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Event-Triggered Communication and Control of Networked Systems for Multi-Agent Consensus
Cameron Nowzari, Eloy Garcia, Jorge Cortes
TL;DR
The paper addresses how networked agents can coordinate efficiently despite implementation constraints in sampling, communication, and control. It surveys event-triggered coordination methods and their technical challenges, highlighting consensus guarantees, non-Zeno behavior, and practical limitations under imperfect information.
Problem
The paper examines how to coordinate networked agents when periodic or continuous sampling, communication, and feedback are impractical or inefficient.
Method
The paper synthesizes motivations, algorithm-design methods, implementation challenges, and distributed-control applications for event-triggered multi-agent consensus.
Results
Event-triggered consensus methods can guarantee convergence and exclude Zeno behavior, while some newer dynamic triggering strategies also guarantee a positive MIET for each agent.
Takeaways & Limitations
The discussion frames aperiodic sampling, communication, and control as opportunities for efficient and adaptive network operation.
Takeaways & Limitations
Uniform quantization may restrict asymptotic convergence to a bounded region around the initial average, and some consensus designs require Laplacian spectral information.
Abstract
from arXiv · showhide
This article provides an introduction to event-triggered coordination for multi-agent average consensus. We provide a comprehensive account of the motivations behind the use of event-triggered strategies for consensus, the methods for algorithm synthesis, the technical challenges involved in establishing desirable properties of the resulting implementations, and their applications in distributed control. We pay special attention to the assumptions on the capabilities of the network agents and the resulting features of the algorithm execution, including the interconnection topology, the evaluation of triggers, and the role of imperfect information. The issues raised in our discussion transcend the specific consensus problem and are indeed characteristic of cooperative algorithms for networked systems that solve other coordination tasks. As our discussion progresses, we make these connections clear, highlighting general challenges and tools to address them widespread in the event-triggered control of networked systems.
1 Introduction
The article introduces event-triggered coordination for networked systems, emphasizing multi-agent consensus and the challenges created by aperiodic execution. It situates the topic within the maturation of event-triggered control and distributed coordination.
- The article focuses on event-triggered coordination of networked systems, particularly multi-agent consensus.
- Consensus requires groups of agents to agree on physical or virtual quantities of interest.
- Event-triggered control replaces periodic or continuous sampling and control with deliberate, opportunistic aperiodic actions to improve efficiency.
- The field has matured toward an independent area of systems and control with distinctive challenges for networked systems.
Why Event-Triggering?
Event-triggered control addresses the difficulty of controlling continuous-time systems with digital controllers when exact periodic sampling is impractical. The section motivates aperiodic sampling as a response to real implementation constraints.
- Event-triggered sampling and control address how digital controllers can control continuous-time systems.
- Periodic sampling is the standard digital-control approach, although real systems generally cannot acquire samples at an exact operating frequency.
- Aperiodic sampling has therefore remained a longstanding systems-and-controls problem involving sampled-data stability.
Aperiodic Sampling as an Opportunity
Event-triggering reframes aperiodic sampling from a disturbance into an opportunity for networked control. However, sampling faster does not universally improve closed-loop performance.
- New event-triggering methods treat aperiodic sampling as an opportunity rather than an inconvenience or disturbance.
- The approach challenges idealized control models that assume exact state information and continuously available feedback.
- Faster sampling can sometimes degrade the performance of a closed-loop feedback control system.
Why Now?
Networked cyber-physical systems make sensing, communication, computation, and actuation shared design resources. Event-triggering is motivated by the need to determine when actions occur while maintaining desired system properties efficiently.
- Wireless communication can make acquiring a sensor sample a resource that must be managed efficiently.
- IoT and other large-scale networks motivate event-triggered rather than periodic coordination.
- IoT devices combine physical-world sensors and actuators with processing, storage, and communication capabilities.
- Event-triggered methods determine when actions such as sensing and wireless communication should occur to efficiently maintain a desired property.
Technical Challenges Specific to Networked Systems
Applying event-triggered control to networked systems introduces distributed implementation and analysis challenges beyond those of either area alone.
- Distributed implementation: Fully distributed settings require new event-triggering methods beyond approaches where a single decision-maker dictates system actions.This extension is needed for applications such as the Internet of Things.
- Technical challenges: Event-triggered coordination automatically introduces asynchronism, making networked-system analysis more difficult.
- Technical challenges: Agents must use local triggering rules based on distributed information while ensuring desired system-level properties.Centralized setups generally make it easier to find such rules.
Why Consensus?
The article uses consensus as a canonical distributed-algorithm problem to examine event-triggered coordination, then connects the discussion to broader networking applications and general coordination challenges.
- Why consensus?: Consensus is chosen as a canonical example because it represents distributed algorithms while remaining broad enough to expose general coordination challenges.The article emphasizes that its methods and reasoning extend beyond consensus.
