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Event-triggered leader-following tracking control for multivariable multi-agent systems
Yi Cheng, V. Ugrinovskii
TL;DR
The paper addresses event-triggered leader-following for multi-agent systems with general linear dynamics, where continuous communication and limited prior system classes constrain practicality. It proposes graph-dependent event-triggering protocols and a combinational-state generation procedure, guaranteeing bounded tracking errors and no Zeno behavior for directed and undirected follower graphs. The error bound can be tuned to a desired small value, with tighter accuracy potentially requiring more initial communication.
Problem
Continuous communication limits practicality, while prior multidimensional leader-following event-triggered methods were often restricted to simple dynamics; the paper targets general linear systems and non-Zeno operation.
Method
The paper designs distributed event-triggered controllers for pinned directed or undirected follower graphs and continuously generates each combinational state from information received at event times.
Results
Bounded leader-tracking errors and no Zeno behavior are guaranteed for both directed and undirected follower graphs, with tunable error bounds.
Takeaways & Limitations
The proposed protocols avoid continuous neighbor-state communication while retaining tunable tracking accuracy, though tighter accuracy can require more frequent initial communication.
Takeaways & Limitations
The analysis assumes at least one follower observes the leader and that the follower graph contains a spanning tree rooted at a pinned node.
Abstract
from arXiv · showhide
The paper considers event-triggered leader-follower tracking control for multi-agent systems with general linear dynamics. For both undirected and directed follower graphs, we propose event triggering rules which guarantee bounded tracking errors. With these rules, we also prove that the systems do not exhibit Zeno behavior, and the bounds on the tracking errors can be tuned to a desired small value. We also show that the combinational state required for the proposed event triggering conditions can be continuously generated from discrete communications between the neighboring agents occurring at event times. The efficacy of the proposed methods is discussed using a simulation example.
1 Introduction
The paper addresses the practicality limits of continuous communication in cooperative multi-agent control by developing event-triggered leader-following methods for general linear dynamics. Its protocols target bounded tracking, tunable accuracy, non-Zeno behavior, and communication without continuous neighbor transmission.
- Continuous communication between neighboring agents limits the practicality of many cooperative multi-agent control techniques.
- Event-triggered control updates actions when a continuously monitored triggering condition is violated, avoiding unnecessary periodic updates after sufficient accuracy is reached.
- General multidimensional event-triggered leader-following problems remain challenging, with prior work often restricted to single- or double-integrator dynamics.
- The proposed method covers general linear follower dynamics, marginally stable or unstable leaders, and both directed and undirected follower interconnections.
- The control protocols guarantee bounded leader-tracking errors with tunable bounds and exclude Zeno behavior.
- Each agent continuously generates its combinational state internally from neighbor information received at communication events, avoiding continuous neighbor communication and additional sampling.
2 Problem formulation and preliminaries
The paper formulates leader-following over a pinned communication graph for general linear agents and seeks distributed event-triggered control with bounded tracking errors and no Zeno behavior. It establishes graph conditions and an event-based control implementation using continuously generated combinational states.
- 2.1 Communication graph: The communication graph contains one leader, followers indexed 1 through N, and directed or undirected communication links among followers.
- 2.1 Communication graph: Follower edge (j, i) means follower i obtains information from neighbor j; A defines adjacency, D contains in-degrees, and L = D − A is the follower Laplacian.
- 2.1 Communication graph: At least one follower observes the leader through a pinned edge, so the pinning matrix G is nonzero.
- 2.1 Communication graph: A spanning tree rooted at a pinned node makes −(L + G) Hurwitz and yields a positive diagonal Θ satisfying H = Θ^-1(L + G) + (L + G)′Θ^-1 > 0.
- 2.2 Problem formulation: Follower and leader agents have n-dimensional states and linear dynamics, while the follower system uses control inputs and a combinational state z_i(t).
- 2.2 Problem formulation: The follower dynamics matrix A is not required to be Hurwitz; the leader may be marginally stable or unstable.
- 2.2 Problem formulation: Each follower updates its control input at event times using samples of z_i, holding the resulting measurement signal constant between updates.
- 2.2 Problem formulation: The design problem is to achieve leader-following errors bounded by a prescribed positive Δ while ensuring only finitely many communication events over every finite interval.
3 Event-triggered leader-following control under a directed graph G
For directed follower graphs, the paper designs event-triggered controllers for general linear leader-following systems. The design guarantees bounded tracking errors and excludes Zeno behavior, with accuracy adjustable through parameters.
- Controller design: The directed-graph design uses Riccati-based conditions and a state-feedback control law to establish leader-following tracking guarantees.The controller requires positive definite matrices satisfying the stated Riccati inequality.
- Tracking guarantee: The resulting tracking errors remain within a bound Δ = Nγ, while the parameter γ can be selected to tune the bound.The design procedure selects γ based on the desired upper bound Δ.
- Implementation: The triggering condition monitors the combinational state, whose continuous generation is addressed by a later computational algorithm.This avoids requiring continuous neighbor communication for implementing the condition.
- Zeno exclusion: The proposed event-triggering rule excludes Zeno behavior by establishing a uniform positive lower bound on inter-event intervals.The proof proceeds after bounding the combinational state and derives a lower bound independent of the event index.
