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Distributed Event-Triggered Control for Global Consensus of Multi-Agent Systems with Input Saturation
Xinlei Yi, Tao Yang, Junfeng Wu, Karl H. Johansson
TL;DR
The paper addresses global consensus for multi-agent systems with input saturation over digraphs, a setting complicated by nonlinear closed-loop dynamics and event-triggered control requirements. It uses Lyapunov methods to show that the distributed consensus protocol achieves consensus under a directed spanning tree, then proposes a distributed event-triggered law. Consensus remains achieved and Zeno behavior is excluded.
Problem
Prior event-triggered consensus work with input saturation considered an undirected graph and did not address excluding Zeno behavior, despite the importance of actuator bounds.
Source: §1 · p. 2Method
Results
The saturated protocol achieves global consensus, with saturation no longer playing a role after finite time; the event-triggered law also achieves consensus without Zeno behavior.
Source: §3.1 · p. 4, §6 · pp. 8–9Takeaways & Limitations
The distributed event-triggered law reduces the need for communication and system updates while retaining consensus and excluding Zeno behavior.
Source: §6 · pp. 8–9The analysis focuses on agents with the same input saturation level, though the authors state it can be extended to differing levels.
Source: §2.2 · p. 3
Abstract
from arXiv · showhide
We consider the global consensus problem for multi-agent systems with input saturation over digraphs. Under a mild connectivity condition that the underlying digraph has a directed spanning tree, we use Lyapunov methods to show that the widely used distributed consensus protocol, which solves the consensus problem for the case without input saturation constraints, also solves the global consensus problem for the case with input saturation constraints. In order to reduce the overall need of communication and system updates, we then propose a distributed event-triggered control law. Global consensus is still realized and Zeno behavior is excluded. Numerical simulations are provided to illustrate the effectiveness of the theoretical results.
1 INTRODUCTION
The paper addresses global consensus with input saturation over digraphs and proposes a distributed event-triggered law to reduce communication and system updates. It targets the challenge of combining actuator bounds with event-triggered control while excluding Zeno behavior.
1 INTRODUCTION
- Consensus protocols for directed graphs with a spanning tree are established background for the paper’s connectivity setting.The introduction notes that agents update from their own and neighbors’ states to converge to a common value.Source: §1 · p. 1
- Input, output, communication-channel, and sensor constraints create closed-loop nonlinearities that affect agent behavior and convergence.The paper emphasizes actuator bounds as important to study in real applications.Source: §1 · pp. 1–2
- Continuous state exchange in widely used consensus protocols may be impractical, motivating event-triggered control with piecewise-constant inputs between triggering times.Designing the control law and trigger threshold, excluding Zeno behavior, and keeping the controller distributed are identified as challenges.Source: §1 · pp. 1–2
- Prior work on event-triggered consensus with input saturation used an undirected graph and did not consider excluding Zeno behavior.The paper identifies this combination of topology, saturation, and Zeno exclusion as an open challenge.Source: §1 · p. 2
- A directed spanning tree suffices for global consensus under input saturation using the widely used distributed protocol, and the proposed event-triggered law also achieves consensus.The event-triggered law is distributed, requires no a priori global network parameters, and is free from Zeno behavior.Source: §1 · p. 2
2 PRELIMINARIES
The preliminaries define the digraph setting and connectivity conditions, then formulate single-integrator agents with saturated inputs. They also establish graph and saturation properties used in the consensus analysis.
2.1 Algebraic Graph Theory
- A directed spanning tree has a root with a directed path to every other agent, and its Laplacian has rank n − 1 with a simple zero eigenvalue.For an irreducible Laplacian, the passage also gives a positive left eigenvector associated with zero.Source: §2.1 · p. 3
2.2 Multi-Agent Systems with Input Saturation
- The agents are modeled as single integrators, and the analysis focuses on a common positive saturation level while noting it can extend to differing levels.The saturation interval is [−h, h].Source: §2.2 · p. 3
- The paper states that consensus is achieved despite input saturation under the widely used distributed consensus protocol.This is the preliminary result for the saturated multi-agent system and protocol introduced in this section.Source: §2.2 · p. 3
- With a graph containing a directed spanning tree, equal saturated negative-Laplacian outputs imply equal agent states.The stated equivalence is that y1 = · · · = yn if and only if x1 = · · · = xn.Source: §2.2 · p. 3
3 GLOBAL CONSENSUS FOR MULTI-AGENT SYSTEMS WITH INPUT SATURATION
The widely used distributed consensus protocol achieves global consensus despite input saturation on strongly connected digraphs and digraphs with a directed spanning tree. The paper’s Lyapunov analysis also provides a basis for its later event-triggered control design.
