Distributed proper orthogonal decomposition for large-scale networked dynamical systems

Abstract

Recently, dynamical systems in engineering and science problems become drastically larger and too complex. One of the ways to solve the difficulty is to model systems with hierarchical network structures. Proper orthogonal decomposition (POD) is a model reduction method using available data and its singular value decomposition. In this paper, we apply the POD to a networked linear dynamical system and propose the distributed POD which can specify the degree of approximate of each subsystem and preserve the network structure. We also characterize an upper bound of the approximation error and give a criterion to determine the optimal degrees of reduced-order subsystems. As an application to the distributed POD, the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm minimizing POD is also proposed for a construction of a simple network structure which approximates the behavior of the entire system appropriately. A numerical example is provided to demonstrate the distributed and ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm minimizing PODs and to show their efficiency.