- Accuracy and communication: Reducing γ improves tracking precision but may increase communication frequency during the initial interval.After a sufficiently large time, the minimum inter-event time becomes independent of γ.
4 Event-triggered leader-following control under an undirected graph G
For undirected follower graphs, the paper derives an alternative event-triggered design that exploits Laplacian symmetry. It provides bounded tracking, excludes Zeno behavior, and relates performance to graph and controller parameters.
- Undirected design: The undirected-graph treatment uses the symmetry of L + G to derive a control and triggering design separately from the directed-graph case.The undirected problem is also covered by the directed result, but symmetry permits an alternative derivation.
- Parameter selection: The undirected design’s parameters can be selected through a process similar to the directed case, including Riccati or LMI-based computation of Y.The same accuracy–communication observation applies to this case.
- Tracking guarantee: Theorem 2 guarantees leader-following tracking with Δ = Nγ(ρλmin(Y)λmin(F))^-1 under the stated control law and communication conditions.The controller is K = −R^-1B′Y, with parameters constrained as specified in the theorem.
- Zeno exclusion: The undirected-graph event-triggering rule also rules out Zeno behavior.The proof uses a transformed error system and a Lyapunov function expressed through the combinational state.
- Graph information: The Riccati condition depends on the smallest eigenvalue of L + G, contrasting with the second-smallest Laplacian eigenvalue used in the cited alternative design.The smallest eigenvalue can be estimated in a decentralized manner when the topology is not fully known at each node.
5 Generation of the combinational state
The paper provides an algorithm that continuously generates each agent’s combinational state from neighbor information received only at event times. This removes the need for continuous inter-agent monitoring and communication.
- Continuous state generation: Each node generates its combinational state continuously using discrete neighbor communications received when neighboring events occur.The method integrates received information within the controllers rather than requiring continuous transmission.
- Communication pattern: Communication is one-directional at event times, even when the follower graph is undirected.Agents receive information from neighbors that trigger events and send their own information to designated recipients when they trigger.
- Directed implementation: For directed graphs, each agent receives combinational-state information according to the graph’s incoming-neighbor relationships.The same continuous computation procedure applies, but message reception follows the directed topology.
- Algorithm: The algorithm updates local combinational-state calculations and measurement errors between events, then broadcasts the state when its triggering condition is met.The initialization and event loop specify local clocks, counters, received states, state computation, and control updates.
- Illustration: The communication timelines in Figures 1 and 2 illustrate event exchanges for undirected and directed follower networks, respectively.Each figure pairs the follower graph with its corresponding communication events.
6 Example
The example evaluates event-triggered leader-following control on twenty pendulums under directed and undirected communication graphs. Four simulations compare the proposed designs with each other and with a prior event-based strategy.
- System and graphs: The example models twenty identical pendulums, each receiving a control input, together with an identical leader pendulum.The pendulum dynamics are linearized, with control torque realized using a DC motor.
- System and graphs: Both directed and undirected follower graphs are tested, with agents 1, 8, 12, and 15 measuring the leader’s state.In the directed graph, follower i receives from follower i−1 only; in the undirected graph, it can receive from both i−1 and i+1.
- Simulation design: Four simulations compare the proposed directed-graph and undirected-graph controllers with a prior event-based control strategy.The prior strategy is selected because it avoids continuous follower-to-follower communication, enabling a fair comparison.
- Simulation design: The first three simulations target a predicted tracking-error bound of ∆≤ 0.05 using common Q and adjusted R matrices.Simulation 4 uses the same Q matrix and compares against Theorem 3 of with specified design parameters.
- Results: Simulation 2 triggered events less frequently on average than Simulation 1, although its minimum inter-event intervals were smaller.The result suggests that the undirected design based on Theorem 2 may use communication resources more efficiently in this example.
- Results: Compared with Simulation 2, Simulation 3 produced more communication events and smaller minimum inter-event intervals under the same undirected graph.The authors relate this difference to the use of the symmetry property of L + G in Theorem 2.
- Results: Compared with Theorem 3 of, the proposed method produced smaller minimum intervals but fewer total events on the same directed graph.When the comparator was tuned to an almost identical gain, it produced E[0,20] = 5553 events versus E[0,20] = 2857 for Simulation 4.
7 Conclusions
The paper establishes event-triggered leader-following control for general linear multi-agent dynamics over directed and undirected graphs. The proposed rules bound tracking errors, exclude Zeno behavior, and generate the required combinational state without continuous neighbor-state monitoring.
- Conclusions: The proposed event-triggered schemes guarantee bounded leader-tracking errors for multi-agent systems with directed and undirected follower graphs.The guaranteed upper bound can be tuned through the triggering-condition parameters.
- Conclusions: The tracking-error upper bound can be tuned to a desired small value at the expense of more frequent initial communications.The proposed conditions retain this tuning property for both follower-graph types.
- Conclusions: The proposed triggering rules do not produce Zeno behavior, even under tight tracking-accuracy requirements.After a sufficiently large time, tighter accuracy requirements do not affect the inter-event intervals.
- Conclusions: Nodes can continuously generate the combinational state from information received during discrete neighbor communication events.This avoids continuously monitoring neighboring states while implementing the event-triggering schemes.
- Conclusions: A simulation example demonstrates the efficacy of the proposed combinational-state generation algorithm.The paper identifies robustness analysis as future work.