- Article approach: The article classifies consensus problems and event-triggered solutions into five categories, illustrating each with recently proposed algorithms.Most discussed results are proved in the Appendix.
- Broader scope: The discussion also considers networking areas that may benefit directly or indirectly from event-triggered coordination ideas.The article closes with an outlook beyond multi-agent consensus.
2 What is Event-Triggered Consensus?
Event-triggered consensus replaces continuous control updates with sampled, opportunistic updates designed to preserve convergence through a correctness certificate. The section develops centralized and decentralized consensus mechanisms, emphasizing implementability, Zeno behavior, and classification of problem-solution properties.
- 2.1 A primer on event-triggered control: Event-triggered control uses sampled states and holds the control input constant between event times while seeking convergence to the desired state.The sampled state replaces continuous state feedback, and updates occur only at discrete event times.
- 2.1.1 Deadlocks, Zeno behavior, and Minimum Inter-Event Time: Under mild regularity conditions, a positive minimum inter-event time guarantees both closed-loop stability and operation within finite hardware action frequencies.The MIET satisfies t_ℓ+1−t_ℓ ≥ τ_min = c > 0 and is more appropriate for physical implementation than merely ruling out Zeno behavior.
- 2.1 A primer on event-triggered control: A triggering function compares an error-dependent quantity with a threshold so the sampled state is updated while the Lyapunov certificate remains effective.The error is e = x̂ − x; the trigger is designed so the relevant error term stays below a threshold and the Lyapunov derivative supports stability.
- 2.2 Multi-agent average consensus: For connected undirected graphs, the continuous controller asymptotically drives all agent states to the average of their initial conditions.This controller provides the baseline consensus mechanism that the event-triggered designs seek to implement.
- 2.3 Centralized event-triggered control: Centralized event-triggered control achieves average consensus and rules out Zeno behavior through a positive MIET, but it requires simultaneous network-wide control updates.The centralized design uses a central decision maker and coordinated update times.
- 2.4 Decentralized event-triggered control: Decentralized triggering achieves average consensus for all non-Zeno trajectories using only local and neighboring state information, while Zeno behavior remains unresolved for all trajectories.The decentralized result is therefore weaker than the centralized result, which establishes a positive MIET.
- 2.5 Classification and broader scope: The article classifies event-triggered consensus problem-solution pairs by their assumptions and execution properties, then surveys extensions and applications beyond consensus.The classification addresses shortcomings of the canonical decentralized formulation and organizes the broader literature and networking applications.
3 Event-Triggered Consensus Algorithms
This section surveys event-triggered consensus algorithms by examining trigger responses, event detection, trigger dependence, and the information available to agents. It contrasts centralized and decentralized designs that use broadcast information to achieve average consensus under connected, undirected communication graphs.
- Classification: The surveyed classification distinguishes Dynamics, Topology, Trigger Response, Event Detection, and Trigger Dependence.These categories separate physical problem structure from communication and computation capabilities.
- Trigger Response: Event-triggered coordination extends event-triggering beyond control updates to determine when agents broadcast information to neighbors.This addresses the need to avoid continuously available neighbor-state information for control-trigger evaluation.
- Trigger Dependence: Decentralized triggers can use each agent’s state and neighbors’ last broadcast states rather than their exact current states.The resulting controller updates using broadcast states, including when a neighboring agent triggers an event.
- Trigger Response: A second decentralized coordination design also achieves multi-agent average consensus for all non-Zeno trajectories under the same graph setting.The two solutions share the same agent-capability assumptions but use different design approaches.
- Limitations: The decentralized designs remain technically incomplete because the cited results do not by themselves rule out Zeno behavior.The discussion notes that the original design adds an additional trigger to address this issue.
3.2 Event Detection
Event detection is a central design choice: continuous monitoring raises implementation and Zeno concerns, while sampled-data schemes impose a positive minimum inter-event time. Trigger designs trade exact convergence, convergence speed, event frequency, and implementability.
- Continuous event detection: Continuous event detection is technically impossible for cyber-physical systems, and existing algorithms may not guarantee avoidance of Zeno behavior.Zeno behavior means infinitely many events can occur within a finite time period.
- Sampled-data detection: Sampled-data triggering evaluates broadcast decisions every h seconds, matching fixed-frequency implementations and automatically ensuring inter-event times are at least h.Events can occur only at discrete times in {0, h, 2h, ...}.
- Sampled-data detection: Theorems 3.4 and 3.5 guarantee average consensus while ruling out Zeno executions through the positive MIET τ_min = h.The guarantees apply under their respective triggering conditions.
