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Finite-Time Consensus Problems for Networks of Dynamic Agents
Long Wang, Feng Xiao
TL;DR
The paper asks how continuous-time multi-agent systems can reach consensus in finite time rather than only asymptotically. It proposes two distributed protocols and proves them using finite-time Lyapunov stability. The results include weighted-average consensus, application to switching topologies, topology-dependent convergence, and simulation-supported effectiveness.
Problem
Finite-time consensus is desirable because conventional linear consensus protocols cannot reach state agreement in finite time.
Method
The paper proposes two continuous distributed protocols and analyzes them with finite-time Lyapunov stability, including a common Lyapunov function for switching topology.
Results
The two protocols solve finite-time consensus problems; one also supports weighted-average consensus and dynamically changing topologies.
Takeaways & Limitations
Convergence time depends on communication topology, with larger algebraic connectivity greatly reducing convergence time in the undirected case.
Abstract
from arXiv · showhide
In this paper, finite-time state consensus problems for continuous-time multi-agent systems are discussed, and two distributive protocols, which ensure that the states of agents reach an agreement in a finite time, are presented. By employing the method of finite-time Lyapunov functions, we derive conditions that guarantee the two protocols to solve the finite-time consensus problems respectively. Moreover, one of the two protocols solves the finite-time weighted-average consensus problem and can be successively applied to the systems with switching topology. Upper bounds of convergence times are also established. Simulations are presented to show the effectiveness of our results.
I. INTRODUCTION
The paper addresses finite-time consensus in continuous-time multi-agent networks, motivated by the limitations of asymptotic linear protocols. It develops two distributed protocols, analyzes their convergence, and extends one to weighted-average consensus and switching topologies.
- Motivation: Consensus theory studies how local interaction rules produce agreement in networks of dynamic agents.The paper situates this problem in decentralized control and applications including unmanned vehicles, mobile robots, communication networks, and sensor networks.
- Motivation: Conventional linear consensus protocols can improve convergence rate but cannot achieve state consensus in finite time.Their convergence speed is related to graph algebraic connectivity, yet consensus remains asymptotic.
- Contributions: The paper presents two continuous distributed protocols that solve consensus problems in finite times.The protocols are continuous state feedbacks that do not satisfy the Lipschitz condition at agreement states, and finite-time Lyapunov stability is used for analysis.
- Performance analysis: Convergence time is closely related to communication topology, especially algebraic connectivity in the undirected case.Larger algebraic connectivity can greatly reduce convergence time.
- Performance analysis: Protocol parameters can be adjusted according to agents’ state differences because different parameter choices favor large or small disagreements.One system converges faster when states differ substantially, while another is faster when states differ slightly.
- Switching topology: The same method is used to study dynamically changing communication topologies.The paper also reports that one protocol applies when the topology changes dynamically.
II. PRELIMINARIES
The preliminaries define the graph, matrix, and Laplacian concepts used to analyze consensus networks. They establish connectivity and spectral properties that support subsequent Lyapunov arguments.
- Graph definitions: A directed graph models communication among agents, with vertices representing agents and edges representing communication links.Weighted directed graphs add a nonnegative edge-weight matrix, while symmetric weights define undirected graphs.
- Graph definitions: A spanning tree is a directed tree covering all vertices, and strong connectivity requires a directed path between every pair of distinct vertices.Strongly connected components partition the vertices of a directed graph.
- Laplacian properties: For connected undirected graphs, the Laplacian is positive semidefinite and its second-smallest eigenvalue is the positive algebraic connectivity.Algebraic connectivity is characterized through a constrained minimum involving vectors orthogonal to 1.
- Spectral analysis: The shifted matrix −L(A)+dI is analyzed using nonnegative-matrix and spectral arguments, yielding spectral radius ρ(−L(A)+dI)=d.The construction uses Gershgorin disks and Perron–Frobenius theory.
- Laplacian properties: For strongly connected directed graphs, a positive weighting transforms the Laplacian into the Laplacian of an undirected weighted graph.The resulting matrix is semipositive definite with a simple zero eigenvalue.
- Matrix properties: If G(A) is undirected and connected and b ≥ 0 with b ≠ 0, then L(A)+diag(b) is positive definite.The proof uses semipositive definiteness and shows that a zero quadratic form forces the vector to vanish.
