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Distributed Consensus of Linear Multi-Agent Systems with Adaptive Dynamic Protocols
Zhongkui Li, Xiangdong Liu, Wei Ren, Lihua Xie
TL;DR
The paper addresses distributed consensus for multi-agent systems with general linear dynamics when neighboring-state information and global graph knowledge are problematic. It proposes two adaptive dynamic protocols, including an edge-based protocol applicable to arbitrary switching connected communication graphs.
Problem
Distributed consensus is considered for multi-agent systems with general dynamics, while neighboring-agent states may not be available in many circumstances.
Method
The paper proposes two distributed adaptive dynamic consensus protocols: one assigns adaptive coupling weights to edges, and the other assigns them to nodes.
Results
The edge-based protocol is applicable to arbitrary switching connected communication graphs.
Takeaways & Limitations
The proposed protocols are designed for fully distributed implementation using local information rather than requiring each agent to know the entire communication graph.
Takeaways & Limitations
The distributed implementation relies on local information, while neighboring-agent states may be unavailable in some circumstances.
Abstract
from arXiv · showhide
This paper considers the distributed consensus problem of multi-agent systems with general continuous-time linear dynamics. Two distributed adaptive dynamic consensus protocols are proposed, based on the relative output information of neighboring agents. One protocol assigns an adaptive coupling weight to each edge in the communication graph while the other uses an adaptive coupling weight for each node. These two adaptive protocols are designed to ensure that consensus is reached in a fully distributed fashion for any undirected connected communication graphs without using any global information. A sufficient condition for the existence of these adaptive protocols is that each agent is stabilizable and detectable. The cases with leader-follower and switching communication graphs are also studied.
1 Introduction
The paper addresses consensus for multi-agent systems with general continuous-time linear dynamics, where prior protocols often require restrictive dynamics or global graph information. It proposes two adaptive dynamic protocols using relative outputs and establishes distributed consensus under connected graphs.
- Consensus research has often assumed first-, second-, or high-order integrator dynamics, which can be restrictive.
- The paper studies general continuous-time linear dynamics using accessible relative outputs rather than relative states.
- Prior protocols may require the smallest nonzero Laplacian eigenvalue, a global quantity unavailable from only local neighbor information.
- Two adaptive dynamic protocols assign coupling weights either to communication edges or to nodes.
- The protocols ensure consensus for any undirected connected communication graph without global information, assuming each agent is stabilizable and detectable.
- The paper also studies leader-follower and switching graphs, showing edge-weight adaptation applies to arbitrary switching connected graphs.
2 Notation and Graph Theory
This section introduces matrix notation and graph-theoretic definitions used to analyze consensus. It defines directed and undirected connectivity, adjacency and Laplacian matrices, and key Laplacian spectral properties.
- The notation defines real matrices, transposes, identity matrices, compatible dimensions, positive definiteness, positive semidefiniteness, and Kronecker products.
- A graph consists of nodes and ordered edges; undirected graphs contain both orientations of every edge.
- A graph is connected when a path exists between every pair of distinct nodes.
- A directed spanning tree has a root with no parent and directed paths to all other nodes.
- The adjacency matrix records directed edges, while the Laplacian uses weighted in-degrees on the diagonal and negative adjacency entries off-diagonal.
- For an undirected graph, λ2(L) is the smallest nonzero Laplacian eigenvalue; zero is simple exactly when the graph has a directed spanning tree.
3 Consensus with Undirected Communication Graphs
The paper studies consensus for identical agents with general linear dynamics over undirected connected graphs, addressing protocols that require global graph information. It proposes two adaptive dynamic protocols using relative outputs and establishes consensus under suitable gain and system conditions.
- Motivation: Existing protocols may require the Laplacian eigenvalue λ2, which is global information unavailable from only local agent and neighbor information.Consequently, those protocols cannot be implemented fully distributedly under that information constraint.
- Protocols: The paper proposes two adaptive dynamic protocols based on neighboring agents’ relative outputs rather than relative states.One protocol dynamically updates coupling weights for communication edges; the other assigns an adaptive coupling weight to each node.
- Protocols: The edge-based protocol uses time-varying edge weights c_ij(t), while the node-based protocol uses agent weights d_i(t) with distributed protocol states.The edge weights are initialized symmetrically, and the protocol parameters include positive constants and gain matrices to be determined.
- Problem: Consensus is formulated as lim_{t→∞}∥x_i(t)−x_j(t)∥=0 for all agents.Each agent has state x_i, control input u_i, and measured output y_i governed by compatible constant matrices A, B, and C.