- Trigger conditions: Theorem 3.4 uses less conservative conditions requiring algebraic connectivity, whereas Theorem 3.5 uses stricter conditions that agents can compute more easily.Both conditions guarantee convergence.
- Asynchrony: Periodic solutions require every agent to use the same period h, imposing synchronous action and motivating asynchronous or aperiodic event detection.The article identifies only a few works investigating this issue.
- Trigger dependence: Time-dependent triggers can increase convergence speed at the cost of more events, while larger c0 reduces events but prevents exact convergence to the initial average.Parameters can be tuned to balance performance and efficiency.
- Trigger dependence: For c0 = 0, exact asymptotic consensus requires α < λ2(L); for c0 > 0, Zeno behavior is avoided for any α but exact asymptotic convergence is lost.The c0 = 0 condition requires global knowledge of λ2(L).
- Dynamic triggering: Dynamic triggering can solve the distributed consensus problem without global information and avoid Zeno behavior, but positive MIET requires the later dynamic strategy of Berneburg and Nowzari.Theorem 3.7 does not itself guarantee a positive MIET; Theorem 3.8 does.
3.4 Topology
Topology determines what information agents can use and which consensus guarantees are available. Extending decentralized event-triggered coordination from undirected to directed graphs requires weight balance and different Lyapunov and triggering analyses.
- Topology scope: The solutions discussed initially share single-integrator dynamics and undirected communication topologies, motivating extensions to more complex settings.The article then considers directed, time-varying, and state-dependent topologies.
- Directed graphs: In directed graphs, agents may send to in-neighbors and receive from out-neighbors, whose sets need not coincide.The resulting Laplacian is no longer symmetric.
- Directed graphs: For directed graphs, the original Lyapunov function cannot produce a locally computable trigger because its derivative depends on unavailable in-neighbor information.A different Lyapunov function is introduced to address this obstacle.
- Directed graphs: Weight balance makes the Lyapunov derivative analysis work and supports a decentralized trigger for which all non-Zeno trajectories achieve multi-agent average consensus.Theorem 3.11 applies to weight-balanced communication graphs.
- Directed graphs: The decentralized directed-graph trigger in Theorem 3.11 guarantees average consensus for all non-Zeno trajectories.Its event times are determined locally for each agent.
- Topology limitations: Exact convergence to the initial-state average is not established for general directed graphs; rooted spanning trees may yield agreement without determining its value.Some directed graphs may not permit agreement at all.
- Topology limitations: State-dependent interaction graphs add the challenge of preserving network connectivity while agents perform consensus.Time-varying topologies may also require modified algorithms.
3.5 Dynamics
The article extends event-triggered consensus from single-integrator agents to double-integrator and general linear dynamics. These designs achieve synchronization under graph and triggering conditions, but exact convergence and Zeno exclusion can require restrictive assumptions or parameter choices.
- 3.5.1 Double-integrator systems: Double-integrator consensus requires synchronization, meaning agents’ states converge together rather than necessarily reaching a stationary point.The continuous controller achieves this objective on connected, undirected graphs.
- 3.5.1 Double-integrator systems: Event-triggered implementation replaces continuous neighbor-state information with access to neighbors’ last broadcast states.The resulting controller uses locally generated events to update broadcasts and control actions.
- 3.5.1 Double-integrator systems: Time-dependent triggering with a first-order-hold controller drives all non-Zeno trajectories into a bounded neighborhood of consensus.The threshold design includes constants c0, c1 ≥ 0 with c0 + c1 > 0.
- 3.5.1 Double-integrator systems: Zeno behavior is excluded when c0 > 0 or 0 < α < λ3(Γ), but c0 > 0 sacrifices exact consensus and α requires global network knowledge.The parameters therefore balance convergence precision, implementability, and event-separation guarantees.
- 3.5.2 Linear systems: For homogeneous linear dynamics, matrix conditions involving Laplacian eigenvalues guarantee asymptotic synchronization along non-Zeno trajectories.Less restrictive controller designs can use the smallest nonzero Laplacian eigenvalue or an estimate of it.
- 3.5.2 Linear systems: Linear-system triggers can exclude Zeno behavior for α < λ∗, while adding a fixed trigger parameter may replace exact synchronization with convergence to a neighborhood.The latter trade-off is explicitly noted for a modified algorithm.
3.6 Uncertainty
The article examines uncertainty from disturbances and imperfect wireless communication. Quantization can change asymptotic average consensus into practical consensus, while robust event-separation remains a critical implementation concern.
- Disturbances and event separation: The discussion assumes no disturbances or uncertainties, leaving robustness of the algorithms as an important unresolved consideration.The article identifies uncertainty accommodation as a necessary direction for these methods.
- Disturbances and event separation: Arbitrarily small disturbances can destroy a positive minimum inter-event time even when the disturbance-free controller guarantees one.The article distinguishes local event-separation from stronger semi-global and global robustness notions.
