission speeds of a signal in the form of energy, mass, or information. In stochastic systems the resulting lagged dynamics challenge our understanding due to the rich behavioral repertoire encompassing monotonic, oscillatory, and unstable evolution. Despite the vast literature, quantifying this rich behavior is limited by a lack of explicit analytic studies of high-dimensional stochastic delay systems. Here we fill this gap for systems governed by a linear Langevin equation of any number of delays and spatial dimensions with additive Gaussian noise. By exploiting Laplace transforms we are able to derive an exact time-dependent analytic solution of the Langevin equation. By using characteristic functionals we are able to construct the full time dependence of the multivariate probability distribution of the stochastic process as a function of the delayed and nondelayed random variables. As an application we consider interactions in animal collective movement that go beyond the traditional assumption of instantaneous alignment. We propose models for coordinated maneuvers of comoving agents applicable to recent empirical findings in pigeons and bats whereby individuals copy the heading of their neighbors with some delay. We highlight possible strategies that individual pairs may adopt to reduce the variance in their velocity difference and/or in their spatial separation. We also show that a minimum in the variance of the spatial separation at long times can be achieved with certain ratios of measurement to reaction delay.