随机耦合的时滞二阶多智能体系统的一致性研究

consensus of multi-agent systems has attracted considerable attention because of its broad applications in many fields. Because multi-agent systems are inevitably subject to noise and communication time delays, it is important to understand the effct of noise and time delays on the consensus of multi-agent systems. In the past few years, the consensus problem of multi-agent systems in noise environments with or without time delays has been extensively studied. In this project we focus on two basic questions: Can consensus be achieved merely with noise coupling? Can consensus be achieved in finite time with noise coupling and communications time delays? These question will be addressed in this project. In addition, the second-order consensus problem is more complicated and challenging than the first-order case which has been extensively studied. The main goal of this project is to provide an analytical treatment for the consensus problem of second-order multi-agent systems with noise coupling and time delays. First, some analytically sufficient conditions for noise-induced consensus in second-order multi-agent systems is established based on the stochastic stability theory. And the effect of time delays and network topologies on the noise-induced consensus is investigated numerically. Further，the finite-time consensus of second-order multi-agent systems is investigated by using the finite-time stability theory, the optimization of convergence time is also considered. The successful implementation of this project will reveal the underlying mechanisms responsible for the effect of noise and time delays on consensus, and might make some contribution to the development of the theory of consensus in multi-agent systems.

多智能体系统的一致性在许多领域中有着广泛的应用，近年来受到众多研究者的关注。多智能体系统不可避免会受到噪声和通信时滞的影响。尽管噪声环境中时滞或非时滞多智能体系统的一致性问题已有研究，然而尚有许多问题有待解决。如，多智能体系统能否仅通过噪声耦合达到一致？随机耦合的时滞多智能体系统能否在有限时间内达到一致？此外，与一阶系统相比二阶系统的一致性问题是一个更加复杂而富有挑战性的研究课题。本项目将利用严格的理论分析研究随机耦合的时滞二阶多智能体系统的一致性问题。首先，利用随机稳定性理论给出噪声耦合诱导系统实现一致性的条件，并分析时滞和网络拓扑对噪声作用的影响；然后利用有限时间稳定性理论研究噪声耦合的时滞二阶多智能体系统的有限时间一致性问题，并研究系统收敛时间的优化问题。本项目的实施将阐明多智能体系统趋于一致的过程中噪声和时滞效应的产生机制，对推动多智能体系统一致性理论的发展具有十分重要的意义。

环境噪声与通信时滞对多智能体系统的收敛速度与收敛时间有着明显的影响。本项目利用严格的理论分析研究了噪声耦合的时滞多智能体系统的平均一致性问题以及有限时间随机一致性问题。首先，我们研究了具有通信时滞以及噪声连接的多智能体系统的平均一致性问题。考虑了两种不同的网络拓扑：固定拓扑与切换拓扑。得到了多智能体系统以概率一趋于一致的充分条件。研究表明：当噪声强度较弱时，多智能体系统可以承受较大的通信时滞；同样如果通信强度较小时系统可以承受较强的噪声扰动，同时研究表明多智能体系统的收敛速度与噪声强度呈反比。进而，研究了具有固定拓扑与切换拓扑的多智能体系统的有限时间一致性问题。为了使多智能体系统在有限时间内趋于一致， 我们构建了一个连续的非Lipschitz的控制器，给出了多智能体系统在有限时间内趋于一致的充分条件以及系统收敛时间上界的解析估计。研究表明：如果多智能体网络拓扑为强连通的且网络图的Laplacian矩阵的第二小特征值与噪声强度矩阵的最大特征值的比值充分大则系统可以在有限时间内趋于一致。且系统的收敛速度与网络图的Laplacian矩阵的第二小特征值成正比，与噪声强度矩阵的最大特征值呈反比。