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Consensus of Multi-Agent Systems with General Linear and Lipschitz Nonlinear Dynamics Using Distributed Adaptive Protocols
Zhongkui Li, Wei Ren, Xiangdong Liu, Mengyin Fu
TL;DR
The paper addresses distributed consensus for multi-agent systems with general linear and Lipschitz nonlinear dynamics without relying on global graph information. It designs adaptive relative-state protocols for both cases, proves consensus on all undirected connected graphs, and studies leader-follower extensions.
Problem
Consensus protocols for general linear and Lipschitz nonlinear agents need a fully distributed design that does not use global communication-graph information.
Method
The paper combines distributed relative-state consensus protocols with adaptive laws that adjust coupling weights between neighboring agents.
Results
Consensus is reached in both linear and Lipschitz nonlinear cases for all undirected connected communication graphs, with extensions to leader-follower graphs.
Takeaways & Limitations
The adaptive protocols can be implemented by each agent in a fully distributed fashion without using global information.
Abstract
from arXiv · showhide
This paper considers the distributed consensus problems for multi-agent systems with general linear and Lipschitz nonlinear dynamics. Distributed relative-state consensus protocols with an adaptive law for adjusting the coupling weights between neighboring agents are designed for both the linear and nonlinear cases, under which consensus is reached for all undirected connected communication graphs. Extensions to the case with a leader-follower communication graph are further studied. In contrast to the existing results in the literature, the adaptive consensus protocols here can be implemented by each agent in a fully distributed fashion without using any global information.
I. INTRODUCTION
The paper addresses distributed consensus for agents with general linear and Lipschitz nonlinear dynamics, where existing protocols require global graph information. It proposes adaptive relative-state protocols that achieve consensus across connected undirected graphs and extend to leader-follower settings.
- Motivation: Existing protocols require each agent to know the communication graph’s Laplacian, preventing fully distributed implementation using only local information.The required coupling-weight bound depends on the smallest nonzero Laplacian eigenvalue.
- Proposed approach: The paper proposes relative-state consensus protocols with adaptive laws that adjust coupling weights between neighboring agents.The protocols are designed for both general linear and Lipschitz nonlinear multi-agent systems.
- Main results: Consensus is reached for both linear and nonlinear agents under any undirected connected communication graph.The linear-case protocol has a sufficient existence condition that each agent is stabilizable.
- Main results: The nonlinear-case results reduce to the linear-case results when the Lipschitz nonlinearity is absent.The paper also discusses existence conditions for the nonlinear adaptive protocol.
- Extensions: The results are extended to leader-follower communication graphs.The extension is studied after the linear and nonlinear consensus cases.
II. ADAPTIVE CONSENSUS FOR MULTI-AGENT SYSTEMS WITH GENERAL LINEAR DYNAMICS
For identical agents with general linear dynamics, the paper replaces graph-dependent coupling design with a relative-state adaptive protocol. For every connected undirected communication graph, the protocol achieves consensus while coupling weights converge to finite steady-state values.
- Motivation: Existing protocols require the coupling weight or feedback design to use λ2 or the Laplacian, which depends on the entire communication graph.This prevents fully distributed implementation using only each agent’s local information.
- Protocol: The proposed protocol combines neighboring agents’ relative states with an adaptive law that adjusts time-varying coupling weights.The weights remain symmetric when κij = κji and cij(0) = cji(0).
- Convergence: Each coupling weight cij is monotonically increasing and converges to a finite steady-state value.Boundedness follows from the nonincreasing Lyapunov function, while LaSalle’s invariance principle yields e(t) → 0.
- Distributed implementation: The adaptive protocol can be computed and implemented by each agent in a fully distributed way for all connected communication topologies.A sufficient condition for the LMI-based protocol is that (A, B) is stabilizable.
III. ADAPTIVE CONSENSUS FOR MULTI-AGENT SYSTEMS WITH LIPSCHITZ NONLINEARITY
For identical agents with Lipschitz nonlinear dynamics, the paper extends the adaptive relative-state protocol through an LMI-based design. Under the stated feasibility condition, consensus is achieved for connected communication graphs and coupling weights converge to finite values.
