二阶网络系统的结构保持降维方法
Reduction of Second-Order Network Systems with Structure Preservation
AI总结：
本文提出基于图聚类的二阶网络系统结构保持降维框架，通过贪心分层聚类算法将相似动力学的顶点分组，并利用Petrov-Galerkin方法生成降阶模型，同时推广了网络可控性Gramian定义并计算H2范数及误差。
摘要：
本文提出了一种基于图聚类的二阶网络系统结构保持降维通用框架。在该方法中，顶点动力学通过输入到个体状态的传递函数捕捉，顶点差异性通过传递函数差异的H2范数量化。提出一种贪心分层聚类算法将具有相似动力学的顶点分到同一簇。然后通过Petrov-Galerkin方法生成降阶模型，其中投影由所得网络聚类的特征矩阵形成。证明简化系统保持互联结构，即可再次解释为在缩减图上演化的二阶系统。此外，本文将网络可控性Gramian的定义推广到二阶网络系统。基于此，开发了高效的H2范数计算方法并推导了全阶与降阶模型之间的近似误差。最后，通过小世界网络的例子说明了该方法。
英文摘要：
This paper proposes a general framework for structure-preserving model reduction of a secondorder network system based on graph clustering. In this approach, vertex dynamics are captured by the transfer functions from inputs to individual states, and the dissimilarities of vertices are quantified by the H2-norms of the transfer function discrepancies. A greedy hierarchical clustering algorithm is proposed to place those vertices with similar dynamics into same clusters. Then, the reduced-order model is generated by the Petrov-Galerkin method, where the projection is formed by the characteristic matrix of the resulting network clustering. It is shown that the simplified system preserves an interconnection structure, i.e., it can be again interpreted as a second-order system evolving over a reduced graph. Furthermore, this paper generalizes the definition of network controllability Gramian to second-order network systems. Based on it, we develop an efficient method to compute H2-norms and derive the approximation error between the full-order and reduced-order models. Finally, the approach is illustrated by the example of a small-world network.
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