矩阵加权网络的强结构可控性
On the Strong Structural Controllability of Matrix-Weighted Networks
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中文总结 AI 辅助
本文针对矩阵加权网络的强结构可控性问题，提出基分解、层特定距离划分等方法，构建更紧的压缩定理，设计多项式时间自动发现算法，通过数值算例验证了相关定理。
中文摘要 AI 辅助
本文研究多智能体网络的强结构可控性。基于均匀划分的定义，建立了强结构可控子空间（SSCS）的上界；为反映状态维度大于1时矩阵权重的物理意义，采用高阶动力学对多智能体系统建模；针对矩阵奇异性与非对称耦合问题，提出矩阵空间基分解方法，将矩阵加权网络转化为分层标量网络；将该基分解扩展至下界估计，引入层特定距离划分（LDP），该公式建立了更紧的压缩定理，通过捕捉层特定结构延迟缩小可控子空间的数学边界；为系统识别最小化边界间隙的最优基，提出基于零空间投影的代数算法；引入模式矩阵与通用秩，严格证明该最优基在参数空间中几乎处处存在，与强结构可控性定义完全契合；为突破手动预定义目标的NP难组合瓶颈，提出基于多层Weisfeiler-Lehman（WL）颜色细化的多项式时间自动发现算法；最后评估网络的强结构可观测性与不变属性，通过含非对称矩阵权重及有向多层拓扑的数值算例验证所得定理。
英文摘要
This paper investigates the strong structural controllability of multi-agent networks. Based on the definition of equitable partitions, an upper bound for the strong structural controllable subspace (SSCS) is established. To reflect the physical significance of matrix weights where the state dimension is greater than one, the multi-agent system is modeled using higher-order dynamics. Furthermore, to address matrix singularity and asymmetric couplings, a matrix space basis decomposition method is proposed to transform the matrix-weighted network into layered scalar networks. Additionally, by extending this basis decomposition to the lower bound estimation, a layer-specific distance partition (LDP) is introduced. This formulation establishes a tighter Squeeze Theorem, narrowing the mathematical boundaries for the controllable subspace by capturing layer-specific structural delays. To systematically identify the optimal basis that minimizes the bounds gap, an algebraic algorithm based on null-space projection is formulated. Furthermore, by introducing pattern matrices and generic rank, the almost-everywhere existence of this optimal basis in the parameter space is rigorously proved, perfectly aligning with the definition of strong structural controllability. To break the NP-hard combinatorial bottleneck of manually pre-defining the targets, a polynomial-time automated discovery algorithm based on the multi-layer Weisfeiler-Lehman (WL) color refinement is proposed. Finally, the strong structural observability and invariant attributes of the network are evaluated. Numerical examples with asymmetric matrix weights and directed multi-layer topologies are provided to verify the derived theorems.
