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Finite-time Consensus for Multi-agent Networks with Unknown Inherent Nonlinear Dynamics
Yongcan Cao, Wei Ren
TL;DR
Finite-time consensus with unknown inherent nonlinear dynamics is harder to analyze than consensus for known linear systems. The paper introduces a generalized comparison tool and applies it to a nonlinear consensus algorithm, showing finite-time convergence under directed switching graphs with directed spanning trees.
Problem
Unknown inherent nonlinear dynamics make finite-time consensus analysis more challenging because the dynamics are unavailable for direct cancellation and the consensus value generally is not constant.
Method
The paper proposes a stability tool based on a generalized comparison lemma and compares the original closed-loop system with specially designed closed-loop systems.
Results
Finite-time consensus is guaranteed when each directed switching interaction graph has a directed spanning tree and the controller parameter satisfies β ≥ γ+ε_1.
Takeaways & Limitations
The comparison tool connects stability and finite-time convergence of the proposed algorithm under general directed switching graphs to specially designed systems under special directed switching graphs.
Abstract
from arXiv · showhide
This paper focuses on analyzing the finite-time convergence of a nonlinear consensus algorithm for multi-agent networks with unknown inherent nonlinear dynamics. Due to the existence of the unknown inherent nonlinear dynamics, the stability analysis and the finite-time convergence analysis of the closed-loop system under the proposed consensus algorithm are more challenging than those under the well-studied consensus algorithms for known linear systems. For this purpose, we propose a novel stability tool based on a generalized comparison lemma. With the aid of the novel stability tool, it is shown that the proposed nonlinear consensus algorithm can guarantee finite-time convergence if the directed switching interaction graph has a directed spanning tree at each time interval. Specifically, the finite-time convergence is shown by comparing the closed-loop system under the proposed consensus algorithm with some well-designed closed-loop system whose stability properties are easier to obtain. Moreover, the stability and the finite-time convergence of the closed-loop system using the proposed consensus algorithm under a (general) directed switching interaction graph can even be guaranteed by the stability and the finite-time convergence of some special well-designed nonlinear closed-loop system under some special directed switching interaction graph, where each agent has at most one neighbor whose state is either the maximum of those states that are smaller than its own state or the minimum of those states that are larger than its own state. This provides a stimulating example for the potential applications of the proposed novel stability tool in the stability analysis of linear/nonlinear closed-loop systems by making use of known results in linear/nonlinear systems. For illustration of the theoretical result, we provide a simulation example.
I. INTRODUCTION
The paper studies finite-time consensus for multi-agent networks with unknown inherent nonlinear dynamics, extending prior work beyond settings without inherent dynamics and beyond undirected graphs.
- Research context: Finite-time consensus requires agents to reach agreement in finite time, building on earlier results for undirected fixed or switching interaction graphs.Prior work also considered continuous nonlinear algorithms and finite-time convergence for single-integrator agents.
- Motivation: Unknown inherent dynamics occur in practical systems and make finite-time consensus analysis more challenging than consensus for known linear systems.The paper contrasts finite-time consensus with unknown dynamics against linear finite-time and unknown-dynamics asymptotic consensus.
- Main approach: The proposed stability tool uses a generalized comparison lemma to analyze the nonlinear consensus algorithm by comparing it with a better-understood closed-loop system.The introduction presents this comparison as the basis for the stability and convergence analysis.
- Main result: For a general directed switching graph, the algorithm’s stability and finite-time convergence can be guaranteed through a specially designed nonlinear system under a special switching graph.The special graph limits each agent to at most one neighbor selected from the nearest state below or above it.
- Main result: Without unknown inherent dynamics, the algorithm guarantees finite-time convergence when the directed switching interaction graph has a directed spanning tree at each time interval.This extends finite-time consensus from single-integrator kinematics to a more general case with a milder graph condition.
