Consensus problems in networks of agents with switching topology and time-delays

In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results

Ultrafast Consensus in Small-World Networks

In this paper, we demonstrate a phase transition phenomenon in algebraic connectivity of small-world networks. Algebraic connectivity of a graph is the second smallest eigenvalue of its Laplacian matrix and a measure of speed of solving consensus problems in networks. We demonstrate that it is possible to dramatically increase the algebraic connectivity of a regular complex network by 1000 times or more without adding new links or nodes to the network. This implies that a consensus problem can be solved incredibly fast on certain small-world networks giving rise to a network design algorithm for ultra fast information networks. Our study relies on a procedure called "random rewiring" due to Watts & Strogatz (Nature, 1998). Extensive numerical results are provided to support our claims and conjectures. We prove that the mean of the bulk Laplacian spectrum of a complex network remains invariant under random rewiring. The same property only asymptotically holds for scale-free networks. A relationship between increasing the algebraic connectivity of complex networks and robustness to link and node failures is also shown. This is an alternative approach to the use of percolation theory for analysis of network robustness. We also show some connections between our conjectures and certain open problems in the theory of random matrices

A Sub-optimal Algorithm to Synthesize Control Laws for a Network of Dynamic Agents

We study the synthesis problem of an LQR controller when the matrix describing the control law is constrained to lie in a particular vector space. Our motivation is the use of such control laws to stabilize networks of autonomous agents in a decentralized fashion; with the information flow being dictated by the constraints of a pre-specified topology. In this paper, we consider the finite-horizon version of the problem and provide both a computationally intensive optimal solution and a sub-optimal solution that is computationally more tractable. Then we apply the technique to the decentralized vehicle formation control problem and show that the loss in performance due to the use of the sub-optimal solution is not huge; however the topology can have a large effect on performance

Drifting instabilities of cavity solitons in vertical cavity surface-emitting lasers with frequency selective feedback

In this paper we study the formation and dynamics of self-propelled cavity solitons (CSs) in a model for vertical cavity surface-emitting lasers (VCSELs) subjected to external frequency selective feedback (FSF), and build their bifurcation diagram for the case where carrier dynamics is eliminated. For low pump currents, we find that they emerge from the modulational instability point of the trivial solution, where traveling waves with a critical wavenumber are formed. For large currents, the branch of self-propelled solitons merges with the branch of resting solitons via a pitchfork bifurcation. We also show that a feedback phase variation of 2\pi can transform a CS (whether resting or moving) into a different one associated to an adjacent longitudinal external cavity mode. Finally, we investigate the influence of the carrier dynamics, relevant for VCSELs. We find and analyze qualitative changes in the stability properties of resting CSs when increasing the carrier relaxation time. In addition to a drifting instability of resting CSs, a new kind of instability appears for certain ranges of carrier lifetime, leading to a swinging motion of the CS center position. Furthermore, for carrier relaxation times typical of VCSELs the system can display multistability of CSs.Comment: 11 pages, 12 figure

Agreement Problems in Networks with Directed Graphs and Switching Topology

In this paper, we provide tools for convergence and performance analysis of an agreement protocol for a network of integrator agents with directed information flow. Moreover, we analyze algorithmic robustness of this consensus protocol for the case of a network with mobile nodes and switching topology. We establish a connection between the Fiedler eigenvalue of the graph Laplacian and the performance of this agreement protocol. We demostrate that a class of directed graphs, called balanced graphs, have a crucial role in solving average-consensus problems. Based on the properties of balanced graphs, a group disagreement function (i.e. Lyapunov function) is proposed for convergence analysis of this agreement protocol for networks with directed graphs. This group disagreement function is later used for convergence analysis for the agreement problem in networks with switching topology. We provide simulation results that are consistent with our theoretical results and demonstrate the effectiveness of the proposed analytical tools

Delay analysis of two batch-service queueing models with batch arrivals: Geo(X)/Geo(c)/1

In this paper, we compute the probability generating functions (PGF's) of the customer delay for two batch-service queueing models with batch arrivals. In the first model, the available server starts a new service whenever the system is not empty (without waiting to fill the capacity), while the server waits until he can serve at full capacity in the second model. Moments can then be obtained from these PGF's, through which we study and compare both systems. We pay special attention to the influence of the distribution of the arrival batch sizes. The main observation is that the difference between the two policies depends highly on this distribution. Another conclusion is that the results are considerably different as compared to Bernoulli (single) arrivals, which are frequently considered in the literature. This demonstrates the necessity of modeling the arrivals as batches

Partitioning and Invariance of AIMD Dynamics in Synchronised Networks.

In this paper we present new results on the dynamics of networks of AIMD flows. The results reveal an invariance and partitioning property that indicates potential for the design of soundly-based adaptive AIMD strategies