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Designing Fully Distributed Consensus Protocols for Linear Multi-agent Systems with Directed Graphs
Zhongkui Li, Guanghui Wen, Zhisheng Duan, Wei Ren
TL;DR
Fully distributed adaptive consensus for general directed graphs is challenging when follower subgraphs are not restricted to being undirected. The paper designs a protocol using agent dynamics and neighboring agents’ relative states, and establishes leader-follower consensus under the stated graph assumption.
Problem
Fully distributed adaptive consensus protocols for general directed graphs remain challenging beyond settings where follower subgraphs are undirected.
Method
The paper designs a distributed adaptive consensus protocol based only on agent dynamics and neighboring agents’ relative state information.
Results
Under Assumption 1, the leader-follower consensus problem is solved.
Takeaways & Limitations
For communication graphs containing a directed spanning tree with the leader as root, the protocol is fully distributed and also addresses multiple leaders.
Takeaways & Limitations
The leader’s control input is assumed in the technical note.
Abstract
from arXiv · showhide
This paper addresses the distributed consensus protocol design problem for multi-agent systems with general linear dynamics and directed communication graphs. Existing works usually design consensus protocols using the smallest real part of the nonzero eigenvalues of the Laplacian matrix associated with the communication graph, which however is global information. In this paper, based on only the agent dynamics and the relative states of neighboring agents, a distributed adaptive consensus protocol is designed to achieve leader-follower consensus for any communication graph containing a directed spanning tree with the leader as the root node. The proposed adaptive protocol is independent of any global information of the communication graph and thereby is fully distributed. Extensions to the case with multiple leaders are further studied.
I. INTRODUCTION
The paper targets fully distributed adaptive consensus for general linear multi-agent systems over directed graphs, addressing protocols that require global Laplacian eigenvalue information. It designs a local-information protocol for leader-follower consensus and extends the framework to multiple leaders.
- Motivation: Existing protocols commonly require the smallest real part of nonzero Laplacian eigenvalues, which is global communication-graph information.For undirected graphs, the relevant quantity is the smallest nonzero eigenvalue; for directed graphs, it is the smallest real part of the nonzero eigenvalues.
- Motivation: Because each agent must know the entire graph to compute this information, those protocols are not fully distributed.Fully distributed design uses only each agent’s local information and that of its neighbors.
- Problem: Fully distributed adaptive consensus for general directed graphs is challenging because directed-graph Laplacians are generally asymmetric.The asymmetry complicates both adaptive-protocol construction and Lyapunov-function selection.
- Contribution: The paper constructs an adaptive protocol using agent dynamics and neighboring agents’ relative states for leader-follower consensus.A novel Lyapunov function is used to show consensus for communication graphs containing a directed spanning tree rooted at the leader.
- Contribution: The protocol is independent of global graph information and is therefore fully distributed.Its design uses only local information available to each agent and its neighbors.
- Extension: For multiple zero-input leaders, the proposed adaptive protocol addresses containment control when each follower has a directed path from at least one leader.Follower states are driven into the convex hull spanned by the leader states under this connectivity condition.
II. MATHEMATICAL PRELIMINARIES
The preliminaries define directed-graph, matrix, vector, and spectral notation used in the consensus analysis. They also state the spanning-tree condition that characterizes a simple zero Laplacian eigenvalue.
- Notation: The notation includes real and complex matrix sets, identity matrices, block-diagonal matrices, positive-(semi-)definite ordering, and entrywise nonnegative vectors.The Kronecker product and the real-part operator are also defined for subsequent matrix analysis.
- Graph definitions: A directed graph consists of nodes and ordered edges; a directed spanning tree has a root with directed paths to all other nodes.The edge orientation identifies parent and child nodes, and undirected graphs contain reciprocal edges.
- Graph matrices: The adjacency matrix assigns positive weights to incoming edges, while the Laplacian has diagonal row sums and off-diagonal entries L_ij = -a_ij.If edge weights are irrelevant, adjacency entries are set to 1 for existing edges.
- Spectral property: For a directed graph Laplacian, zero is an eigenvalue with the all-ones right eigenvector, and every nonzero eigenvalue has positive real part.Zero is simple exactly when the graph has a directed spanning tree.
- Analytical tool: Young’s inequality is stated for nonnegative a and b with positive conjugate exponents satisfying 1/p + 1/q = 1.This inequality is included among the analytical tools used in the paper.
III. PROBLEM STATEMENT AND MOTIVATIONS
The paper formulates leader-follower consensus for identical agents with general linear dynamics over directed graphs containing a leader-rooted spanning tree. Existing protocols may require global Laplacian spectral information, preventing fully distributed implementation.
- Problem formulation: The agents follow identical general linear dynamics, with one leader and N followers whose states should converge to the leader's state.The leader has zero control input and receives no information from followers.
