Source-linked AI summary
Distributed event-triggered coordination for average consensus on weight-balanced digraphs
Cameron Nowzari, Jorge Cortes
TL;DR
The paper addresses average consensus without continuous neighbor-state access or prior global-parameter knowledge. It develops distributed event-triggered communication and control, proving no Zeno behavior and exponential convergence on weight-balanced strongly connected digraphs, with asymptotic convergence under specified switching-topology conditions.
Problem
Existing average-consensus implementations often require continuous or periodic neighbor information, motivating asynchronous event-triggered coordination to reduce implementation demands.
Method
A Lyapunov-based distributed event-triggered design uses locally available information to determine communication and control updates while accounting for broadcast-induced discontinuities.
Results
The executions exclude Zeno behavior and converge exponentially to the initial-state average on weight-balanced strongly connected digraphs; under stated switching-connectivity conditions, convergence is asymptotic.
Takeaways & Limitations
The strategy avoids continuous neighbor-state access and prior knowledge of global network parameters, and the paper also characterizes switching-topology convergence.
Takeaways & Limitations
The authors establish that the additional trigger is sufficient to prevent Zeno behavior, but whether it is necessary remains open.
Abstract
from arXiv · showhide
This paper proposes a novel distributed event-triggered algorithmic solution to the multi-agent average consensus problem for networks whose communication topology is described by weight-balanced, strongly connected digraphs. The proposed event-triggered communication and control strategy does not rely on individual agents having continuous or periodic access to information about the state of their neighbors. In addition, it does not require the agents to have a priori knowledge of any global parameter. We show that, under the proposed law, events cannot be triggered an infinite number of times in any finite period (i.e., no Zeno behavior), and that the resulting network executions provably converge to the average of the initial agents' states exponentially fast. We also provide weaker conditions on connectivity under which convergence is guaranteed when the communication topology is switching. Simulations illustrate our results.
1 Introduction
The paper targets average consensus without continuous or periodic neighbor-state access by designing a distributed event-triggered strategy for weight-balanced digraphs. It establishes convergence and non-Zeno execution while extending the treatment to switching topologies.
- Motivation: The problem is motivated by the inefficiency of continuous or periodic neighbor-information requirements in energy, bandwidth, congestion, and processor usage.The proposed strategy aims to preserve average consensus in asynchronous executions.
- Contributions: The paper designs event-triggered broadcasting and controller updates for average consensus over weight-balanced digraphs.The manuscript extends prior work from undirected graphs to weight-balanced directed graphs.
- Contributions: The strategy avoids continuous neighbor-state access and requires no a priori knowledge of global network parameters.Agents evaluate triggers using locally available information.
- Contributions: The analysis proves finite event counts on every finite time interval and exponential convergence to the initial average on weight-balanced, strongly connected digraphs.The paper also provides a lower bound on the exponential convergence rate.
- Contributions: For switching topologies, the paper characterizes convergence under weight balance and joint strong connectivity.A periodic implementation is also proposed, with a sampling-period condition guaranteeing correctness.
2 Preliminaries
The preliminaries define weighted digraphs, connectivity, degree balance, and the Laplacian framework used to analyze consensus. Weight balance gives the structural properties needed for average preservation and spectral arguments.
- Graph-theoretic preliminaries: A weighted digraph consists of vertices, directed edges, and a nonnegative weighted adjacency matrix.Positive adjacency entries correspond to edges, while zero entries represent absent edges.
- Graph-theoretic preliminaries: Strong connectivity means that a directed path exists from every vertex to every other vertex.The definition is stated in terms of paths over the directed edges.
- Graph-theoretic preliminaries: A digraph is weight-balanced when its out-degree and in-degree matrices are equal.The weighted Laplacian is defined as L = D_out − W.
- Laplacian properties: For a strongly connected weight-balanced digraph, the symmetric Laplacian has a simple zero eigenvalue and the remaining eigenvalues are ordered positive values.This spectral structure supports later convergence analysis.
3 Problem statement
The problem statement considers single-integrator agents communicating over a weight-balanced, strongly connected digraph and seeks opportunistic broadcasts that preserve the initial average while achieving agreement. The event-triggered implementation replaces continuous neighbor information and controller updates with locally timed actions.
- Network model: The network contains N agents with single-integrator states and a weight-balanced, strongly connected communication digraph.Each agent receives information from designated neighbors and maintains its own state continuously.
- Motivation: The continuous consensus controller requires agents to continuously access neighbor states and continuously update their control laws.The paper seeks an opportunistic implementation that relaxes both requirements.
- Event-triggered implementation: In the event-triggered implementation, neighbors receive an agent’s state only when that agent broadcasts, then update their control laws using the latest broadcast state.Each agent has continuous access to its own state, while neighbor information is sampled through broadcasts.
- Average preservation: Using the last broadcast state in the controller preserves the average of the agents’ initial states throughout the network evolution.This design choice distinguishes the communicated state from the agent’s continuously available current state.
- Consensus objective: The objective is agreement at the average of the agents’ initial states, with average preservation reducing the remaining requirement to state agreement.The average is conserved by the continuous dynamics under weight balance.
4 Distributed trigger design
The paper designs distributed triggers for broadcasting states and updating controls using locally available information, while accounting for discontinuities caused by neighbor broadcasts. The resulting law combines multiple triggers to support event-driven average-consensus implementation.
- Trigger synthesis: The design uses network-disagreement evolution and a candidate Lyapunov function to synthesize distributed broadcast and controller-update triggers.Agents evaluate the triggers using locally available information rather than continuous neighbor-state access.