III. PROBLEM FORMULATION
The paper formulates consensus for continuous-time agents communicating over weighted directed graphs and introduces two distributed protocols designed to achieve agreement in finite time.
- System model: The system consists of n autonomous agents with scalar states x_i and control inputs u_i designed from neighbor-state information.Agents communicate through a weighted directed graph G(A), where neighbors are agents whose information is directly received.
- Topology: The communication topology may be switching when its adjacency matrix A depends on time, although A is assumed time-invariant unless specified otherwise.The switching-topology case is discussed separately in the paper.
- Consensus objective: A finite-time consensus problem requires all agent states to become equal to a common value κ after some finite time t* and remain equal thereafter.Ordinary consensus requires pairwise state differences to converge to zero, whereas finite-time consensus additionally requires exact agreement after t*.
A. Basic Properties and Lemmas
The basic analysis establishes continuity, global solution existence, boundedness, and the characterization of consensus equilibria for the proposed protocols.
- Continuity: Protocols (5) and (6) are continuous with respect to all state variables.This follows from continuity of sig(r)^α for α > 0, including at r = 0.
- Existence: Solutions exist on [0, ∞) for any initial state under either protocol.The proof extends local solutions globally using boundedness and an extension theorem.
- Boundedness: The maximum agent state is non-increasing, the minimum is non-decreasing, and ||x(t)||∞ remains bounded by ||x(0)||∞.These monotonicity properties prevent finite escape and support global existence.
- Equilibria: If G(A) has a spanning tree, the equilibrium point set of either protocol is exactly span(1), the set of consensus states.The proof shows both that equilibria must have equal components and that every equal-state vector is an equilibrium.
- Finite-time tool: The finite-time Lyapunov lemma states that V(t) reaches zero no later than V(0)^(1−α)/[K(1−α)] when V̇(t) ≤ −K V(t)^α with 0 < α < 1.Once V reaches zero, it remains zero.
A. Networks Under Protocol (5)
For protocol (5), the paper proves finite-time consensus when the communication graph has a spanning tree, using Lyapunov arguments across strongly connected components and leader-follower reductions.
- Theorem 1: Theorem 1 states that protocol (5) solves a finite-time consensus problem whenever G(A) has a spanning tree.The proof handles strongly connected graphs, a single-leader follower network, and general spanning-tree topologies.
- Strongly connected topology: For strongly connected graphs, a weighted disagreement Lyapunov function reaches zero in finite time, implying y(t) = 0 and consensus.The finite-time conclusion follows from the differential inequality and Lemma 3.
- Leader-follower topology: With one leader and strongly connected follower topology, the leader remains time-invariant and followers reach consensus with the leader in finite time.The reduced disagreement function V2(t) vanishes exactly when the full state lies in span(1).
- General spanning-tree topology: For a general spanning-tree graph, strongly connected components are collapsed successively into virtual agents as each component reaches consensus.Induction over the directed component tree yields finite-time agreement of every component with the leaders’ final state.
- Weighted invariance: Protocol (5) preserves a weighted quantity ω^T x(t), so under the stated zero-initial-condition relation, agents reach zero in finite time.The weighted invariance is used to reduce the dynamics to the disagreement system analyzed in Theorem 1.
- Scope boundary: Protocol (5) cannot reach finite-time consensus at agreement points other than the origin under the stated Lipschitz condition.The paper notes that the system is Lipschitz on span(1) except at the origin, preventing finite-time arrival at other agreement points.
B. Networks Under Protocol (6)
Protocol (6) is shown to achieve finite-time consensus under several graph conditions, including undirected, detail-balanced, and spanning-tree settings. Under appropriate assumptions, the final state can equal the average or a leader’s state.
- Protocol (6) solves finite-time consensus on undirected connected communication topologies.The analysis uses a Lyapunov function whose disagreement measure reaches zero in finite time.
- The final common state is the average of the agents’ initial states under the undirected connected case.
- Protocol (6) reaches the leader’s state when followers have an undirected connected local topology and symmetric protocol parameters.
