二阶多智能体系统的弹性平均一致性算法及其应用

方崇荣; 还约辉; 郑文喆; 包贤晨; 李政

doi:10.11999/JEIT251155

二阶多智能体系统的弹性平均一致性算法及其应用

doi: 10.11999/JEIT251155 cstr: 32379.14.JEIT251155

1.
上海交通大学自动化与感知学院上海，200400
2.
浙江国利网安科技有限公司宁波 315000

基金项目: 浙江省“尖兵领雁+X”研发攻关计划(2024SJCZX0003, 2024SJCZX0004)

详细信息

作者简介:
方崇荣：男，副教授，研究方向为信息物理系统安全

还约辉：男，总经理，研究方向为工业控制系统安全

郑文喆：男，硕士，研究方向为多智能体系统安全

包贤晨：男，技术经理，研究方向为工业信息安全

李政：男，博士在读，研究方向为信息物理系统攻击检测

通讯作者:
还约辉　huanyuehui@guolisec.com

中图分类号: TP278
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- 被引次数: 0
出版历程
- 修回日期: 2025-12-31
- 录用日期: 2025-12-31
- 网络出版日期: 2026-01-15

Resilient Average Consensus for Second-Order Multi-Agent Systems: Algorithms and Application

1.
School of Automation and Intelligent Sensing, Shanghai Jiao Tong University , Shanghai, 200400, China
2.
Zhejiang Guoli Security Technology Co. Ltd., Ningbo, 315000, China

Funds: Pioneer Leading Goose+X R&D Program of Zhejiang (2024SJCZX0003, 2024SJCZX0004)

摘要

摘要: 多智能体系统是实现协同协作的重要途径，而弹性一致性则是其安全支撑技术之一。该文针对二阶多智能体系统，研究在存在异常行为节点(包括恶意攻击和意外故障)情况下保证系统协同性能的弹性平均一致性问题。该问题面临双重挑战：如何实现分布式异常检测，以及如何通过一维加速度输入精确补偿二维状态误差。为解决该问题，该文首先推导了二阶平均一致性实现的充分条件。基于此，通过引入两跳通信信息设计了分布式检测机制，并提出能够精确补偿有限控制输入误差的方案。针对输入、速度、位置维度可能遭受的持续攻击，进一步提出具有容错机制的扩展算法。理论证明表明，所提算法能使节点在存在异常节点情况下渐近实现二阶平均一致性。最后，通过大量数值仿真和实验验证了所提方法的有效性。
- 二阶多智能体系统 /
- 弹性一致性 /
- 攻击检测 /
- 误差补偿.
Abstract: Objective Multi-agent systems (MASs) are pivotal for collaborative tasks in dynamic environments, with consensus algorithms serving as a cornerstone for applications like formation control. However, MASs are vulnerable to misbehaviors (e.g., malicious attacks or accidental faults), which can disrupt consensus and compromise system performance. While resilient consensus methods exist for first-order systems, they are inadequate for second-order MASs, where agents’ dynamics involve both position and velocity. This work addresses the gap by developing a resilient average consensus framework for second-order MASs that ensures accurate collaboration under misbehaviors. The primary challenges include distributed error detection and compensating two-dimensional state errors (position and velocity) using one-dimensional acceleration inputs. Methods The study first derives sufficient conditions for second-order average consensus under misbehaviors, leveraging graph theory and Lyapunov stability analysis. The system is modeled as an undirected graph $ \mathcal{G}=(\mathcal{V},\mathcal{E}) $, where agents follow double-integrator dynamics. Two algorithms are proposed: Finite Input-Errors Detection-Compensation (FIDC): For finite control input errors, detection strategies (1 and 2) use two-hop communication information to identify discrepancies in neighbors’ states or control inputs. Compensation Scheme I designs input sequences to satisfy consensus conditions (Corollary 1). Infinite Attack Detection-Compensation (IADC): For infinite errors in control input, velocity, and position, detection strategies are extended to identify falsified data. Compensation Schemes 2 and 3 mitigate errors, while an exponentially decaying error bound isolates persistent attackers. The algorithms are distributed and require no global knowledge. Results and Discussions Simulations on a 10-agent network validate the algorithms’ efficacy. Under FIDC, agents achieve exact average consensus despite finite input errors from malicious and faulty agents (Fig. 5). IADC ensures consensus among normal agents after isolating malicious ones exceeding the error bound (Fig. 6). Experimental evaluations on a multi-robot platform demonstrate resilience against real-world faults (e.g., actuator failures) and attacks (e.g., false data injection). In fault scenarios, FIDC reduces formation center deviation from 180mm to 34mm (Fig. 8). For attacks, IADC isolates malicious robots, allowing normal agents to converge correctly (Fig. 9). Discussions on relaxing Assumption 1 (non-adjacent misbehaving agents) reveal that Detection Strategy 3 and majority voting can handle certain connected malicious topologies (Fig. 3–Fig. 4), though complex cases require further study. Conclusions This work proposes a novel resilient average consensus framework for second-order MASs. Theoretically, sufficient conditions ensure consensus under misbehaviors, while FIDC and IADC algorithms enable distributed detection, compensation, and isolation of errors. Simulations and physical experiments confirm that the methods achieve accurate average consensus against both finite and infinite errors. Future work will explore extensions to directed networks, time-varying topologies, and higher-dimensional systems.
- Second-order multi-agent system /
- resilient consensus /
- attack detection /
- compensation

