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Adaptive Leader-Following Consensus for a Class of Higher-Order Nonlinear Multi-Agent Systems with Directed Switching Networks
Wei Liu, Jie Huang
TL;DR
The paper studies leader-following consensus for higher-order uncertain nonlinear multi-agent systems with constant parameter uncertainties and external disturbances over jointly connected directed switching networks. It strengthens adaptive distributed-observer convergence from asymptotic to exponential and combines that observer with adaptive control in a distributed state-feedback law. The resulting design solves the consensus problem under the stated assumptions, with simulations illustrating its effectiveness.
Problem
The paper addresses leader-following consensus when higher-order nonlinear agents have constant parameter uncertainties and external disturbances and communicate over jointly connected directed switching networks.
Method
The paper combines an adaptive distributed observer with conventional adaptive control to construct a distributed adaptive state-feedback control law.
Results
The designed distributed control law solves the leader-following consensus problem, with follower states converging to the leader states under the stated assumptions.
Takeaways & Limitations
Exponential convergence of the adaptive distributed observer supplies the property needed for the distributed adaptive controller to achieve consensus despite switching and disturbances.
Abstract
from arXiv · showhide
In this paper, we study the leader-following consensus problem for a class of uncertain nonlinear multi-agent systems under jointly connected directed switching networks. The uncertainty includes constant unbounded parameters and external disturbances. We first extend the recent result on the adaptive distributed observer from global asymptotical convergence to global exponential convergence. Then, by integrating the conventional adaptive control technique with the adaptive distributed observer, we present our solution by a distributed adaptive state feedback control law. Our result is illustrated by the leader-following consensus problem for a group of van der Pol oscillators.
I. INTRODUCTION
The paper addresses leader-following consensus for uncertain nonlinear multi-agent systems under jointly connected directed switching networks. It combines adaptive observation and control to handle higher-order dynamics, constant uncertainties, disturbances, and networks that may disconnect temporarily.
- The study targets leader-following consensus for uncertain nonlinear multi-agent systems with generic order and jointly connected switching networks.The networks may be disconnected at any time, while the result is global and achieves exact consensus.
- The nonlinear systems need not satisfy a global Lipschitz-like condition, allowing benchmark systems such as van der Pol and Duffing systems.Consequently, linear control techniques used in earlier work do not apply to the considered class.
- The model includes constant uncertain parameters that may take any constant value and external disturbances.These features distinguish the setting from approaches addressing only parameter uncertainty or restricted uncertainty models.
- The paper develops a global solution that achieves exact leader-following consensus for the stated class of systems.The paper presents the problem formulation, adaptive distributed observer, main result, example, and concluding remarks.
- The communication network is extended from undirected jointly connected graphs to directed jointly connected graphs.The jointly connected condition includes static and always-connected switching networks as special cases.
II. PROBLEM FORMULATION
The formulation models followers with higher-order nonlinear dynamics, unknown constant parameters, and disturbances generated by an exosystem, while a leader generates the reference. The objective is distributed control achieving asymptotic agreement with the leader under jointly connected switching communication.
- System model: Each follower has a higher-order state x_i, scalar input u_i, known locally Lipschitz nonlinearities, an unknown constant parameter vector θ_i, and disturbance d_i(w).The disturbance is a known C1 function of the exosystem state w.
- Exosystems: The leader reference and disturbance are generated by linear exosystems, combined into v = col(x_0, w) with S = diag(S_a, S_b).The leader system is treated as the reference-generating component of the overall multi-agent system.
- Communication graph: The time-varying digraph contains one leader node and N follower nodes, with edges encoding which neighboring information each follower can use.The resulting control law is distributed because each input depends only on the follower and its neighbors.
- Assumptions: The leader exosystem matrix S has distinct eigenvalues with zero real parts, permitting constant and multi-tone sinusoidal leader signals.The associated exosystem trajectories remain bounded for initial conditions in a compact set.
- Assumptions: Joint connectivity requires that a union of switching graphs over suitable intervals contain a directed spanning tree rooted at the leader.The condition allows the network to be disconnected at individual time instants.
III. ADAPTIVE DISTRIBUTED OBSERVER
The adaptive distributed observer estimates both the leader’s state and system matrix using neighboring information. Under the stated exosystem and joint-connectivity assumptions, the paper strengthens state-estimation convergence from asymptotic to exponential convergence.
- Observer design: The adaptive distributed observer estimates the leader state v and matrix S through follower variables v̂_i and Ŝ_i.This avoids requiring every follower to know S directly, and Ŝ_i depends on S only when the leader is a neighbor.
- Error dynamics: The observer errors are defined as ṽ_i = v̂_i − v and S̃_i = Ŝ_i − S, then stacked into global error vectors.The compact representation uses a block-diagonal matrix for the individual matrix-estimation errors.