3.1 Strongly Connected Digraphs
- For strongly connected digraphs, the distributed consensus protocol achieves global consensus despite input saturation.After a finite time, the saturation function no longer affects the dynamics.Source: §3.1 · pp. 3–4
3.2 Digraphs Having A Spanning Tree
- For digraphs with a directed spanning tree, the saturated multi-agent system achieves global consensus under the distributed consensus protocol.The proof proceeds from the final strongly connected component through preceding components, which follow the same consensus value as leaders.Source: §3.2 · p. 5
- The spanning-tree analysis uses a Lyapunov function different from prior work, which the authors say facilitates event-triggered control design.The cited comparison is to Li et al. (2011).Source: §3.2 · p. 5
4 EVENT-TRIGGERED CONTROL FOR MULTIAGENT SYSTEMS WITH INPUT SATURATION
The paper equips the saturated distributed consensus protocol with event-triggered communication and control updates. For strongly connected digraphs and digraphs with a directed spanning tree, the resulting law achieves global consensus without Zeno behavior.
4 EVENT-TRIGGERED CONTROL FOR MULTIAGENT SYSTEMS WITH INPUT SATURATION
- The event-triggered protocol updates control signals only at triggering times, reducing continuous inter-agent communication and actuator updates.The law is described as distributed because it needs only each agent’s own state information, without prior knowledge of global parameters.Source: §4 · pp. 5–6, §4.1 · p. 8
4.1 Strongly Connected Digraphs
- For strongly connected digraphs, the event-triggered law achieves global consensus and excludes Zeno behavior.The authors also show that the saturation function ceases to affect the dynamics after a finite time.Source: §4.1 · pp. 6, 8
4.2 Digraphs Having A Spanning Tree
- For digraphs with a directed spanning tree, the event-triggered protocol achieves global consensus without Zeno behavior.Source: §4.2 · p. 8
5 SIMULATIONS
A seven-agent numerical example illustrates consensus under saturation with both the distributed and event-triggered protocols.
5 SIMULATIONS
- The example uses saturation parameter h = 10 and event-trigger parameters α_i = 10 and β_i = 1.Source: §5 · p. 8
- In a seven-agent example with two strongly connected components, both protocols achieve global consensus while saturated inputs remain within the saturation level.The initial agent values are selected randomly within [−10, 10].Source: §5 · p. 8
- With the event-triggered protocol, agents broadcast their states to neighbors only at triggering times, avoiding continuous broadcasting in the simulation.Source: §5 · p. 8
6 CONCLUSION
The paper concludes that a directed spanning tree suffices for global consensus with input saturation, and that its distributed event-triggered law reduces communication and system updates while excluding Zeno behavior.
6 CONCLUSION
- The authors use a novel Lyapunov function to show that saturation ceases to affect the system after finite time and global consensus is achieved.The setting is multi-agent systems with input saturation and directed communication graphs containing a directed spanning tree.Source: §6 · p. 8
- The distributed event-triggered control law reduces the overall need for communication and system updates while preserving global consensus and excluding Zeno behavior.The conclusion states that the event-triggered law was shown to be free of Zeno behavior.Source: §6 · pp. 8–9
- The simulations depict the communication topology, state evolution with the distributed protocol, and state evolution, saturated inputs, and triggering times with event-triggered control.The figures identify the plotted system and protocol for each simulation.Source: Figure 3 · p. 9, Figure 2 · p. 9, Figure 1 · p. 9
A Proof of Theorem 2
The proof establishes global consensus under input saturation by treating strongly connected components and using Lyapunov arguments. It considers the case of two components explicitly, with the authors stating that more components can be handled similarly.
A Proof of Theorem 2
- For the proof, the authors take M = 2 and state that the case M > 2 can be treated similarly.The proof first treats the second strongly connected component, which does not depend on agents in the first component, as a strongly connected directed graph.Source: A Proof of Theorem 2 · p. 11
- The proof then considers the first strongly connected component using a Lyapunov function and its derivative along the distributed consensus protocol.The text defines a related Lyapunov candidate and establishes nonnegativity before analyzing its derivative.Source: A Proof of Theorem 2 · p. 11
- The argument introduces an auxiliary exponentially decaying state, constructs a Lyapunov candidate, and invokes LaSalle’s Invariance Principle.The derivative calculation is for the trajectories under the distributed consensus control protocol.Source: A Proof of Theorem 2 · pp. 11–12
- After finite time, saturation no longer affects the system, and global consensus is achieved.The result follows the preceding Lyapunov and invariance arguments for the directed spanning-tree case.Source: A Proof of Theorem 2 · p. 12
B Proof of Theorem 4
The proof extends the Lyapunov argument to event-triggered control, treating two strongly connected components in turn. It establishes finite-time removal of saturation’s effect and global consensus, and states that Zeno behavior is excluded.
B Proof of Theorem 4
- The authors omit the Zeno-exclusion proof because it is the same as the corresponding proof for Theorem 3.Source: B Proof of Theorem 4 · p. 12
- The proof takes M = 2 and treats the second strongly connected component as a strongly connected directed graph because it does not depend on the first component.It then considers the first component using a Lyapunov derivative argument.Source: B Proof of Theorem 4 · p. 12
- The proof constructs Lyapunov candidates, analyzes their derivatives along event-triggered trajectories, and applies LaSalle’s Invariance Principle.It also introduces auxiliary states, including one specified to decay exponentially over time.Source: B Proof of Theorem 4 · p. 13
- After finite time, saturation no longer affects the system, and global consensus is achieved under the event-triggered protocol.The proof applies the argument to the event-triggered control protocol and its trajectories.Source: B Proof of Theorem 4 · p. 13