Lur'e类复杂动态网络优化牵制控制的几个重要问题研究

The pinning control problem for complex networks whose nodal dynamics satisfies the so-called QUAD conditions has been intensively studied in the past few decades. However, the pinning control problem of Lur'e type complex networks with nodal dynamics satisfying the sector conditions has not yet been addressed until present. Considering the fact that Lur'e systems play important roles in the field of control engineering, this project deeply investigates sevral important problems in the pinning control of Lur'e complex networks with directed topologies. The main research contents of this project include the following three aspects. First, some robust and reliable pinning control algorithms are designed for Lur'e networks described by ordinary differential equations and Lur'e networks with reaction-diffusion terms. Based on M-matrix theory, S-procedure and other related techniques, some easy-verified sufficient conditions are derived to guarantee that the Lur'e network can be pinned to a homogenous state. Second, by utilizing the set of maximum matching and some features of the network topology, we classify the nodes in a directed network topology into four categories including critical nodes, important nodes, ordinary nodes, and redundant nodes, and rearrange them in the descending order according to some prescribed rules. Third, we study the optimal combinations of pinned nodes and pinning feedback gains such that the Lur'e network can be pinned with a least number of pinned nodes and the smallest pinning feedback gains. For some particular networks, it may be necessary to perturb the network topology by adding very few critical links at suitable locations to reduce the number of pinned nodes. The results of this project can enrich the theoretical studies on the pinning control of complex networks, and also provide some potential application values for the optimized control of network systems.

节点动力学满足QUAD条件复杂网络的牵制控制研究已较深入，而节点非线性项满足扇区条件Lur'e类复杂网络的牵制控制研究近年才刚起步。考虑到Lur'e系统在控制领域中的重要地位，本项目针对有向拓扑下Lur'e复杂网络牵制控制的几个重要问题展开深入研究，主要包括：1）以常微分方程描述的Lur'e网络及具有反应扩散项的偏微分Lur'e网络为研究对象，设计鲁棒可靠的牵制控制算法，利用M矩阵和S过程等工具，给出易于验证的条件确保Lur'e网络的牵制控制；2）根据最大匹配集及拓扑结构特征，把有向网络的节点分为关键、重要、普通和冗余四类，并依据某些性能指标对节点进行排序；3）研究牵制节点与牵制增益的优化组合，必要时对拓扑采用微扰方式加少量关键边，以期用最少数目的节点和最小的增益实现Lur'e网络的牵制控制。本项目的成果可丰富完善复杂网络牵制控制的研究内容，同时对网络系统的优化控制也有一定的潜在应用价值。

本项目对Lur'e类复杂网络的牵制控制及多智能体系统的协同控制进行了深入研究，项目按计划书进展顺利，较好地完成了预定目标, 取得了一些研究成果，主要包括：1）研究了有向图拓扑下Lur'e复杂网络的牵制控制，利用M矩阵和S过程等工具，基于线性反馈和状态反馈设计了网络的牵制控制算法，给出了易于验证的条件以确保Lur'e网络的牵制控制；2) 考虑到节点全维信息的测量比较困难，本项目基于网络节点的输出信息研究了Lur'e网络的牵制控制问题，设计了分布式输出反馈控制器，实现了Lur'e复杂网络的牵制控制；3) 研究了节点动力学满足单边Lipschitz条件复杂网络的牵制控制问题，采用状态反馈和输出反馈策略，给出了一些充分条件以确保单边Lipschitz网络的牵制控制同步；4) 提出了两种蜂拥网络模型，设计了长距离吸引及短距离排斥的有界势能函数，研究了二阶多智能体系统的蜂拥控制问题；5) 二阶多智能体网络系统的一致性算法大都需要个体的速度信息，我们基于位置信息设计了基于降维观测器的一致性算法，并考虑有/无通讯时滞的情形，给出了实现网络一致性的充分必要条件；6）针对具有多个领导者与执行器饱和受限的二阶多智能体网络系统，基于M矩阵和牵制控制研究了网络的半全局及全局包含控制问题。在项目资助下，到目前为止，已发表了SCI期刊论文6篇（均标注了本项目的批准号），授权专利3项，软件著作权2项，培养了相关青年教师和本科专业学生。本项目组较为圆满地完成了既定任务，研究成果为解决Lur'e类复杂网络的牵制控制问题提供了一些新思路和新方法，并丰富了网络系统协同控制的研究内容。