4 Consensus with Leader-Follower Communication Graphs
The leader-follower extension assumes an undirected follower subgraph and a leader-rooted directed spanning tree. Two adaptive protocols use relative neighboring outputs and guarantee leader-follower consensus, with protocol states vanishing and adaptive weights remaining finite.
- Assumptions: The leader-follower setting requires an undirected follower subgraph and a directed spanning tree rooted at the leader.The leader has no neighbors and followers obtain relative outputs from their neighbors.
- Problem: Leader-follower consensus is defined by follower states converging to the leader state.The required condition is limt→∞∥x_i(t) − x_1(t)∥ = 0 for every follower.
- Edge-adaptive protocol: The first protocol assigns dynamic coupling weights to edges and uses relative output information from neighboring agents.The edge weights d_ij are initialized symmetrically, with positive adaptation constants and protocol gains specified through F, L, and Γ.
- Edge-adaptive protocol: The edge-adaptive protocol achieves leader-follower consensus, while protocol states converge to zero and edge weights converge to finite steady-state values.The result follows from the stated theorem under the leader-follower graph assumption.
- Node-adaptive protocol: The second protocol assigns a time-varying coupling weight to each follower rather than to each edge.Each follower has a protocol state and an adaptive weight d̂_i, with positive adaptation constants.
- Node-adaptive protocol: The node-adaptive protocol also achieves leader-follower consensus, with protocol states and follower coupling weights converging to finite limits.Theorem 4 states that the protocol states converge to zero and each d̂_i converges to some finite steady-state value.
5 Extensions to Switching Communication Graphs
The paper extends its edge-adaptive protocol to switching undirected connected graphs, motivated by time-varying communication. A common Lyapunov analysis establishes consensus while adaptive edge weights remain finite.
- Motivation: The switching extension addresses communication graphs that may change over time because of communication constraints and link variations.The protocol is analyzed for switching communication graphs rather than only fixed graphs.
- Switching model: The switching signal selects graphs from the finite set of all undirected connected N-node graphs.The graph G_σ(t) is piecewise constant, and its adjacency matrix varies with the active graph.
- Consensus result: For arbitrary switching graphs in this connected-graph set, the edge-adaptive protocol achieves consensus and drives all protocol states to zero.Theorem 5 states this result for protocol (33) with F, L, and Γ chosen as in Theorem 1.
- Consensus result: The adaptive edge weights converge to finite values under arbitrary switching among graphs that remain connected at every time instant.The proof uses boundedness and monotonicity of the adapted weights, followed by a LaSalle-Yoshizawa argument for the consensus errors.
- Scope boundary: The switching result is specific to the edge-adaptive protocol because the node-adaptive protocol’s Lyapunov function depends explicitly on the communication graph.The paper states that this dependence prevents using that function as a feasible common Lyapunov function; switching leader-follower graphs are only discussed as similar.
6 Simulation Examples
Simulations use third-order integrators on fixed and randomly switching connected graphs. The reported consensus errors vanish in both adaptive-protocol cases, while coupling weights approach finite steady-state values.
- Setup: Figure 1 shows the two undirected communication graphs G1 and G2 used in the simulations.For the switching case, G1 and G2 are both connected and the graph switches randomly every 0.1 second.
- Setup: The simulations evaluate third-order integrators with gains selected so that A + BF is Hurwitz.The gain F is chosen as −[3 6.5 4.5], and L is obtained by solving the stated LMI.
- Observed outcomes: Consensus is achieved in both fixed-graph cases, and the edge- and node-associated coupling weights converge to finite steady-state values.The paper attributes these observations to Figures 2–5.
- Edge-adaptive protocol: Figure 2 reports consensus errors x_i − x_1 for third-order integrators under protocol (33).The corresponding edge coupling weights c_ij are shown in Figure 3.
- Node-adaptive protocol: Figure 4 reports consensus errors x_i − x_1 under protocol (3), while Figure 5 shows the node coupling weights d_i.These figures correspond to the node-adaptive protocol simulation.
7 Conclusion
The paper addresses consensus for identical general linear multi-agent dynamics using relative outputs from neighboring agents. It proposes two distributed adaptive dynamic protocols, with adaptive coupling weights assigned either to edges or nodes, achieving consensus without global information.
- The consensus problem is formulated for multi-agent systems with identical general linear dynamics using neighboring agents’ relative output information.
- Two distributed adaptive dynamic consensus protocols are proposed: one adapts coupling weights on communication-graph edges, and the other adapts weights for nodes.
- The protocols ensure consensus for any undirected connected communication graph in a fully distributed fashion without using global information.