In network science, real world networks have been extensively studied on the basis of statistics and graph theory, and some universal properties of real world networks, such as the small-world property and the power law of degree distribution, have been found out. Major examples of research topics in this area include community detection, which is formulated as a problem to divide a given network into some meaningful subnetworks in the sense of, e.g., personal human relationship.

In our work, we take an approach to the community detection problem from a viewpoint of control theory. More specifically, assigning a state variable to each node of networks, we construct a set of clusters according to the similarity of state behavior for input signals. As a result, as shown in the figure below, we can detect a set of similar nodes, and can visualize meaningful inter-cluster connections in the sense of input-to-state mapping. In addition, by aggregating the state of each cluster into a lower-dimensional variable, we can realize model reduction that preserves interconnection structure among clusters.

There are a number of systems having an internal state that does not escape from the nonnegative orthant. Example of such systems include spatially-discrete reaction-diffusion systems, electric circuit systems, Markovian processes, and so forth. In linear systems theory, this class of systems is called positive systems, and it is known that the positivity of systems is characterized by nonnegativity of system matrices. Based on this characterization, a model reduction problem preserving the system positivity can be formulated as a problem to approximate the input-to-output mapping of systems while preserving the nonnegativity of system matrices. In our work, in addition to the positivity-preserving model reduction problem, we formulate a dissipativity-preserving model reduction problem, and give a solution to it based on generalized singular perturbation.

A number of systematic control design methods, such as robust control, have been developed, and the efficiency of those methods has been verified from various viewpoints. However, it is known that, for large-scale systems, such a sophisticated design method inevitably makes a resultant controller and observer high-dimensional. In this sense, the implementation of control design methods for large-scale systems is not necessarily straightforward, and it is crucial to address the approximation problem of controllers and observers with a specified error precision.

The controller and observer reduction problem is formulated as a structured model reduction problem to find a lower-dimensional controller and observer preserving the signal transmission structure shown in the figures below. Even though this kind of structured model reduction problems is known as one of difficult problems in systems theory, we successfully give a solution to it by utilizing our dissipativity-preserving model reduction.