Modeling and control of physical processes using proper orthogonal decomposition

The proper orthogonal decomposition (POD) technique (or the Karhunan Loeve procedure) has been used to obtain low-dimensional dynamical models of many applications in engineering and science. In principle, the idea is to star t with an ensemble of data, called snapshots, collected from an experiment or a numerical procedure of a physical system. The POD technique is then us ed to produce a set of basis functions which spans the snapshot collection. When these basis functions are used in a Galerkin procedure, they yield a finite-dimensional dynamical system with the smallest possible degrees of f reedom. In this context, it is assumed that the physical system has a mathe matical model, which may not be available for many physical and/or industri al applications. In this paper, we consider the steady-state Rayleigh-Benar d convection whose mathematical model is assumed to be unknown, but numeric al data are available. The aim of the paper is to show that, using the obta ined ensemble of data, POD can be used to model accurately the natural conv ection. Furthermore, this approach is very efficient in the sense that it u ses the smallest possible number of parameters, and thus, is suited for pro cess control. Particularly, we consider two boundary control problems (a) tracking problem, and (b) avoiding hot spot in a certain region of the domain. (C) 2001 Elsevier Science Ltd. All rights reserved.