- Paper organization: The paper develops the stability tool, analyzes the directed switching case, and illustrates the theoretical result with a simulation example.The stated organization places the stability tool before the switching-graph analysis and simulation.
A. Notations
This section introduces real-vector notation, the componentwise signed-power function, upper Dini derivatives, and local Hölder continuity.
- Basic notation: The norm symbol ∥·∥ denotes the 2-norm of a vector.
- Special functions: sig(x)_α is defined componentwise as sgn(x)|x|^α and is continuous in x when α > 0.
- Derivatives: The upper Dini derivative is defined using the limsup of forward difference quotients for functions of time or state trajectories.The section gives corresponding definitions for f(t) and f(x(t)).
- Regularity: A function is locally χ-Lipschitz in x when its local variation is bounded by C∥x−y∥^χ for positive constants C, χ, and ε.The bound applies for y in a ball B(x, ε).
B. Graph Theory Notions
The graph-theoretic model represents information flow among agents using directed edges, neighbor sets, paths, spanning trees, adjacency matrices, and Laplacian matrices.
- Directed graphs: A directed edge (i, j) means agent j obtains information from agent i, so agent i is a neighbor of agent j.
- Directed graphs: A directed spanning tree exists when at least one agent has directed paths to all other agents.
- Matrix representations: The adjacency matrix records positive edge weights, while absent edges receive zero entries according to the graph orientation.The matrix uses a_ij > 0 when (j, i) belongs to the edge set.
- Matrix representations: The Laplacian has diagonal entries equal to the sum of incident adjacency weights and off-diagonal entries equal to negative adjacency weights.It has at least one zero eigenvalue with right eigenvector 1_n.
C. Problem Statement
The problem is to design a nonlinear distributed controller that achieves finite-time consensus for agents with unknown Lipschitz nonlinear dynamics over directed switching graphs.
- Agent dynamics: Each agent follows ṙ_i = φ(t, r_i) + u_i, where φ(t, r_i) is unknown inherent Lipschitz nonlinear dynamics and u_i is the control input.
- Objective: The objective is to ensure ∥r_i(t)−r_j(t)∥→0 in finite time for every pair of agents.The consensus value generally is not constant when unknown inherent dynamics are present.
- Objective: Unknown dynamics cannot be canceled directly because φ(t, r_i) is unavailable to the controller.
- Consensus algorithm: The proposed controller is nonlinear because linear algorithms normally cannot guarantee finite-time convergence.
- Consensus algorithm: The algorithm uses relative states for stabilization at distances at least one and signed-power terms with α⋆ ∈ (0, 1) for finite-time convergence below one.
- Switching assumptions: The interaction graph switches at specified instants, with each graph held for at least t_L and positive adjacency weights bounded below.The nonlinear term remains continuous at the distance threshold, while switching causes discontinuities only at distinct switching instants.
III. A NOVEL STABILITY TOOL BY COMPARISON
The paper introduces a generalized-comparison stability tool that establishes finite-time or asymptotic convergence of one nonlinear system through a suitably designed comparison system. Its proof repeatedly matches trajectories while preserving an initial-function bound, then transfers the comparison system’s convergence to the original system.
- Theorem 3.1: Theorem 3.1 uses a nonnegative function F(z) and comparison dynamics f(t,z) to infer convergence of G(x) for the original system.The theorem requires locally Lipschitz dynamics, continuity in time, comparison inequalities, existence of solutions, and convergence of F(z).
- Proof mechanism: The proof constructs successive intervals [t_i,t_i+1) and comparison initial states that maintain G(x) ≤ max_z∈Π F(z).At each interval, the comparison trajectory is reinitialized so that it meets the original trajectory while retaining F(ẑ_i(0)) ≤ F(x(0)).
- Convergence transfer: When the comparison function F(z) reaches zero in finite time, the original function G(x) also reaches zero in finite time.The same transfer applies to asymptotic convergence when F(z) tends to zero only as t →∞.