- Trigger synthesis: The error e_i(t)=bxi(t)−xi(t) measures the difference between agent i’s last broadcast state and its current state.This error enters the Lyapunov-based analysis and trigger construction.
- Trigger objective: The average is preserved by the control law, so trigger design only needs to enforce eventual agreement among agents.Agreement at the average corresponds to the consensus state characterized in the section.
- Trigger objective: The triggering function’s parameter σ_i controls a trade-off: values closer to 1 enable less frequent triggering but contribute less to Lyapunov decrease.Each agent selects σ_i in (0,1) as part of the trigger design.
- Event implementation: Neighbor broadcasts can discontinuously change a triggering function, so the event rule includes a condition preventing agents from missing newly enabled events.An additional trigger also avoids requiring continuous broadcasts when the last broadcast state already agrees with received neighbor states.
- Event implementation: Triggers (10)–(12) form the event-triggered communication and control law: broadcasting agents send current states and update controls, while in-neighbors update controls upon receipt.The law is formally summarized in Table 1 and differs from designs assuming exact neighbor information is continuously available.
5 Analysis of the event-triggered communication and control law
The analysis proves that the event-triggered law avoids Zeno behavior and achieves exponential convergence to average consensus on weight-balanced strongly connected digraphs. It also extends convergence guarantees to switching topologies under a uniformly bounded joint-connectivity condition.
- No Zeno behavior: The proposed executions do not require agents to communicate an infinite number of times within any finite time period.This is the paper’s no-Zeno guarantee for weight-balanced strongly connected digraphs.
- No Zeno behavior: The additional trigger (12) supports the no-Zeno proof by organizing potentially synchronized broadcasts and preventing infinitely many finite-time communications with finitely many agents.The proof repeatedly expands the synchronized set; an infinite expansion is impossible because the network has finitely many agents.
- Exponential convergence: Theorem 5.3 establishes exponential convergence of all agents to the average of their initial states under the event-triggered law.The proof bounds the Lyapunov function by an exponentially decaying expression.
- Exponential convergence: The exponential estimate uses σ_max=max_i σ_i to bound the Lyapunov derivative and obtain decay toward zero.The resulting decay of the disagreement function yields convergence of the network trajectories to the average state.
- Switching topologies: A topology-independent Lyapunov function serves as a common Lyapunov function for time-varying communication graphs.This enables the switching-topology result under additional connectivity and neighbor-awareness assumptions.
- Switching topologies: For switching digraphs, convergence is guaranteed when the union of communication graphs is strongly connected over infinitely many contiguous, nonempty, uniformly bounded intervals.The proposition also assumes agents know their current neighbors.
6 Periodically checked event-triggered coordination
The periodic event-triggered law evaluates broadcast conditions only at fixed sampling times, making implementation compatible with periodic real-time operation and automatically excluding Zeno behavior. Under a suitable sampling-period condition, agents exponentially converge to the initial-state average over weight-balanced strongly connected digraphs.
- Periodic strategy: Agents evaluate broadcast triggers only at times separated by a fixed sampling period h, rather than continuously.This design matches implementations whose components operate at a specified frequency.
- Periodic strategy: Because trigger evaluations are separated by h, inter-event times are lower bounded by h and the additional Zeno-prevention trigger is unnecessary.
- Periodic strategy: At each sampling time, an agent broadcasts when its trigger condition is positive, or when it is zero while the auxiliary quantity φ_i(t) is nonzero.Receiving new neighbor information also causes the agent to update its control signal.
- Convergence guarantee: A sampling period h satisfying the theorem’s condition guarantees correctness of the periodic design.The theorem specifies the admissible condition on h.
- Convergence guarantee: All agents exponentially converge to the average of the initial states under the periodic law on a weight-balanced strongly connected digraph.
7 Simulations
Simulations examine the proposed event-triggered laws on undirected and weight-balanced directed networks, and compare periodic triggering with periodic Laplacian consensus. They also show that increasing the triggering parameter reduces communication but slows convergence.
- Undirected and directed networks: Figure 1 compares the proposed event-triggered law with Garcia et al. (2013) on a five-agent undirected network.The comparison plots Lyapunov-function evolution and the total event count, using σ_i = 0.999 and a = 0.2, respectively.
- Undirected and directed networks: Figure 2 illustrates the proposed event-triggered law on a five-agent weight-balanced digraph with the stated edge weights and initial condition.It does not compare against Garcia et al. (2013), whose algorithm is designed only for undirected graphs.
- Periodic comparison: Figure 3 compares periodic event-triggered coordination with periodic Laplacian consensus at timestep h = 0.1.Both Lyapunov-function evolution and total event count are plotted, with varying σ for the event-triggered law.
- Periodic comparison: Higher σ results in less communication but slower convergence than smaller σ in the periodic event-triggered executions.
8 Conclusions
The paper proposes distributed event-triggered coordination for average consensus over weight-balanced directed topologies without continuous neighbor-state access or prior global-parameter knowledge. It establishes exclusion of Zeno behavior and exponential convergence, including weaker connectivity conditions for switching topologies.
- Contributions: The proposed strategies address multi-agent average consensus over weight-balanced directed communication topologies.
- Contributions: Agents need not continuously access neighbors’ states or know global network parameters a priori to execute the algorithm.
- Convergence guarantees: The algorithms exclude Zeno behavior and establish exponential convergence to agreement on the initial average under identified conditions.
- Convergence guarantees: The paper also provides a lower bound on convergence rate and characterizes convergence under switching topologies with weaker connectivity assumptions.