- Strong connectivity and detail balance extend finite-time consensus results beyond purely undirected graphs.The paper states results for strongly connected detail-balanced graphs and for graphs with a spanning tree whose strongly connected components are detail-balanced.
- Protocol (6) can also solve finite-time average-consensus under strongly connected, detail-balanced conditions with symmetric αij parameters.
C. Performance Analysis
The paper relates finite-time convergence to disagreement, protocol parameters, and graph connectivity. Larger initial disagreement and parameters approaching one increase convergence time, whereas greater algebraic connectivity shortens it.
- V1, V2, V3, and V4 measure disagreement among agents’ states or from their common final state.The paper identifies V1 as measuring the length of L(A)x and V3 and V4 as measuring disagreement from the final state.
- Larger initial values of V1, V2, V3, or V4 result in longer convergence times.
- The convergence-time bound tends to infinity as all αi or αij approach 1.At αi or αij = 1, the system is described as achieving consensus at most asymptotically.
- Larger algebraic connectivity of G(A) or G(B) leads to shorter convergence time under the derived differential inequalities.
- Smaller protocol parameters are better when state differences are small, while larger parameters are better when differences are large.The paper therefore suggests changing αi or αij as the system evolves.
D. Networks With Switching Topology
Protocol (6) is extended to switching topologies using a topology-independent Lyapunov function. Under continuously connected switching graphs and symmetric parameters, it achieves finite-time average agreement, but connectivity at every time remains essential for the stated theorem.
- Finite-time convergence analysis for protocol (5) under switching topology is left for future research.
- The topology-independent Lyapunov function V3 is proposed as a common function for switching-topology convergence analysis.
- Protocol (6) solves finite-time average agreement when the switching graph remains undirected and connected and αij(t) is symmetric with values in a finite set.
- The theorem gives a finite convergence bound t∗= 21−α0V3(0).
- The stated switching-topology result cannot conclude finite-time consensus when the graph is disconnected at some times, even if its union is connected over an interval.
V. SIMULATIONS
Simulations illustrate finite-time trajectories, parameter-dependent convergence rates, connectivity effects, and switching-topology behavior. The reported examples include an average-agreement bound of 11.2999 seconds under periodic switching.
- Under protocol (5), the trajectories show faster convergence when αi is relatively small for initially similar agent states.
- The simulations use six agents and four connected undirected graphs with edge weights equal to 2.
- For protocol (5), α=0.3 and α=0.8 are compared under topology G1 to examine convergence rates.
- For topologies G2 and G4, the estimated protocol (5) convergence-time upper bounds are 6.0638 and 16.2819, respectively.
- Under periodic switching among G1, G2, G3, and G4, protocol (6) achieves average agreement with estimated upper bound t3 = 11.2999.
VI. CONCLUSION
The paper presents two effective continuous finite-time consensus protocols and analyzes how convergence time depends on topology, initial states, and protocol parameters. Simulations demonstrate effectiveness, while several extensions remain open.
- Two effective continuous finite-time consensus protocols are presented for multi-agent systems.
- The relationship between convergence time, communication topology, initial states, and protocol parameters α_i or α_ij is analyzed.
- Several simulations demonstrate the effectiveness of the theoretical results.
- The work is described as a first step toward finite-time consensus analysis, with additional topics still to be addressed.
- Open questions include whether protocol (6) works with a spanning-tree topology and whether similar results hold under switching topologies with communication time-delays.
- Further open problems concern other effective finite-time protocols.
APPENDIX I
Appendix I collects mathematical results used in the paper, including eigenvalue localization, existence and extension of differential-equation solutions, comparison, and nonnegative-matrix properties.
- Gershgorin’s Disk Theorem locates all eigenvalues of a matrix within a union of discs.
- For a nonnegative matrix, strong connectivity is included among the stated equivalent conditions.
- The Perron-Frobenius theorem gives a positive eigenvector associated with the spectral radius and establishes its algebraic and geometric simplicity.
- Peano’s Existence Theorem guarantees at least one local solution to a differential equation under continuity and boundedness conditions.
- The Extension Theorem states that a solution can extend over a maximal interval and approaches the boundary of the domain at finite endpoints.
- The Comparison Principle provides conditions for comparing a differential-equation solution with a continuous function satisfying a right-derivative inequality.