- Required conditions: Condition 5 keeps comparison initial states within a bounded set or bounds their state range, preventing the comparison construction from becoming unbounded.This boundedness is used together with the convergence condition for F(z).
- Relation to Lyapunov analysis: Unlike Lyapunov analysis, the tool compares derivatives of nonnegative functions across systems and does not require the original function to have a negative-semidefinite derivative.Consequently, it can analyze systems for which constructing a Lyapunov function is difficult or a Lyapunov function may not exist.
IV. FINITE-TIME CONSENSUS UNDER A DIRECTED SWITCHING INTERACTION GRAPH
The section establishes finite-time consensus for nonlinear multi-agent systems with directed switching graphs by comparing the proposed dynamics with specially designed systems. Under a directed spanning tree on every interval, the state spread converges to zero in finite time.
- Main theorem: Theorem 4.1 guarantees finite-time consensus when every directed interaction graph has a directed spanning tree and β ≥ γ+ε1/q_n−1+ε2.Here ε1 and ε2 are arbitrary positive constants, and q_n is the maximal positive number satisfying the stated inequality.
- Stability comparison: The analysis compares the proposed closed-loop system with a specially designed system whose state spread is easier to prove finite-time convergent.This comparison is implemented through a generalized comparison framework and an auxiliary disagreement function.
- Disagreement analysis: The proof tracks G(r)=max_i r_i−min_i r_i and uses upper Dini derivatives to bound its evolution under switching interactions.The analysis separately handles neighbors of maximum- and minimum-state agents and exploits the directed spanning-tree condition.
- Proof feature: The disagreement function G(r) is not necessarily nonincreasing, so the proof does not rely on monotonic decrease of the original state spread.Instead, the comparison construction transfers finite-time convergence from the auxiliary dynamics.
- Auxiliary system: The auxiliary system reaches zero state spread in finite time, which yields finite-time convergence of the original system through the comparison result.The auxiliary spread is shown to converge through a nonnegative comparison function and a scalar differential inequality.
- Additional algorithm: A second nonlinear consensus algorithm also achieves finite-time consensus under a directed spanning tree when its gain satisfies k > γ.The section further notes that linear and nonlinear terms can be mutually helpful or harmless in different dynamics settings.
V. SIMULATION
The simulation tests the proposed consensus algorithm on four one-dimensional agents under a switching interaction graph, showing finite-time consensus despite unknown nonlinear dynamics. The disagreement measure can increase during some intervals, so it is not a Lyapunov function.
- Setup: The simulation uses four one-dimensional agents with an interaction graph switching between G(1) and G(2) every 0.5 seconds.The graphs are the directed graphs shown in Fig. 1.
- Results: The trajectories and disagreement plots show that all agents reach consensus in finite time under the proposed algorithm.Consensus corresponds to max_i r_i − min_i r_i = 0, equivalently log(1 + max_i r_i − min_i r_i) = 0.
- Results: The disagreement measure G(r) = max_i r_i − min_i r_i may increase over some intervals and therefore is not a Lyapunov function.The logarithmic transformation is used to better display finite-time convergence.
VI. CONCLUSION
The paper develops a generalized-comparison-based stability tool for finite-time consensus with unknown inherent nonlinear dynamics. Its conclusions relate the general directed switching case to a special system and confirm finite-time consensus in the absence of unknown dynamics.
- Contributions: The paper proposes a novel stability tool based on a generalized comparison lemma for analyzing the closed-loop consensus system.The tool compares the original closed-loop system with a well-designed system having finite-time consensus properties.
- Contributions: Stability and finite-time convergence under a general directed switching graph are guaranteed by corresponding properties of a special nonlinear system under a special switching graph.The special graph restricts each agent to at most one neighbor selected by an extremal-state rule.
- Conclusions: Without unknown inherent nonlinear dynamics, the proposed nonlinear consensus algorithm still guarantees finite-time consensus under a directed switching interaction graph.