Clustered Model Reduction of Large - Scale Complex Networks ∗

Large-scale dynamical networks (dynamical systems on large-scale complex networks), whose behaviors are described by the interaction of a large number of subsystems, have been extensively studied over the past decades. Examples of such dynamical networks include biological networks, power networks, traffic networks; see [1, 2] for an overview. In order to control large-scale systems, the model reduction techniques will be indispensable in general. Although many useful model reduction techniques such as the balanced truncation, the Hankel norm approximation, and the Krylov projection method have been developed so far (see e.g.,[3]), they do not preserve the network topology among subsystems, in other words, each state of the reduced order model is in general obtained by a linear combination of all original states. This inhibits to use the reduced order model when one addresses, e.g., the distributed control problem and the sensor allocation problem. To deal with this kind of issue, the papers [4] and [5] have proposed a structured balanced truncation method and a Krylov-based model reduction method for interconnected systems, respectively, which require a priori knowledge on decomposition of the whole system into subsystems. As related approaches, structure preserving model reduction methods, e.g., [6, 7, 8], and state aggregation-based singular perturbation methods, e.g., [9, 10, 11], have been intensively developed. In the former methods, reduced order models are derived based on the Krylov projection so as to preserve the underlying structure of systems such as the Lagrange structure