Source-linked AI summary
Distributed velocity-constrained consensus of discrete-time multi-agent systems with nonconvex constraints, switching topologies, and delays
Peng Lin, Wei Ren, Huijun Gao
TL;DR
The paper addresses consensus for discrete-time multi-agent systems whose velocities lie in nonconvex sets under switching directed communication and bounded delays. It proposes a distributed constrained-control algorithm using local information and establishes exponentially fast position consensus while preserving velocity constraints.
Problem
The paper studies how to achieve position consensus when discrete-time agents have nonconvex velocity constraints, switching directed communication, and bounded delays.
Method
A distributed constrained-control algorithm uses local information and constraint operators instead of projection control, with individually adjustable agent gains.
Results
All agents reach position consensus exponentially fast while their velocities remain in their corresponding constraint sets.
Takeaways & Limitations
The proposed approach handles arbitrarily bounded communication delays and arbitrarily switching directed graphs when graph unions contain directed spanning trees over each specified interval.
Abstract
from arXiv · showhide
In this paper, a distributed velocity-constrained consensus problem is studied for discrete-time multi-agent systems, where each agent's velocity is constrained to lie in a nonconvex set. A distributed constrained control algorithm is proposed to enable all agents to converge to a common point using only local information. {The gains of the algorithm for all agents need not to be the same or predesigned and can be adjusted by each agent itself based on its own and neighbors' information.} It is shown that the algorithm is robust to arbitrarily bounded communication delays and arbitrarily switching communication graphs provided that the union of the graphs has directed spanning trees among each certain time interval. The analysis approach is based on multiple novel model transformations, proper control parameter selections, boundedness analysis of state-dependent stochastic matrices, exploitation of the convexity of stochastic matrices, and the joint connectivity of the communication graphs. Numerical examples are included to illustrate the theoretical results.
I. INTRODUCTION
Consensus research has largely focused on unconstrained agents or restricted constraint and communication settings. This paper targets discrete-time velocity-constrained consensus with nonconvex sets, directed switching graphs, and nonuniform delays using locally adjustable gains.
- Consensus has important applications including formation control, satellite attitude alignment, and flocking.
- Existing constrained-consensus results often assume continuous-time dynamics, hypercube constraints, or undirected communication graphs.
- Projection-based results for switching discrete-time systems assume agents remain in convex constraint sets, limiting direct application to nonconvex sets.
- The paper addresses discrete-time velocity constraints in nonconvex sets with directed, arbitrarily switching graphs and nonuniform bounded delays.
- Its distributed algorithm uses local information, while agent gains can be adjusted individually rather than being uniform or predesigned.
II. NOTATIONS AND PRELIMINARIES
The preliminaries establish notation for vectors, matrices, products, norms, graphs, paths, spanning trees, and stochastic matrices used in the analysis.
- The notation defines real vectors, identity matrices, integer indices, Kronecker products, transposes, block-diagonal matrices, infima, and matrix products.
- A directed graph consists of nodes and ordered edges, where edge direction specifies which agent receives information from another.
- The preliminaries define neighbors, edge weights, the directed-graph Laplacian, graph unions, directed paths, and strong connectivity.
- A directed spanning tree exists when at least one node can reach the other nodes through directed paths.
- A nonnegative square matrix is stochastic when multiplying it by an all-ones vector returns that vector.
III. MODEL AND PROBLEM STATEMENT
The model is a discrete-time double-integrator multi-agent system whose velocities lie in agent-specific, bounded, closed, potentially nonconvex sets. The objective is position consensus with velocities converging to zero while constraints remain satisfied.
- Each agent has discrete-time position and velocity dynamics, with control input determining the next velocity.
- Agent velocity v_i(k) must remain in a nonempty constraint set V_i known only to that agent.
- The velocity constraint sets may be nonconvex because driving-force limits can differ across directions.
- Each V_i is nonempty, bounded, and closed, contains zero, and satisfies positive operator bounds specified by Assumption 1.
- The constraint operator selects the largest vector aligned with x whose magnitude does not exceed x and whose radial segment lies in V_i.
- The goal is for all positions to converge to a common vector while every velocity converges to zero under the velocity constraints.
IV. MAIN RESULTS
The paper develops a distributed control algorithm for velocity-constrained discrete-time multi-agent systems with switching directed topologies and bounded communication delays. Under suitable parameter and connectivity conditions, agents preserve their nonconvex velocity constraints and reach position consensus exponentially fast.
- Assumptions: The analysis assumes bounded communication delays and graph unions containing directed spanning trees within every interval of at most η time steps.The delay model bounds each communication delay by M, while the joint-connectivity condition is imposed over switching graph unions.
- Control algorithm: The algorithm uses a constraint operator rather than projection to handle nonconvex velocity sets alongside coupled position–velocity dynamics and velocity delays.The operator preserves each agent’s velocity constraint while avoiding nonlinearities that would be difficult to analyze with projection.
- Control algorithm: Each agent can adjust its positive damping gain locally, without requiring identical or predesigned gains across agents.The distributed selection uses each agent’s own and neighbors’ information and satisfies the stated parameter bounds.
