Source-linked AI summary
Periodic Event-Triggered Synchronization of Linear Multi-agent Systems with Communication Delays
Eloy Garcia, Yongcan Cao, David W. Casbeer
TL;DR
The paper addresses limited conditions affecting convergence, including communication delays. It proposes periodic event-triggered control with decentralized transmission decisions and reports strictly positive inter-event times and reduced communication.
Problem
The paper addresses multiple issues affecting convergence to common trajectories, including communication delays.
Method
The paper proposes periodic event-triggered techniques in which agents use decentralized control inputs and decide transmission instants from local information.
Results
Strictly positive inter-event times are designed, while the method reduces communication.
Takeaways & Limitations
The approach provides decentralized transmission timing and limits communication-related implementation demands.
Abstract
from arXiv · showhide
Multi-agent systems cooperation to achieve global goals is usually limited by sensing, actuation, and communication issues. At the local level, continuous measurement and actuation is only approximated by the use of digital mechanisms that measure and process information in order to compute and update new control input values at discrete time instants. Interaction with other agents or subsystems takes place, in general, through a digital communication channel with limited bandwidth where transmission of continuous-time signals is not possible. Additionally, communication channels may be subject to other imperfections such as time-varying delays. This paper considers the problem of consensus (or synchronization of state trajectories) of multi-agent systems that are described by general linear dynamics and are connected using undirected graphs. An event-triggered consensus protocol is proposed, where each agent implements discretized and decoupled models of the states of its neighbors. This approach not only avoids the need for continuous communication between agents but also provides a decentralized method for transmission of information in the presence of time-varying communication delays where each agent decides its own broadcasting time instants based only on local information. This method gives more flexibility for scheduling information broadcasting compared to periodic and sampled-data implementations. The use of discretized models by each agent allows for a periodic event-triggered strategy where continuous actuation and continuous measurement of the states are not necessary.
I. INTRODUCTION
The paper addresses consensus for general linear multi-agent systems under communication delays and constrained sensing, actuation, and computation. It proposes periodic event-triggered strategies using asynchronous, decentralized transmissions and neighbor-state models.
- Periodic approaches can require synchronized transmissions and conservative sampling periods for worst-case situations.
- The paper replaces periodic communication with asynchronous event-triggered broadcasting that does not require synchronization.
- The proposed setting includes general linear dynamics, communication delays, and limited sensing, actuation, and event-computation update rates.
- Periodic event-triggered control evaluates measurements, errors, and thresholds only at periodic instants, while sampling and actuation bounds are related to consensus disagreement.
- Each agent uses discretized and decoupled models of neighbors’ states to compute control inputs without continuous communication.
- The paper designs decentralized thresholds yielding practical consensus and strictly positive inter-event times, with lower bounds independent of particular system trajectories.
A. Graph Theory
The paper models agent interactions with undirected graphs and uses Laplacian properties to characterize connectivity and consensus. For connected graphs, the Laplacian has one zero eigenvalue and positive nonzero eigenvalues.
- An undirected graph represents agents as nodes connected by symmetric edges, with neighboring nodes joined in both directions.
- The adjacency matrix records whether node pairs are adjacent, while the degree matrix contains each node’s number of neighbors.
- A graph is connected when every pair of distinct nodes has a path through consecutive adjacent nodes.
- The graph Laplacian is L = D − A; for undirected graphs it is symmetric, positive semi-definite, and has zero row sums.
- For a connected graph, zero is the unique zero eigenvalue and all remaining Laplacian eigenvalues are positive.
- Consensus can be characterized through a weighted Laplacian condition involving ˆL = L ⊗ Q and agent state vectors χ_i.
B. Problem Statement
The paper formulates consensus for linear agents using periodic sampling, discretized neighbor models, and event-triggered broadcasts. Agents sample and compute locally at fixed intervals, while communication intervals vary and need not be periodic.
- Each agent has linear dynamics with state x_i and control input u_i, while F and c are design parameters.
- The discretized neighbor model uses sampling period h, with G = e^Ah, while agent states are sampled every h time units.
- Sampling is periodic, but each agent’s communication intervals are not constant and are determined by event-triggered rules.
- At each sampling instant, an agent updates its control input, computes a local state error, and decides whether to transmit its current state.
- Periodic event-triggered control performs measurements and error computations every h time units instead of continuously, and discrete-time models remove the need for continuous actuation.
- Agents implement models of themselves and their neighbors, updating neighbor models when a local event broadcasts the current state.
- Communication delays make the model state of an agent differ depending on whether it is viewed locally or by a neighboring agent.
- The decentralized control input uses the sampled local state and discretized neighbor models, so continuous or periodic neighbor-state access is unnecessary.
III. CONSENSUS WITH CONTINUOUS MEASUREMENTS
The continuous-measurement consensus analysis establishes stability conditions for controllable linear agents connected by an undirected graph. A Lyapunov argument shows that disagreement decreases until the agents’ state trajectories agree.
- The analysis assumes continuous actuation and requires agents to estimate the second-smallest Laplacian eigenvalue as the only global information.
- Theorem 1 states that the relevant symmetric matrix has n zero eigenvalues and all remaining eigenvalues negative, with its zero-eigenvalue subspace aligned with that of ˆL.
