$H^2$-Optimal Reduction of Positive Networks using Riemannian Augmented Lagrangian Method

In this study, we formulate the model reduction problem of a stable and positive network system as a constrained Riemannian optimization problem with the $H^2$-error objective function of the original and reduced network systems. We improve the reduction performance of the clustering-based method, which is one of the most known methods for model reduction of positive network systems, by using the output of the clustering-based method as the initial point for the proposed method. The proposed method reduces the dimension of the network system while preserving the properties of stability, positivity, and interconnection structure by applying the Riemannian augmented Lagrangian method (RALM) and deriving the Riemannian gradient of the Lagrangian. To check the efficiency of our method, we conduct a numerical experiment and compare it with the clustering-based method in the sense of $H^2$-error and $H^\infty$-error.