- Problem formulation: The nonlinear agents combine general linear dynamics with a nonlinear function satisfying a Lipschitz condition with constant γ > 0.The protocol is analyzed for dynamics of the form ˙xi = Axi + D1f(xi) + Bui.
- Protocol design: The nonlinear adaptive protocol is designed by solving an LMI for Q > 0, τ > 0, and a diagonal scaling matrix T > 0.The resulting feedback choices are F = −BTQ^−1 and Γ = Q^−1BB^TQ^−1.
- Convergence: Under the LMI-based design, the nonlinear agents reach global consensus for any connected communication graph.The proof reduces consensus to convergence of disagreement coordinates and applies a Lyapunov argument with LaSalle-Yoshizawa reasoning.
- Convergence: The adaptive nonlinear protocol makes each coupling weight converge to a finite steady-state value.The shifted weights are bounded and monotonically increasing, which implies finite limits for cij.
- Feasibility: Feasibility of the nonlinear LMI is linked to the distance to unobservability of (A, B) being larger than γ.A diagonal scaling matrix T is introduced to reduce conservatism.
- Relation to linear case: When the Lipschitz nonlinearity vanishes, the nonlinear theorem reduces to the general linear result.Setting D1 = 0 and choosing T sufficiently large transforms the nonlinear LMI into the linear condition.
IV. EXTENSIONS
The paper extends adaptive consensus protocols to leader-follower networks, where followers asymptotically approach a virtual leader and adaptive coupling weights remain finite.
- Leader-follower setting: The extension considers a network of N followers and one virtual leader with an undirected communication graph among followers.The leader receives no follower information and has control input u0 = 0.
- Distributed protocol: The leader’s state is available to only a subset of followers, represented by positive pinning gains for the first q followers and zero gains for the rest.The protocol includes adaptive coupling to neighboring agents and, for pinned followers, adaptive coupling to the virtual leader.
- Leader tracking: The design objective is to choose feedback gain matrices so every follower’s state asymptotically approaches the leader’s state.The target condition is limt→∞∥xi(t)−x0(t)∥= 0 for all followers.
- Leader tracking: Under the proposed protocol, connected follower networks with at least one follower accessing the leader achieve asymptotic leader-following consensus.The stated gains are ˆF = −BTP −1 and ˆΓ = P −1BBT P −1, with P > 0 solving (3).
- Adaptive weights: The adaptive coupling weights between followers and between followers and the leader converge to finite values.The proof uses a Lyapunov function and LaSalle-Yoshizawa theorem; the nonlinear case is discussed similarly.
- Comparison: Unlike an earlier adaptive pinning scheme using an identity inner coupling matrix, protocol (23) is described as more general.The comparison appears in Remark 5.
V. SIMULATION EXAMPLES
A simulation example evaluates the theory on eight single-link manipulators with revolute joints driven by DC motors and Lipschitz nonlinear dynamics.
- Simulation setup: The simulation considers a network of single-link manipulators with revolute joints actuated by DC motors.The manipulator dynamics are described by model (12).
- Simulation figures: Figure 1 depicts the communication topology used in the simulation.The figure is captioned as “The communication topology.”
- Simulation setup: The manipulator nonlinearity satisfies the Lipschitz condition with constant γ = 0.333.The feedback gain matrices are obtained by solving LMI (14) using Matlab’s LMI toolbox.
- Simulation figures: Figure 2 presents the states of the eight manipulators under protocol (4).The supplied caption identifies the plotted agents and protocol.
VI. CONCLUSION
The paper concludes that fully distributed adaptive relative-state protocols solve consensus for linear and Lipschitz nonlinear agents on undirected connected graphs, with a leader-follower extension.
- Conclusion: The paper addresses distributed consensus for multi-agent systems with general linear and Lipschitz nonlinear dynamics.This is the stated scope of the concluding section.
- Conclusion: The proposed protocols use relative states and adaptive laws to adjust coupling weights between neighboring agents.The protocols are designed for both the linear and nonlinear cases.
- Conclusion: Consensus is reached for all undirected connected communication graphs in both the linear and nonlinear cases.The conclusion states this result without restricting it to a particular graph instance.
- Conclusion: The paper also studies extensions to communication graphs with a leader-follower structure.This extends the considered consensus setting beyond follower-only networks.