Modeling and control of physical processes using proper orthogonal decomposition

• Published in: Mathematical and computer modelling Vol. 33; no. 1; pp. 223 - 236
• Main Authors: Ly, Hung V., Tran, Hien T.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Oxford Elsevier Ltd 2001; Elsevier Science
• Subjects: Applied sciences; Boundary control; Computer science; control theory; systems; Control theory. Systems; Exact sciences and technology; Modelling and identification; Proper orthogonal decomposition; Rayleigh-Bénard convection; System theory; Rayleigh-Bénard convection; Boundary control; Proper orthogonal decomposition; Freedom degree; Turbulent flow; Tracking; Image processing; Industrial application; Data compression; Natural convection; Decomposition; Dynamical system; Benard convection; Convection; Engineering; Numerical computation; Numerical data; Controllability; System identification; Mathematical model; Observability; System structure; Karhunen Loeve transformation; Process control; Steady state; Orthogonal transformation; Rayleigh Bernard convection; Tracking(movable target); Rayleigh Bénard instability; Galerkin method; Yield
• ISSN: 0895-7177; 1872-9479
• DOI: 10.1016/S0895-7177(00)00240-5

The proper orthogonal decomposition (POD) technique (or the Karhunan Loève procedure) has been used to obtain low-dimensional dynamical models of many applications in engineering and science. In principle, the idea is to start with an ensemble of data, called
snapshots, collected from an experiment or a numerical procedure of a physical system. The POD technique is then used 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 freedom. In this context, it is assumed that the physical system has a mathematical model, which may not be available for many physical and/or industrial applications. In this paper, we consider the steady-state Rayleigh-Bénard convection whose mathematical model is assumed to be unknown, but numerical data are available. The aim of the paper is to show that, using the obtained ensemble of data, POD can be used to model accurately the natural convection. Furthermore, this approach is very efficient in the sense that it uses the smallest possible number of parameters, and thus, is suited for process control. Particularly, we consider two boundary control problems
1.
(a) tracking problem, and
2.
(b) avoiding hot spot in a certain region of the domain.