A new controllability Gramian for semistable systems and its application to approximation of directed networks

In this paper, we propose a new definition of a controllability Gramian for semistable systems, which is then used for model reduction of network systems. The system under consideration is modeled as single integrators that interconnected with each other according to a connected directed network. In the proposed method, the complexity of the network is reduced through a graph clustering method which aggregates the vertices if they respond similarly with respect to external inputs. Here, the similarity of the vertices is computed based on the new Gramian. The reduced-order model is obtained by a Petrov-Galerkin projection where the projection matrices are constructed from the resulting clustering. The reduced system preserves the network structure, and the approximation error between the full-order and reduced-order models is shown to be always bounded. Finally, the proposed approach is illustrated by an example.