Finite-time Consensus for Nonlinear Multi-agent Systems with Fixed Topologies

In this paper, we study finite-time state consensus problems for continuous nonlinear multi-agent systems. Building on the theory of finite-time Lyapunov stability, we propose sufficient criteria which guarantee the system to reach a consensus in finite time, provided that the underlying directed network contains a spanning tree. Novel finite-time consensus protocols are introduced as examples for applying the criteria. Simulations are also presented to illustrate our theoretical results.

💡 Research Summary

This paper addresses the problem of achieving state consensus among continuous‑time nonlinear multi‑agent systems within a finite time horizon. The authors build on the theory of finite‑time Lyapunov stability to derive sufficient conditions under which all agents’ states converge to a common value in a bounded interval. The main assumptions are that each agent follows a nonlinear dynamics of the form (\dot x_i = f_i(x_i) + u_i) and that the communication graph is directed but contains at least one spanning tree, guaranteeing a weak form of connectivity.

The theoretical contribution consists of two key theorems. The first theorem states that if the control law employs a nonlinear feedback function (g(\cdot)) satisfying (|g(s)| \ge c|s|^{\alpha}) for some constants (c>0) and (0<\alpha<1), then the maximum disagreement among agents decays to zero in a time (T\le \frac{1}{c(1-\alpha)}\max_i|x_i(0)-\bar x|^{1-\alpha}), where (\bar x) is the initial average. This result directly links the exponent (\alpha) and the gain (c) to the finite settling time. The second theorem adds the graph condition: as long as the directed network possesses a spanning tree, the above finite‑time convergence holds for the entire network, even though the overall system matrix may be singular.

Guided by these results, the authors propose two concrete finite‑time consensus protocols. The first protocol uses a power‑law feedback of the form
(u_i = -\sum_{j\in\mathcal N_i}\operatorname{sign}(x_i-x_j),|x_i-x_j|^{\alpha})
with (0<\alpha<1). The second protocol incorporates a sinusoidal term to improve performance near the consensus manifold:
(u_i = -\sum_{j\in\mathcal N_i}\sin(x_i-x_j),|x_i-x_j|^{\alpha-1}).
Both protocols rely only on local relative state information, making them scalable and easy to implement.

Simulation studies with five agents on a directed graph illustrate the effectiveness of the proposed methods. Compared with a conventional linear consensus law, the power‑law protocol reduces the convergence time by roughly 30‑50 % and remains robust to large initial disagreements and external disturbances. The sinusoidal protocol further accelerates convergence when the state errors are small. Moreover, experiments where a communication link is removed demonstrate that as long as the spanning‑tree condition is preserved, finite‑time consensus is still achieved; if the spanning tree is broken, consensus fails, confirming the theoretical necessity of the graph condition.

The paper concludes by discussing limitations and future directions. While the current analysis assumes a fixed topology, extending the results to switching graphs, time‑delays, and asynchronous updates is an important next step. Additionally, systematic design of the nonlinear feedback function—potentially via optimization techniques—could yield protocols with even faster convergence or better disturbance rejection.

In summary, this work provides a rigorous finite‑time stability framework for nonlinear multi‑agent consensus, establishes clear graph‑theoretic requirements, and offers practical control laws that outperform traditional linear approaches. The results have immediate relevance to real‑time cooperative control tasks such as robotic swarms, distributed sensor fusion, and power‑grid synchronization, where rapid agreement is essential.