- Prior result: Earlier results established exponential convergence of S̃ and only asymptotic convergence of ṽ under the relevant assumptions.The paper strengthens the latter result because exponential convergence is required to handle the external disturbance.
- Main observer result: The paper proves that ṽ(t) converges to zero exponentially under Assumptions 2.1 and 2.2 for any positive observer gains.The proof treats the observer error system as an exponentially stable switched system with an exponentially decaying perturbation.
- Proof mechanism: The proof constructs a positive definite bounded time-varying matrix P(t) and applies a Lyapunov function to establish exponential decay of the observer error.The argument accounts for the piecewise-continuous switching dynamics and exponentially vanishing estimation terms.
IV. MAIN RESULT
The paper combines an adaptive distributed observer with adaptive state feedback to solve leader-following consensus for higher-order nonlinear agents under jointly connected switching networks. The observer estimates the leader state and exogenous signals, while the resulting stability analysis establishes consensus under the stated assumptions.
- Adaptive distributed observer: The adaptive distributed observer estimates the leader state and exogenous signal for each follower under the jointly connected switching network.The observer is decomposed into estimates of the leader-state components and exogenous-system matrices.
- Adaptive distributed observer: The observer estimates converge to the leader state, with lim t→+∞(ˆxki(t) −xk0(t)) = 0 for k = 1, · · · , r.This follows from convergence of the observer errors ˜Si and ˜vi under the assumptions.
- Distributed adaptive control: The proposed distributed control law combines estimated leader information with adaptive parameter updates and stabilizing feedback for the higher-order nonlinear agents.The construction replaces unavailable leader signals with their estimates, yielding a distributed rather than decentralized implementation.
- Main result: Under Assumptions 2.1 and 2.2, Theorem 4.1 states that the leader-following consensus problem is solvable by the distributed control law.The proof uses a continuous Lyapunov function and the generalized Barbalat lemma for the piecewise-continuous closed-loop system.
- Consensus conclusion: The proof establishes limt→+∞si(t) = 0 for every follower and then shows convergence of the follower states to the leader state in every component.Boundedness of the relevant signals supports the Lyapunov analysis and the generalized Barbalat argument.
- Proof dependence: The stability proof relies critically on exponential convergence of the observer error; asymptotic convergence alone would not guarantee boundedness of V(t).This dependence is identified explicitly in Remark 4.3.
V. AN EXAMPLE
A van der Pol oscillator example illustrates the distributed control design under switching communication. The observer errors and follower tracking errors approach zero despite graphs that are individually disconnected.
- The example considers leader-following consensus for a group of four van der Pol systems.
- The leader and exosystem are represented in the forms required by the general system model.
- The switching communication graph satisfies the connectivity assumption even though all four individual digraphs are disconnected.
- Theorem 4.1 yields a distributed control law with μ1 = 3, μ2 = 12, β1 = 1, and ki = 3 for each follower.
- The observer estimation errors approach zero, and all follower states asymptotically approach the leader states.
VI. CONCLUSION
The paper solves leader-following consensus for higher-order nonlinear multi-agent systems with constant parameter uncertainties and external disturbances over jointly connected switching networks. It combines adaptive control with an adaptive distributed observer in a distributed state feedback law.
- The paper addresses higher-order nonlinear systems with constant parameter uncertainties and external disturbances under jointly connected switching networks.
- The proposed solution combines adaptive control with an established adaptive distributed observer in a distributed state feedback control law.
A. Digraph
This section defines digraphs, their adjacency representations, unions, and switching-graph construction. These definitions formalize the directed communication networks used in the paper.
- A digraph consists of a finite node set V and an edge set E contained in V × V.
- A directed spanning tree is a directed tree whose node set equals the full node set of the graph.
- The union of several digraphs retains their common node set and combines all constituent edge sets.
- A weighted adjacency matrix is nonnegative, has zero diagonal, and encodes an edge from j to i through aij > 0.
- A switching graph is generated by a piecewise constant switching signal selecting among a set of digraphs and their adjacency matrices.
B. Existence of the limit limt→+∞V (t)
The proof combines Lyapunov functions for observer and auxiliary error dynamics into a composite function. Positive definiteness, boundedness, and nonpositive derivatives establish convergence of the relevant limits.
- The observer component ˜z converges to zero exponentially by Lemma 3.1.
- The observer error dynamics are written in a compact form using the stacked state ˜X = col(˜z, ˜v) and function f(t, ˜X).
- A Lyapunov function V3 combines V1 and V2 through V3(t, ˜X) = l5V1(t, ˜z) + V2(t, ˜v).
- The bounded leader state allows the disturbance-related term to be bounded by a constant multiple of the observer error norm.
- The composite Lyapunov function ¯V3 is proper and positive definite, and its nonpositive derivative implies that limt→+∞¯V3(t) exists.
- Using U = V + ¯V3, the proof similarly concludes that limt→+∞U(t) exists because U is lower bounded and its derivative is nonpositive.