HTML全文

图 1 弹性平均一致性方法的整体设计框架

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图 2 典型拓扑结构

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图 3 FIDC算法下的弹性一致性性能表现

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图 4 IADC算法下的弹性一致性性能表现

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图 5 多机器人平台架构^[23]

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图 6 故障存在时多机器人系统节点运动轨迹

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图 7 攻击存在时多机器人系统节点运动轨迹

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1 FIDC

1：初始化：设置补偿周期$ {T}_{c} $，并交换各节点的邻居数量$ \|{\mathcal{N}}_{j}\| $信息
2：$ \rm{f}\rm{o}\rm{r} $$ k=0\colon {\mathrm{Max}}\_ {\mathrm{time}} $ $ \rm{d}\rm{o} $
3：　　$ \rm{f}\rm{o}\rm{r} $ $ i\in {\mathcal{N}}_{j} $ $ \rm{d}\rm{o} $
4：　　　节点$ j $基于公式计算节点$ i $引起的误差
5：　　　$ \rm{i}\rm{f} $ $ \varepsilon _{i}^{j}(k)\neq 0 $
6：　　　　通过公式计算补偿序列$ {\varepsilon }_{j}(k+l\|k),l=1,2,\cdots ,{T}_{c} $
7：　　$ \rm{e}\rm{n}\rm{d} $ $ \rm{i}\rm{f} $
8：　$ \rm{e}\rm{n}\rm{d} $$ \boldsymbol{f}\boldsymbol{o}\boldsymbol{r} $
9：　通过公式和公式(6)分别计算$ {\varepsilon }_{j}(k+1) $和控制输入并施加
10：更新$ {v}_{j}(k+2) $和$ {\tilde{r}}_{j}(k+2) $状态值
11：$ \rm{e}\rm{n}\rm{d} $ $ \rm{f}\rm{o}\rm{r} $

下载: 导出CSV

2 IADC

1：初始化：设置补偿周期$ {T}_{c} $，并交换各节点的邻居数量$ \|{\mathcal{N}}_{j}\| $信息
2：$ \rm{f}\rm{o}\rm{r} $$ k=0\colon {\mathrm{Max}}\_ {\mathrm{time}} $ $ \rm{d}\rm{o} $
3：　$ \rm{f}\rm{o}\rm{r} $ $ i\in {\mathcal{N}}_{j} $ $ \rm{d}\rm{o} $
4：　　通过公式计算$ i $在控制输入、速度和位置上的误差。
5：　　$ \rm{i}\rm{f} $ $ {\omega }_{i}(k)\neq 0 $或$ {\delta }_{i}(k)\neq 0 $或$ \varepsilon _{i}^{j}(k)\neq 0 $
6：　　　根据公式和获取补偿序列。
7：　　$ \rm{e}\rm{n}\rm{d} $ $ \rm{i}\rm{f} $
8：　　$ \rm{i}\rm{f} $ $ \varepsilon _{i}^{j}(k)\geq {\alpha }_{j}\rho _{j}^{k} $或$ {\omega }_{i}(k)\geq {\alpha }_{j}\rho _{j}^{k} $或$ {\delta }_{i}(k)\geq {\alpha }_{j}\rho _{j}^{k} $
9：　　　节点$ j $隔离节点$ i $，通过公式计算补偿序列。
10：　 $ \rm{e}\rm{n}\rm{d} $ $ \rm{i}\rm{f} $
11： $ \rm{e}\rm{n}\rm{d} $$ \rm{f}\rm{o}\rm{r} $
12：通过公式计算$ {\varepsilon }_{j}(k+1) $，基于公式计算控制输入并施加
13：更新$ {v}_{j}(k+2) $和$ {\tilde{r}}_{j}(k+2) $状态值
14：$ \rm{e}\rm{n}\rm{d} $ $ \rm{f}\rm{o}\rm{r} $

下载: 导出CSV

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