- Graph assumption: The communication graph is assumed to contain a directed spanning tree rooted at the leader.Equivalently, the leader has directed paths to all followers.
- Objective: The paper therefore targets fully distributed consensus protocols for directed communication graphs satisfying the leader-rooted spanning-tree assumption.The proposed design is motivated by the need to avoid global graph information.
- Existing protocols: Prior consensus protocols use coupling gains tied to the smallest real part of the nonzero Laplacian eigenvalues.For critically unstable agents, such as double integrators, this spectral quantity is generally required for protocol design.
- Motivation: The required Laplacian spectral quantity is global because each follower must know the entire communication graph to compute it.Consequently, these protocols cannot be designed using only each agent's own and neighboring information.
IV. DISTRIBUTED ADAPTIVE CONSENSUS PROTOCOL DESIGN
The paper introduces an adaptive protocol based on neighboring relative states and agent dynamics, with time-varying coupling weights. Under the graph assumption and suitable LMI conditions, it achieves leader-follower consensus without Laplacian eigenvalue information.
- Protocol design: The protocol uses relative states of neighboring agents to construct a distributed adaptive consensus law.Its coupling weights vary over time and are adjusted by local error-dependent dynamics.
- Convergence: Each coupling weight c_i converges to a finite steady-state value while the consensus error asymptotically converges to zero.The proof establishes bounded, monotonically increasing coupling weights and uses LaSalle's invariance principle.
- Design condition: A sufficient condition for the protocol design is stabilizability of the pair (A, B).The gain parameters can alternatively be obtained through an algebraic Riccati equation.
- Distributed implementation: The adaptive protocol depends only on agent dynamics and neighboring relative states, so each agent can compute and implement it fully distributed.Unlike prior protocols, it does not require eigenvalues of the asymmetric Laplacian matrix.
- Scope and design feature: The protocol applies to general directed leader-follower communication graphs, beyond protocols restricted to undirected graphs or undirected follower subgraphs.The functions ρ_i provide additional design freedom and converge to 1 as the consensus error vanishes.
V. EXTENSIONS TO THE CASE WITH MULTIPLE LEADERS
The paper extends the adaptive protocol to networks with multiple leaders and studies containment control. Under the corresponding graph assumption, followers converge into the leaders' convex hull and coupling weights remain finite.
- Multiple-leader setting: With multiple leaders, the problem becomes containment control, requiring followers to enter the convex hull of the leaders' states.Leaders have zero control inputs, and each follower must have a directed path from at least one leader.
- Graph properties: Under the multiple-leader graph assumption, the relevant follower Laplacian block has eigenvalues with positive real parts.The associated matrix structure supports reduction of containment control to the stability of the follower error.
- Extension result: The adaptive protocol from Theorem 1 solves the containment control problem for multiple-leader networks.The result follows through an error representation involving the leaders' convex hull.
- Convergence: Each coupling weight c_i converges to a finite steady-state value in the multiple-leader case.This preserves the finite-weight property established for a single leader.
- Relation to prior results: The multiple-leader theorem reduces to the single-leader theorem when only one leader exists.Compared with a prior static controller relying on Laplacian eigenvalues, the adaptive protocol uses only local information and remains fully distributed.
VI. SIMULATION EXAMPLE
A third-order-integrator simulation uses a directed graph with the leader accessible only to one node and verifies consensus under the adaptive protocol. The adaptive coupling weights converge to finite steady-state values.
- Simulation setup: The simulation considers third-order integrators communicating over a directed graph whose leader is accessible only to node 1.The graph weights are randomly chosen within (0, 3), and the graph satisfies the stated assumption.
- Consensus outcome: The consensus errors x_i − x_0 for the followers demonstrate that leader-follower consensus is achieved.The errors are plotted for the third-order-integrator simulation under adaptive protocol (5).
- Adaptive gains: The adaptive coupling weights c_i converge to finite steady-state values.Their trajectories are shown in the coupling-gain plot associated with protocol (5).
VII. CONCLUSION
The paper designs a fully distributed adaptive consensus protocol for general linear multi-agent systems with directed leader-follower graphs. It applies to graphs containing a leader-rooted directed spanning tree, discusses multiple leaders, and identifies extensions beyond the zero-input-leader setting.
- The paper addresses consensus for multi-agent systems with general linear dynamics and directed leader-follower communication graphs.
- For any graph containing a directed spanning tree rooted at the leader, the paper designs an adaptive consensus protocol.
- The protocol depends only on agent dynamics and neighboring agents’ relative state information, making it fully distributed.
- The paper also discusses the case with multiple leaders.
- The analysis assumes the leader’s control input is zero.
- Future work includes leaders with bounded control inputs or leaders that are arbitrary reference signals, and graphs without a leader or with only relative output information.