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Leader-following coordination of multi-agent systems with coupling time delays
Jiangping Hu, Yiguang Hong
TL;DR
The paper addresses leader-following consensus for autonomous agents with time-varying coupling delays, especially in directed networks. It analyzes fixed and switched coupling topologies using convergence tools, proving a necessary and sufficient condition for fixed directed graphs and proposing a sufficient condition for switched balanced graphs. Numerical examples verify the theoretical analysis.
Problem
The paper studies leader-following consensus when agents have time-varying coupling delays and directed interconnection topologies.
Method
The paper analyzes fixed directed and switched balanced topologies using Lyapunov-Razumikhin functions and linear matrix inequalities.
Results
For fixed directed topology, the paper proves a necessary and sufficient condition; for switched balanced topology, it proposes a sufficient condition.
Takeaways & Limitations
The results establish conditions under which followers converge to the leader despite time-varying coupling delays, with numerical examples verifying the theoretical analysis.
Takeaways & Limitations
The switched-topology analysis assumes the coupling graph G_σ is balanced.
Abstract
from arXiv · showhide
In this paper, we consider a leader-following consensus problem of a group of autonomous agents with time-varying coupling delays. Two different cases of coupling topologies are investigated. At first, a necessary and sufficient condition is proved in the case when the interconnection topology is fixed and directed. Then a sufficient condition is proposed in the case when the coupling topology is switched and balanced. Numerical examples are also given to illustrate our results.
1 Introduction
The paper studies leader-following coordination in multi-agent systems, focusing on second-order agents, directed interconnections, and time-varying coupling delays. It analyzes fixed and switched topologies using Lyapunov-Razumikhin functions and linear matrix inequalities.
- Motivation: Multi-agent coordination research addresses cooperative behavior, interconnection rules, and applications in physics, biology, ecology, and engineering.Examples include collective particle motion, aggregation, flocking, schooling, and robot control.
- Motivation: Coupling delays matter because agents cannot instantly obtain information from other agents or the leader.The paper motivates delayed models through practical problems involving communication and coupling delays.
- Problem setting: The paper considers second-order agents with time-varying coupling delays and directed interconnection graphs.This setting differs from earlier first-order models with undirected topologies.
- Problem setting: Directed graphs and time delays make convergence analysis more challenging than analysis of undirected, delay-free settings.The introduction identifies graph complexity and delayed dynamics as sources of analytical difficulty.
- Approach: The analysis treats fixed and switched coupling topologies using Lyapunov-Razumikhin functions together with linear matrix inequalities.The two topology cases are analyzed in separate sections before numerical examples and concluding remarks.
2 Model Description
The model contains one leader and n follower-agents represented by weighted digraphs and leader adjacency matrices. Coupling uses delayed neighbor and leader information under either fixed or switching topologies, with convergence defined by followers approaching the leader's position and velocity.
- Agent model: The system contains n+1 identical agents, with agent 0 designated as the leader and agents 1 through n as followers.The leader moves independently, while follower motion is influenced by the leader and other followers.
- Graph representation: Follower interconnections are represented by a weighted digraph G with node set V, arc set E, and nonnegative adjacency matrix A.The Laplacian is constructed from the graph's degree and adjacency matrices.
- Graph representation: The leader adjacency matrix B records which followers directly receive information from the leader.A positive diagonal entry indicates a leader-to-follower arc, while zero indicates no direct leader neighbor.
- Graph properties: A leader is globally reachable when every follower has a path to node 0, a condition weaker than strong connectedness.The paper also relates global reachability to the Laplacian having a simple zero eigenvalue.
- Delayed coupling: The coupling delay r(t) is time-varying and continuously differentiable, while a switching signal selects graphs from a finite collection.A constant switching signal yields a fixed interconnection topology.
3 Fixed Coupling Topology
For fixed directed coupling topology, the paper links leader-following consensus to graph reachability and establishes stability conditions for systems with time-varying delays. The analysis uses Lyapunov-Razumikhin arguments and derives a finite admissible delay bound.
- 3 Fixed Coupling Topology: The fixed-topology analysis studies convergence of dynamic multi-agent systems with coupling delays.The section focuses on convergence analysis for dynamic agents and introduces time-delay-system preliminaries.
- 3 Fixed Coupling Topology: The Lyapunov-Razumikhin theorem provides uniform asymptotic stability when the derivative condition holds under delayed-state constraints.The theorem requires a positive-definite Lyapunov function satisfying the stated Razumikhin inequalities.
- 3 Fixed Coupling Topology: H = L + B is positive stable if and only if the leader node 0 is globally reachable in the augmented graph.This graph condition is established through the relationship between H and graph connectedness.
- 3 Fixed Coupling Topology: Theorem 1 gives a necessary and sufficient consensus condition: node 0 must be globally reachable in the augmented graph.The proof uses positive stability of H, a positive-definite matrix, and Lyapunov-Razumikhin analysis.
- 3 Fixed Coupling Topology: The analysis obtains a finite upper bound on the considered time-varying delay, rather than only requiring that the delay be sufficiently small.The paper also notes that the result remains valid for constant delays and connects the delay-free rule to nearest-neighbor rules.
- 3 Fixed Coupling Topology: In simulation, four agents use k = 3 and the time-varying delay r(t) = 0.0300| cos(t)|; errors and trajectories are shown in Figs. 3 and 4.Figure 3 reports position and velocity errors, while Figure 4 displays agent and leader trajectories.
4 Switched Coupling Topology
For switched coupling topologies, the analysis studies leader-following errors under switching graphs and time-varying delays, deriving stability conditions and illustrating them numerically. The approach uses globally reachable leaders, balanced graphs, matrix positivity, and Lyapunov arguments.
- The switched-topology analysis considers leader-following errors under a switching signal and uses the transformed error system.The switching signal selects among possible digraphs, while the analysis works with errors relative to the leader.
- A globally reachable leader node is assumed in the augmented switching graph.This assumption supports the block-structured Laplacian analysis and the resulting stability argument.
- The Laplacian is organized by strong components, with each corresponding diagonal block associated with a nonzero leader-coupling matrix.The nodes are renumbered so the Laplacian has strong-component blocks, and the assumption ensures each associated B_i(σ) is nonzero.
- The proof establishes positive definiteness of the relevant matrices and uses a Lyapunov function whose derivative becomes negative definite under the stated condition.Q_σ is positive definite for every switching graph, and the minimum eigenvalue across possible Q_σ matrices enters the conclusion.
- For switched balanced graphs, the paper presents a sufficient stability condition, while noting that balancedness is not necessary for the stability result in Theorem 2.The numerical example reports stability even when the coupling topology is sometimes not balanced.
- A simulation switches between two coupling topologies with order {Ḡ1, Ḡ2, Ḡ1, Ḡ2, ...} and uses k = 9 with delay r(t) = 0.0150|cos(t)|.The resulting leader-following errors are shown in Fig. 5.
5 Conclusions
The paper addresses leader-following coordination with time-varying coupling delays. It gives a necessary and sufficient condition for fixed directed topologies, a sufficient condition for switched balanced topologies, and simulations verifying the analysis.
- The paper studies a multi-agent coordination problem in which follower agents track a leader moving at constant velocity despite time-varying coupling delays.
- For fixed directed coupling topologies, the paper provides a necessary and sufficient condition.
- For switched balanced coupling topologies, the paper presents a sufficient condition.
- Numerical simulations verify the theoretical analysis.