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摘要: 本文研究了混杂多智能体系统在矩阵权重网络上的随机一致性问题.针对此类系统, 提出了一种基于采样信息的分布式随机一致性协议, 该协议采用异步成对更新机制, 有效降低了通信与计算需求.利用期望图理论和随机矩阵稳定性分析, 推出系统达到随机一致性的充分必要条件, 该条件与采样周期和期望拉普拉斯矩阵的零空间有关.特别地, 当反馈增益相同时, 给出了系统达到随机平均一致性的充要条件.此外, 通过分析误差系统的二阶矩收敛性, 借助马尔科夫不等式, 导出其收敛速度的$\epsilon$-一致性时间的解析上界.最后, 数值仿真验证了所提协议的可行性与理论结果的有效性.Abstract: This paper investigates the randomized consensus problem for hybrid multi-agent systems on matrix-weighted networks. For such systems, this paper proposes a distributed randomized consensus protocol based on sampled information, which employs an asynchronous pairwise update mechanism to effectively reduce communication and computational demand. By leveraging expected graph theory and stochastic matrix stability analysis, necessary and sufficient conditions are derived for the system to achieve consensus in expectation, which are related to the sampling period and the null space of the expected Laplacian matrix. Specifically, when the feedback gains are identical, necessary and sufficient conditions are provided for achieving randomized average consensus. Furthermore, by analyzing the second-moment convergence of the error system and using Markov's inequality, an analytical upper bound for the $\epsilon$-consensus time is derived to characterize its convergence speed. Finally, numerical simulations validate the feasibility of the proposed protocol and the effectiveness of the theoretical results.
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图 1 采样周期$\tau=0.35$下的状态轨迹
Fig. 1 State convergence trajectory under sampling period $\tau=0.35$
表 1 符号说明
Table 1 Notation
符号 含义 $\mathbb{Z}^{+}$ 正整数集 $\mathbb{R}^{+}$ 正实数集 $\mathbb{R}^d$ $d$维实列向量的集合 $I_d$ $d \times d$单位矩阵 $\boldsymbol{1}_N$ 元素全为1的$N$维列向量 $\boldsymbol{0}$ 元素全为0的向量 $\mathbf{0}_{d \times d}$ 元素全为0的矩阵 $\| \cdot \|$ 欧几里得范数 ${\cal{I}}_M$ 整数集合$\{1,\; \cdots,\; M\}$ ${\cal{I}}_N \setminus {\cal{I}}_M$ 整数集合$\{M+1,\; \cdots,\; N\}$ ${\cal{N}}(Q)$ 矩阵$Q$的零空间 ${\rm{range}}(Q)$ 矩阵$Q$的值域 ${\rm{diag}}(\cdot)$ 对角矩阵 ${\rm{blkdiag}}(\cdot)$ 分块对角矩阵 $\lambda_i(E)$ 矩阵$E$的第$i$个特征值 $\otimes$ 克罗内克积 
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