- Main theorem: Under Assumptions 1–3, all agents reach position consensus exponentially fast while every velocity remains in its corresponding constraint set.There exist x̄, C > 0, and 0 < μ ≤ 1 such that ∥x_i(k)−x̄∥ ≤ C(1−μ)^k for all agents and k ≥ 0.
- Main theorem: The theorem extends constrained consensus analysis to directed switching communication, nonconvex velocity constraints, double-integrator dynamics, and communication delays.These features distinguish the setting from prior results based on hypercube or convex constraints, undirected graphs, or single-integrator dynamics.
A. Multiple Model Transformations in the Proof of Theorem 1
The proof applies three nonsingular model transformations to convert the delayed, constrained multi-agent dynamics into an augmented system suitable for nonnegative-matrix analysis. The transformations preserve equivalence while separating position–velocity relations and incorporating delayed interactions.
- Transformation strategy: Three model-transformation steps recast the closed-loop system so nonnegative-matrix properties can be used for stability analysis.The transformations are stated to apply without information loss because all introduced matrices are nonsingular.
- Step 1: Constraint handling: The constraint operator preserves the direction of a nonzero vector, allowing the constrained update to be rewritten with agent-dependent scaling terms.The resulting expressions replace the operator by terms involving b_i(k) and e_i(k).
- Step 2: Variable substitution: A state substitution partly decouples each agent’s position and velocity integral relationship before the transformed variables are assembled.The section introduces ξ(k)=Q(k)φ(k) and corresponding block-diagonal matrices.
- Delay representation: Delayed neighbor interactions are represented through matrices Φ_m(k), whose entries encode edge weights according to the corresponding communication delay.The index m identifies the delay associated with edge (j,i).
- Step 3: Augmentation: An augmented state Z(k) stacks transformed states over the current and previous M time steps, producing a block matrix Ψ(k) for the delayed dynamics.The construction includes shifted identity blocks and blocks for each delay level.
B. Consensus Stability Analysis in the Proof of Theorem 1
The stability proof establishes that the transformed delayed dynamics are governed by stochastic transition matrices whose rows converge under joint connectivity. Convex-combination arguments then yield consensus and exponential convergence despite zero diagonal entries and arbitrarily small nonzero weights.
- Stochastic transition structure: The transformed system has a stochastic transition matrix Γ(k,s), enabling each future state component to be represented as a convex combination of earlier components.The proof also establishes stochasticity of the underlying block matrix and its transition products.
- Matrix properties: Parameter inequalities ensure nonnegative stochastic matrices, while the scaling factors and nonzero entries retain uniform positive lower bounds where required.These bounds follow from the relations among p_i(k), b_i(k), and the graph weights.
- Connectivity under delays: Although communication delays create zero diagonal entries and some nonzero entries may approach zero, joint spanning-tree connectivity still yields a uniformly positive column over sufficiently long intervals.Lemma 2 provides a column index and positive bound over intervals of length at least 4n(M+1).
- Consensus of transition rows: Every row of Γ(k,s) converges to a common limiting vector, with the convergence error bounded geometrically by C0(1−μ̂)^(k/η̂).The limiting coefficients sum to one, preserving the stochastic interpretation.
- Consensus conclusion: The agents’ positions converge to a common vector x̄, velocities converge to zero, and position errors satisfy ∥x_i(k)−x̄∥≤C(1−μ)^k.This conclusion follows from the convex-combination representation and the limiting behavior of Γ(k,s).
V. A NUMERICAL EXAMPLE
A four-agent planar example tests the algorithm with nonconvex velocity constraints, a cyclic one-edge-at-a-time switching sequence, and a sampling period of 0.2 s.
- Setup: The four agents move in a plane with velocities constrained to a union of a unit disk and a rectangular region.The same nonempty nonconvex set V_i is used for every agent.
- Switching topology: Only one directed edge transmits at each time, following the repeating sequence (1,2), (2,3), (3,4), (4,1).Each edge has weight 0.5.
- Simulation parameters: The simulation uses sampling period T=0.2 s and zero prehistory velocities, with initial positions specified at time zero.The passage states the initial conditions for all k<0.
VI. CONCLUSIONS
The paper studies distributed velocity-constrained consensus with nonconvex velocity sets under directed, delayed, and arbitrarily switching communication graphs. Its analysis transforms the system to state-dependent stochastic matrices and establishes exponentially aligning transition-matrix rows.
- The problem involves discrete-time agents whose velocities lie in a nonconvex set, with directed communication, bounded delays, and switching graphs.
- A distributed constrained control algorithm addresses the problem using local information under the stated graph-connectivity condition.
- Figure 3 presents the trajectories of all agents.
- The analysis uses model transformations and parameter selection to obtain an equivalent system with state-dependent stochastic matrices.
- An auxiliary matrix establishes a transition matrix with at least one fully positive column over a certain interval.
- Convexity of stochastic matrices is used to show that all transition-matrix rows converge to the same value exponentially.