7,527 research outputs found
Tight Bounds for Asymptotic and Approximate Consensus
We study the performance of asymptotic and approximate consensus algorithms
under harsh environmental conditions. The asymptotic consensus problem requires
a set of agents to repeatedly set their outputs such that the outputs converge
to a common value within the convex hull of initial values. This problem, and
the related approximate consensus problem, are fundamental building blocks in
distributed systems where exact consensus among agents is not required or
possible, e.g., man-made distributed control systems, and have applications in
the analysis of natural distributed systems, such as flocking and opinion
dynamics. We prove tight lower bounds on the contraction rates of asymptotic
consensus algorithms in dynamic networks, from which we deduce bounds on the
time complexity of approximate consensus algorithms. In particular, the
obtained bounds show optimality of asymptotic and approximate consensus
algorithms presented in [Charron-Bost et al., ICALP'16] for certain dynamic
networks, including the weakest dynamic network model in which asymptotic and
approximate consensus are solvable. As a corollary we also obtain
asymptotically tight bounds for asymptotic consensus in the classical
asynchronous model with crashes.
Central to our lower bound proofs is an extended notion of valency, the set
of reachable limits of an asymptotic consensus algorithm starting from a given
configuration. We further relate topological properties of valencies to the
solvability of exact consensus, shedding some light on the relation of these
three fundamental problems in dynamic networks
Distributed Optimization: Convergence Conditions from a Dynamical System Perspective
This paper explores the fundamental properties of distributed minimization of
a sum of functions with each function only known to one node, and a
pre-specified level of node knowledge and computational capacity. We define the
optimization information each node receives from its objective function, the
neighboring information each node receives from its neighbors, and the
computational capacity each node can take advantage of in controlling its
state. It is proven that there exist a neighboring information way and a
control law that guarantee global optimal consensus if and only if the solution
sets of the local objective functions admit a nonempty intersection set for
fixed strongly connected graphs. Then we show that for any tolerated error, we
can find a control law that guarantees global optimal consensus within this
error for fixed, bidirectional, and connected graphs under mild conditions. For
time-varying graphs, we show that optimal consensus can always be achieved as
long as the graph is uniformly jointly strongly connected and the nonempty
intersection condition holds. The results illustrate that nonempty intersection
for the local optimal solution sets is a critical condition for successful
distributed optimization for a large class of algorithms
Distributed Frequency Control in Power Grids Under Limited Communication
In this paper, we analyze the impact of communication failures on the
performance of optimal distributed frequency control. We consider a
consensus-based control scheme, and show that it does not converge to the
optimal solution when the communication network is disconnected. We propose a
new control scheme that uses the dynamics of power grid to replicate the
information not received from the communication network, and prove that it
achieves the optimal solution under any single communication link failure. In
addition, we show that this control improves cost under multiple communication
link failures. Next, we analyze the impact of discrete-time communication on
the performance of distributed frequency control. In particular, we will show
that the convergence time increases as the time interval between two messages
increases. We propose a new algorithm that uses the dynamics of the power grid,
and show through simulation that it improves the convergence time of the
control scheme significantly.Comment: 8 pages, 7 figure
Distributed Decision Through Self-Synchronizing Sensor Networks in the Presence of Propagation Delays and Asymmetric Channels
In this paper we propose and analyze a distributed algorithm for achieving
globally optimal decisions, either estimation or detection, through a
self-synchronization mechanism among linearly coupled integrators initialized
with local measurements. We model the interaction among the nodes as a directed
graph with weights (possibly) dependent on the radio channels and we pose
special attention to the effect of the propagation delay occurring in the
exchange of data among sensors, as a function of the network geometry. We
derive necessary and sufficient conditions for the proposed system to reach a
consensus on globally optimal decision statistics. One of the major results
proved in this work is that a consensus is reached with exponential convergence
speed for any bounded delay condition if and only if the directed graph is
quasi-strongly connected. We provide a closed form expression for the global
consensus, showing that the effect of delays is, in general, the introduction
of a bias in the final decision. Finally, we exploit our closed form expression
to devise a double-step consensus mechanism able to provide an unbiased
estimate with minimum extra complexity, without the need to know or estimate
the channel parameters.Comment: To be published on IEEE Transactions on Signal Processin
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