Efficient model order reduction for multi-agent systems using QR decomposition-based clustering

11 Reduced-order modeling of large-scale network systems

“…Then a projected gradient algorithm can be employed to solve this optimization problem with mathematical guarantees on its convergence. Related to the work in [58], a combination of the Krylov subspace method with graph clustering is proposed in [54], where a reduced basis is firstly found by the iterative rational Krylov algorithm, and then a graph partition is obtained by the QR decomposition with column pivoting on the projection matrix. An alternative graph-based model reduction method is proposed in [49], which finds a graph clustering based on the edge agreement protocol of a network (see the definition in [79]) and provides a greedy contraction algorithm as a suboptimal solution of graph clustering.…”

“…However, finding almost equitable partitions itself is rather difficult and computationally expensive for general graphs. Clustering can also be found using the QR decomposition with column pivoting on the projection matrix obtained by the Krylov subspace method [54]. For undirected networks with tree topology, an asymptotically stable edge system can be considered, which has a pair of diagonal generalized Gramian matrices for characterizing the importance of edges.…”

Clustering approach to model order reduction of power networks with distributed controllers

“…As an extension, [37] suggests to use the almost equitable partition (AEP) as a clustering of the underlying graph, but finding AEPs is generally a very difficult problem and computationally expensive, which limits this method in practical applications. The results in [9,35] can be regarded as the combination of graph clustering and conventional model reduction techniques, balanced truncation, and Krylov subspace methods, respectively. However, it is still a challenge to apply these ideas to second-order systems, which are common model settings in the context of power networks.…”

“…Compute (C i , C j ) by (35) 8: In principle, hierarchical clustering does not require preliminary knowledge about the number and the size of clusters. In practice, it generates a dendrogram that organizes a hierarchy of clusters in a tree structure.…”

Model reduction of linear multi-agent systems by clustering with $$\varvec{\mathcal {H}_2}$$ H 2 and $$\varvec{\mathcal {H}_\infty }$$ H ∞ error bounds

“…Model reduction techniques specifically for networked multi-agent systems with first-order agents have been proposed in [6,15,16,22]. Extensions to second-order agents have been considered in [7,14] and to more general higher-order agents in [4,17,23,25].…”