Distributed Time-Varying Optimization for Euler-Lagrange Swarm Intelligent Systems over Unbalanced Directed Graphs

ZHANG Yunfei; LUAN Meng; ZHAO Dan; HUANG Di; HUANG Tingwen

doi:10.12263/DZXB.20251219

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Distributed Time-Varying Optimization for Euler-Lagrange Swarm Intelligent Systems over Unbalanced Directed Graphs

The Theory and Application of Swarm Intelligence Technology in the Information\-Rich Era | 更新时间：2026-06-16

- Distributed Time-Varying Optimization for Euler-Lagrange Swarm Intelligent Systems over Unbalanced Directed Graphs
- ACTA ELECTRONICA SINICA Vol. 54, Issue 3, Pages: 938-946(2026)
- 作者机构：
  
  1.辽宁工程技术大学理学院，辽宁阜新 123032
  2.岭南大学数据科学学院，香港特别行政区
  3.东南大学自动化学院，江苏南京 210096
  4.湖北珞珈实验室，湖北武汉 430072
  5.深圳理工大学计算机科学与控制工程学院，广东深圳 518107
- 作者简介：
- 基金信息：
  
  National Natural Science Foundation of China(62325304;U2541220);Basic Research Program of Jiangsu(BK20253020);Key Research and Development Program of Zhejiang Province(2025C01055);Jiangsu Provincial Scientific Research Center of Applied Mathematics(BK20233002)
- DOI：10.12263/DZXB.20251219
  CLC： TP273;
- Received：25 January 2026，
  
  Accepted：24 February 2026，
  
  Published：25 March 2026
- 稿件说明：
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张云飞, 栾萌, 赵丹, 等. 非平衡有向图下欧拉-拉格朗日群体智能系统分布式时变优化[J]. 电子学报, 2026, 54(03): 938-946.

ZHANG Yunfei, LUAN Meng, ZHAO Dan, et al. Distributed Time-Varying Optimization for Euler-Lagrange Swarm Intelligent Systems over Unbalanced Directed Graphs[J]. Acta Electronica Sinica, 2026, 54(03): 938-946.
张云飞, 栾萌, 赵丹, 等. 非平衡有向图下欧拉-拉格朗日群体智能系统分布式时变优化[J]. 电子学报, 2026, 54(03): 938-946. DOI：10.12263/DZXB.20251219

ZHANG Yunfei, LUAN Meng, ZHAO Dan, et al. Distributed Time-Varying Optimization for Euler-Lagrange Swarm Intelligent Systems over Unbalanced Directed Graphs[J]. Acta Electronica Sinica, 2026, 54(03): 938-946. DOI：10.12263/DZXB.20251219

摘要

本文研究非平衡有向通信图下欧拉-拉格朗日群体智能系统的分布式时变优化问题。考虑由多个具有欧拉-拉格朗日动力学的智能体组成的群体智能系统，其信息交互建模为强连通且非平衡的有向图，每个智能体拥有私有的局部时变代价函数，目标是为每个智能体设计控制输入，使所有智能体实时状态达成一致并共同收敛至全局代价函数的时变最优解。实际应用中，通信拓扑非平衡导致的加权一致性偏差、时变代价函数引起的解漂移，与系统固有的非线性及参数不确定性交织耦合，为设计分布式优化算法带来了严峻的理论挑战。为克服上述挑战，本文提出一种融合分布式优化器与自适应控制器的新型双层算法框架。在优化层，设计了一种适用于非平衡有向图的分布式优化器，通过在线估计拉普拉斯矩阵零特征值对应的左特征向量以消除通信拓扑的非平衡性影响，同时引入时变梯度补偿器以动态修正最优轨迹漂移，生成参考速度信号。在控制层，针对每个智能体设计了基于系统参数自适应律的跟踪控制器，利用欧拉-拉格朗日系统的线性参数化性质在线更新未知参数估计值，使智能体的实际速度能够精确跟踪参考速度，从而抑制模型不确定性对跟踪性能的影响。为分析算法的收敛性，首先，利用输入-状态稳定性理论分析优化器的动态演化，证明当跟踪误差有界且收敛时智能体状态能够达成一致；其次，构造Lyapunov函数分析最优跟踪步骤，结合Barbalat引理证明全局梯度之和渐近趋于零；基于此，严格证明了当控制器参数满足给定的不等式条件时，所有智能体的实时状态全局渐近收敛于时变最优轨迹；最后，通过包含十个双连杆机械臂的数值仿真验证了所提算法的有效性，仿真结果表明所有智能体的关节角度轨迹均能快速、准确地跟踪最优轨迹。

Abstract

This paper investigates a distributed time-varying optimization problem for Euler-Lagrange swarm intelligent systems over unbalanced directed communication topologies. Consider a swarm intelligent system composed of multiple agents with Euler-Lagrange dynamics

where information exchange is modeled as a strongly connected and unbalanced directed graph

and each agent possesses its own private local time-varying cost function. The objective is to design control inputs for each agent such that all agents’ real-time states achieve consensus and collectively converge to the time-varying optimal solution of the global cost function. In practical applications

the coupling of weighted consensus bias caused by unbalanced communication topologies

solution drift induced by time-varying cost functions

and inherent system nonlinearities and parameter uncertainties presents severe theoretical challenges for the design of distributed optimization algorithms. To overcome the aforementioned challenges

this paper proposes a novel dual-layer algorithm framework that integrates a distributed optimizer and an adaptive controller. In the optimization layer

a distributed optimizer tailored for unbalanced directed graphs is designed. It eliminates the impact of communication topology imbalance by online estimation of the left eigenvector associated with the zero eigenvalue of the Laplacian matrix

while introducing a time-varying gradient prediction compensation term to dynamically correct the optimal trajectory drift

thus generating a reference velocity signal. In the control layer

a tracking controller with a system parameter adaptation law is designed for each agent. Utilizing the linear parameterization property of Euler-Lagrange systems

it updates estimates of unknown parameters online

enabling the agents’ actual velocities to accurately track the reference velocity

thus mitigating the impact of model uncertainties on tracking performance. To analyze the convergence of the proposed algorithm

first

the input-to-state stability theory is utilized to analyze the optimizer dynamics

demonstrating that the states of agents achieve consensus when the tracking error is bounded and convergent. Second

a Lyapunov function is constructed to analyze the optimal tracking step

and Barbalat’s lemma is applied to prove that the sum of global gradients asymptotically approaches zero. Consequently

it is rigorously proven that the real-time states of all agents globally and asymptotically converge to the time-varying optimal trajectory provided the controller parameters satisfy the certain inequality conditions. Finally

the effectiveness of the proposed algorithm is validated through numerical simulations involving ten two-link manipulators. The results demonstrate that the joint angle trajectories of all agents can accurately track the optimal trajectory.

关键词

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references

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