- Quantization: Uniform quantization generally limits convergence to a bounded region around the initial average instead of guaranteeing asymptotic convergence to that average.The region’s radius depends on the quantization-step size.
- Wireless communication: Wireless networked systems introduce additional uncertainty sources, including quantization, communication delays, and packet drops.These effects require separate analysis when event-triggered algorithms are deployed over wireless links.
- Quantization: Logarithmic quantization still does not generally guarantee convergence to the initial average, although the difference between agent states remains bounded.When the initial average is zero, asymptotic convergence to that average is achieved.
4 Applications of Event-Triggered Consensus
Event-triggered consensus supports applications including formation control, leader-tracking, distributed estimation, clock synchronization, and distributed optimization. Across these settings, consensus propagates shared quantities while event-triggering reduces the need for continuous or periodic communication.
- Scope of applications: The article presents these applications as examples of event-triggered coordination beyond multi-agent consensus itself.Its application discussion is intended as an initial reference rather than an exhaustive survey.
- Formation control: Formation control can be reduced to average consensus on virtual states xi = pi − bi, allowing agents to agree on the formation center.Here bi encodes each agent’s desired relative displacement within the formation.
- Leader-tracking: In leader-tracking, followers use consensus to propagate a leader’s motion through the network when direct access to that motion is unavailable.The leader may move freely or be controlled by a user.
- Distributed estimation: Distributed state estimation uses consensus protocols to help agents communicate and agree on a common state estimate, traditionally with periodic communication.Event-triggered coordination addresses the communication pattern underlying this estimation process.
- Clock synchronization: Clock synchronization applies consensus to virtual clock variables so agents with unknown drifts and biases can synchronize their local clocks.Event-triggered methods reduce communication required while agents update controlled clock drift and bias.
- Distributed optimization: Distributed optimization commonly relies on agents sharing local states or information about the global solution while cooperatively optimizing separable objectives.The passage frames this communication pattern as an application area for event-triggered coordination.
5 A Look Beyond
The article extends event-triggered coordination beyond consensus to distributed stabilization and sensor-based control, while identifying shared challenges from distributed and partial information. It also highlights unresolved needs for performance guarantees and principled algorithm comparisons.
- Event-triggered coordination is presented as a broader networked-systems tutorial, with consensus serving as a specific case study.
- Distributed stabilization: Distributed stabilization replaces continuous neighbor-state access with event-triggered broadcasts while seeking asymptotic convergence.The setup includes interconnected subsystems, local state and input dynamics, coupling, and distributed triggering conditions.
- Extensions: Related extensions address nonlinear systems, network delays, packet dropouts, and decentralized sensors that sample different components of a system state.
- Imperfect information: Output-based coordination targets synchronization or asymptotic convergence when agents use output feedback rather than exact state feedback.
- Imperfect information: Distributed and partial information create technical concerns analogous to Zeno-execution issues in multi-agent consensus.
- Open challenges: The literature still lacks broad performance guarantees and comparisons that reveal transient trade-offs between communication burden and convergence speed.The article notes that several algorithms achieve asymptotic and non-Zeno convergence without clarifying which is better for a given implementation task.
6 Conclusions
The conclusion frames event-triggered coordination as increasingly relevant for large-scale interconnected cyber-physical systems, where periodic strategies do not scale well. It presents aperiodic sampling, communication, and control as an opportunity for efficient and adaptive network operation.
- Growing numbers of interconnected devices make distributed time-triggered coordination difficult to scale in large-scale networks.
- The article positions event-triggered coordination as a response to these scaling limitations by reconsidering periodic control through opportunistic schemes.
- A prior perspective identifies event-triggered sampling as a useful alternative when constant sampling rates are invalid in multi-rate and networked systems.
- The article argues that event-triggered coordination can support efficient and adaptive operation across contemporary network problems.
Appendix
The appendix records proof mechanisms for event-triggered consensus and related networked-system results, using Lyapunov arguments, invariance properties, graph structure, and spectral transformations. These proofs establish convergence and characterize closed-loop stability conditions.
- Lyapunov-function arguments are used to analyze continuous and event-triggered closed-loop dynamics.The proofs evaluate derivatives of Lyapunov functions and bound error-dependent terms using inequalities.
- Average consensus follows by combining convergence toward the agreement set with invariance of the sum of all agent states.
- Proofs use average-state deviations, graph symmetry or balance, and non-Zeno arguments to establish convergence results.
- Periodic event-trigger evaluation requires analyzing Lyapunov evolution between sampling times because continuous negativity is no longer guaranteed.
- A similarity transformation reduces networked closed-loop spectral analysis to the eigenvalues of A + cλ_jBF for Laplacian eigenvalues λ_j.