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Eigenstructure analysis from symmetrical graph motives with application to aggregated controller design

In this paper, we analyze the eigenstructure of network systems having symmetrical graph motives and apply it to reduced order controller design based on their aggregated models. In the eigenstructure analysis, formulating the symmetry of graph motives as the graph automorphism, we show that particular eigenspace decomposition of network systems can be found by analyzing the common eigenspaces of all possible permutation matrices, with regard to the graph automorphism. This eigenspace decomposition explains the appearance of uncontrollable and unobservable subspaces that can be removed by aggregating, i.e., averaging, symmetrical graph motives. Furthermore, it turns out that the resultant aggregated model, whose state behavior tracks a kind of centroids of that of the original network system, has good compatibility with observer-based state feedback controller design. The efficiency of the aggregated controller design method is numerically demonstrated by output regulation of second-order oscillator networks.