Reduced Order Modeling of Diffusively Coupled Network Systems: An Optimal Edge Weighting Approach

Abstract

This paper studies reduced-order modeling of dynamic networks with strongly connected topology. Given a graph clustering of an original complex network, we construct a quotient graph with less number of vertices, where the edge weights are parameters to be determined. The model of the reduced network is thereby obtained with parameterized system matrices, and then an edge weighting procedure is devised, aiming to select an optimal set of edge weights that minimizes the approximation error between the original and the reduced-order network models in terms of H2-norm. The effectiveness of the proposed method is illustrated by a numerical example.

Reduced-Order Modeling of Large-Scale Network Systems

“…Moreover, 𝑃 e := 𝑋 ⊗ 𝐾 −1 and 𝑄 f := 𝑌 ⊗ 𝐾 are a generalized controllability Gramian of Σ e in (28) and a generalized observability Gramian of Σ f in (29), respectively, where 𝐾 satisfies (24) for the passive subsystems.…”

“…Generally, all the existing clustering-based reduction methods fall into the framework of Petrov-Galerkin Projections. In [25,24], an ℋ 2 optimal approach is presented, which does not aim to find a suitable graph clustering. Instead, this approach focuses on how to construct a "good" reduced-order model for a given clustering.…”

“…Then, the pair (𝜓, −𝜙) is a psd-convex-concave decomposition of Φ [24]. The bilinear matrix inequality (47) with the nonlinearity term Ā⊤ 𝑟 Ā𝑟 can be handled using such a decomposition, which can linearize the optimization problem (49) at a stationary point Ŵ [31].…”

“…Let 𝑋 > 0 be the unique solution of (59a), and let 𝑌 > 0 be a solution of (59b). Let 𝐾 𝑚 > 0 and 𝐾 𝑀 > 0 be the minimum and maximum solutions of (24), respectively. Then, the matrices 𝒳 := 𝑋 ⊗ 𝐾 − interpretation in the reduced-order model Σ, the following theorem is provided in [21], which states that there exits a similarity transformation between Γ and an undirected graph Laplacian matrix.…”

“…Subsequent research in [12,22,17,19] show that the dissimilarity-based clustering method can also be extended to second-order networks, controlled power networks, and directed networks. In [25,24], a framework is presented on how to build a reduced-order model from a given clustering. The edge weights in the reduced network are parameterized so that an optimization problem is formulated to minimize the reduction error.…”