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摘要: 研究了节点状态为高维的多层复杂网络系统的能控性问题. 讨论了节点的异质性、层间耦合和层内耦合对网络能控性的影响. 研究发现当节点状态由同质变为异质、内耦合矩阵由相同变为不同时, 对网络能控性均有影响(网络可以由能控变为不能控, 反之亦然). 对层间耦合模式为驱动响应模式和相互依赖模式, 分别给出了网络系统能控的充分条件或必要条件. 相比于直接应用经典的能控性判据, 这些条件更易于验证, 且驱动响应模式比相互依赖模式实现系统完全能控所需的条件更弱.Abstract: This paper studies the controllability of multi-layer complex network systems with high-dimensional node states. We discuss the influence of node heterogeneity, inter-layer coupling, and intra-layer coupling on the controllability of the network. It is found that when the state of the node changes from homogeneous to heterogeneous or the inner-coupling matrix changes from identical to nonidentical, both have an impact on the controllability of the network (the network can be changed from controllable to uncontrollable, and vice versa). The sufficient or necessary conditions for the controllability of the networked system when the inter-layer coupling mode is the drive response mode and the interdependence mode are given, respectively. These conditions are easier to verify than using the classic controllability criteria directly, and the conditions that the drive response mode to achieve fully controllability are weaker than the conditions for the interdependence mode.
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图 1 层间耦合为驱动响应模式的两层网络
Fig. 1 Two-layer networks with the inter-layer drive-response couplings
图 2 层间耦合为相互依赖模式的两层网络
Fig. 2 Two-layer networks with the inter-layer interdependent couplings
图 3 层间耦合为驱动响应模式的M层网络结构示意图
Fig. 3 The illustration of M-layer networks with the inter-layer drive-response couplings
图 4 层间耦合为相互依赖模式的3层网络结构示意图
Fig. 4 The illustration of three-layer networks with the inter-layer interdependent couplings
图 5 驱动层为链网络、响应层为星形网络的两层网络
Fig. 5 Two-layer networks with the topology of the drive-layer is a chain network, and the response layer is a star network
表 1 本文模型中所用的特殊符号
Table 1 Special notations used in the model of this paper
特殊符号 含义 ${\boldsymbol{A}}_i^K$ 第$ K $层网络的第$ i $个节点的状态矩阵 ${\boldsymbol{B}}_i^K$ 第$ K $层网络的第$ i $个节点的输入矩阵 ${\boldsymbol{C}}_i^K$ 第$ K $层网络的第$ i $个节点的输出矩阵 ${\boldsymbol{H}}_i^K$ 第$ K $层网络的第$ i $个节点与该层其他节点之间的内耦合矩阵 ${\boldsymbol{H}}_i^{KJ}$ 第$ J $层网络的第$ i $个节点与第$ K $层网络的其他节点之间的内耦合矩阵 ${{\boldsymbol{W}}^K}$ 第$ K $层的网络拓扑 ${{\boldsymbol{D}}^{KJ} }$ 第$ J $层到第$ K $层的网络拓扑 ${\boldsymbol{x}}$ 整个网络系统的状态 ${{\boldsymbol{x}}^K}$ 第$ K $层网络的状态 ${\boldsymbol{u}}$ 整个网络系统的输入 ${{\boldsymbol{u}}^K}$ 第$ K $层网络的输入 ${\boldsymbol{ \Phi}}$ 整个网络系统的状态矩阵 ${{\boldsymbol{\Phi}} _{KK} }$ 第$ K $层网络的状态矩阵 ${{\boldsymbol{\Phi}} _{KJ} }$ 第$ J $层到第$ K $层网络的状态矩阵 ${\boldsymbol{\Psi}}$ 整个网络系统的输入矩阵 ${\boldsymbol{ \Xi}}$ 整个网络系统的输出矩阵 ${{\boldsymbol{\Lambda}} ^K}$ 以${\boldsymbol{A}}_1^K,\cdots,{\boldsymbol{A}}_N^K$为对角元的分块对角矩阵 ${\boldsymbol{ \Delta}}$ 对角矩阵${ {\text{diag} } } \{ {{\boldsymbol{\Delta}} ^1},\cdots,{{\boldsymbol{\Delta}} ^M}\}$ ${{\boldsymbol{\Delta}} ^K}$ 对角矩阵${ {\text{diag} } } \{ \delta _1^K,\cdots,\delta _N^K\}$ 
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