- The Lyapunov function V = x^T ˆLx has derivative x^T ¯Lx along the system trajectories.
- The derivative is negative during disagreement and equals zero only when the corresponding agent states are in total agreement.
- For general linear dynamics, consensus requires state differences to tend to zero rather than trajectories to converge to a constant value.
IV. DECENTRALIZED PERIODIC EVENT TRIGGERED CONSENSUS
The paper develops a decentralized periodic event-triggered consensus method using discrete-time neighbor models and local thresholds. Under stated controllability and graph assumptions, it bounds consensus errors and excludes Zeno behavior while requiring only sampled measurements and actuation updates.
- Each agent implements discrete-time models of neighbors and evaluates decentralized thresholds using only local information.Errors and communication decisions are computed at sampling instants, using discretization errors before they reset.
- Under controllability, connected-undirected graph, and coupling assumptions, the agents achieve a bounded consensus error for all times.The bound applies to the difference between any two states and is derived from the Lyapunov analysis.
- The analysis separates errors caused by event-based communication from errors caused by sampled actuation and discrete neighbor models.The first error reflects limited communication, while the second reflects reduced actuation update times.
- Inter-event times are greater than the sampling period h, so the event-triggered strategy does not exhibit Zeno behavior.The threshold parameter is selected so the triggering error cannot grow from zero to its threshold within one sampling period.
- The results apply to general linear dynamics, including single- and double-integrator systems, and bound state differences for any t ≥0.The event-triggered algorithm provides a bound beyond sampling instants, not only at sampled times.
V. DECENTRALIZED PERIODIC EVENT TRIGGERED CONSENSUS WITH COMMUNICATION DELAYS
The delayed-communication extension models each neighbor’s state from delayed measurements and designs local triggering rules despite errors unavailable to the receiving agent. With bounded delays and the stated graph and controllability assumptions, it preserves bounded consensus error and excludes Zeno behavior.
- Time-varying communication delays are represented through neighbor-specific model states updated when delayed measurements arrive.A measurement sent at tki is received after hpi(tki), and the receiving agent propagates it through its delayed model.
- Delayed models can differ across agents because communication updates occur at different times.The local and neighbor views coincide only after the delayed measurement has been received and the corresponding model updated.
- Because an agent cannot access or reset its neighbors’ delayed errors, the method estimates those effects using local errors and the worst-case delay.The current local error is propagated forward by the maximum delay and incorporated into the local threshold.
- Local event rules based on eii(tµ) achieve bounded consensus error when delays satisfy di < d under controllability and connected-undirected graph assumptions.The resulting bound applies to the difference between any two states.
- Inter-event times are bounded below by d, and the delayed strategy does not exhibit Zeno behavior.The theorem states this lower bound for every agent under the bounded-delay design.
VI. EXAMPLE
The example applies the delayed event-triggered protocol to four second-order agents with unstable linear dynamics. The agents synchronize both state dimensions while broadcasting less frequently near consensus, with inter-event times exceeding the delay bound.
- The simulation uses four second-order agents with unstable continuous-time dynamics and h = 0.002 seconds.The state matrix has complex eigenvalues λ(A) = 0.125 ± .5i, and the delay bound is d = 0.014 seconds.
- The agents synchronize their states in both dimensions despite the unstable dynamics and communication delays.The reported response shows synchronization in each of the two state dimensions.
- Agents transmit less frequently as they approach consensus, while every inter-event time remains greater than 0.014 seconds.The reported lower bound matches the specified delay bound.
VII. CONCLUSIONS
The paper studies synchronization of linear multi-agent state trajectories under limited sensing, actuation, communication, and time-varying delays. Its periodic event-triggered schemes decentralize both control and transmission timing while using discretized, decoupled models.
- The paper studies synchronization of state trajectories for linear multi-agent systems subject to limited sensing, actuation, communication, and time-varying delays.
- Event-triggered schemes provide decentralized control inputs and let each agent decide when to broadcast its current measurements using local information.
- Global knowledge of communication periods and transmission instants is unnecessary compared with sampled-data approaches.
- Discretized and decoupled neighbor models, combined with periodic event-triggering, provide a framework that limits actuation and sensing update rates.
- The figures depict four-agent synchronization and decentralized broadcasting instants, including transmissions during the first three seconds.
- The method reduces communication while giving each agent freedom to determine its own broadcasting instants.
APPENDIX
The appendix establishes consensus properties through Laplacian-based transformations and a block-diagonal decomposition. It characterizes consensus states as the zero-eigenvalue subspace and links zero Lyapunov value to equal agent states.
- Because Q > 0 and the graph is connected, V = 0 implies ξi = ξj for all agents.
- If consensus is achieved, the Lyapunov expression V = ¯ξT ˆL¯ξ equals zero.
- For an undirected connected graph, the Laplacian admits a similarity transformation to a diagonal form.
- The transformed matrix separates the consensus mode from blocks Ui = A + cλiBF for i = 2...N.
- The transformed matrix has n zero eigenvalues associated with consensus-state eigenvectors of the form ρ = 1N ⊗ς.