Proper Orthogonal Decomposition for Reduced Order Modeling: 2D Heat Flow Mehmet ¨ Onder Efe Collaborative Center of Control Science Department of Electrical Engineering The Ohio State University Columbus, OH 43210, U.S.A. E-mail: onderefe@ieee.org Hitay ¨ Ozbay Dept. of Electrical & Electronics Eng. Bilkent Univ., Ankara TR-06533, Turkey on leave from Dept. of Electrical Eng. The Ohio State University E-mail: ozbay@ee.eng.ohio-state.edu Abstract— Modeling issues of infinite dimensional systems is studied in this paper. Although the modeling problem has been solved to some extent, use of decomposition techniques still pose several difficulties. A prime one of this is the amount of data to be processed. Method of snapshots integrated with POD is a remedy. The second difficulty is the fact that the decomposition followed by a projection yields an autonomous set of finite dimensional ODEs that is not useful for developing a concise understanding of the input operator of the system. A numerical approach to handle this issue is presented in this paper. As the example, we study 2D heat flow problem. The results obtained confirm the theoretical claims of the paper and emphasize that the technique presented here is not only applicable to infinite dimensional linear systems but also to nonlinear ones. I. I NTRODUCTION Although the applications of Proper Orthogonal Decomposi- tion (POD) particularly focus on the extraction of coherent and dominant modes available in aerodynamic flows, the problem of tackling with huge amounts of data and technical difficulties in the obtained model constitute barriers between the stipulated efforts and the thorough understanding of the process, [1-6]. For this reason, we study the modeling problem that addresses the above mentioned difficulties appropriately on a 2D, simple, and a linear dynamics, namely the heat flow. The goal of the paper is to introduce how the data is obtained, how the external stimuli enter into the dynamics, and how the data set is processed so as to obtain a set of orthogonal basis, and finally, how the external stimuli is made explicit in an autonomous set of ODEs. Modeling of systems displaying spatial continuum requires a careful consideration since the physical process under in- vestigation is an infinite dimensional one due to the spatial continuity. Efforts in understanding the behavior of such systems have particularly focused on the low dimensional models capturing the essential behavioral properties with a few Ordinary Differential Equations (ODEs). This has been done by using modal decompositions such as Proper Orthogonal Decomposition (POD) and Singular Value Decomposition (SVD). Although neither the decomposition techniques nor the infinite dimensionality are new issues in this field, obtaining a model having the boundary conditions as external inputs is a major problem in the POD and SVD methods. More explicitly, these approaches result in models where external control input appears in the dynamical equations implicitly, and this is not very useful for controller design. Another difficulty is the presence of modeling uncertainties, which stem from varying internal parameters or hypotheses that are not thoroughly valid. For the heat transport process, imprecise knowledge on thermal diffusivity parameter is a good example to study uncertainties. The use of decomposition techniques in modeling of spa- tially continuous systems has extensively been studied in the field of aerodynamic flow control problems, [1-4]. Since the dynamics of the process under investigation is governed by Navier-Stokes equations, obtaining closed form solutions are very difficult and the modeling studies particularly focus on the real time observations from the process. For systems having two or more spatial dimensions, the POD technique has been utilized with the aid of snapshots method, [1-2]. Alternatively, for single dimensional processes, the same modeling procedure can be followed by exploiting the SVD technique. Procedurally, in both of them, if the numerical data contains coherent modes, the expansion accurately describes the tem- poral modes and the spatial components distributing them over the physical domain of the process. Furthermore, the orthogo- nality of the basis functions, which describe the spatial proper- ties, helps in finding a set of ODEs synthesizing the temporal modes. Although the algorithmic part seems straightforward, the final form of the ODEs depicts an autonomous system having no external input. At this point, several modifications are needed to separate the effect of boundary conditions, which constitute the inputs exciting the process. 2D heat transport problem is therefore a good candidate to study how modeling issues are addressed. A number of variations of this problem has been taken into consideration in former studies, [5-7]. Atwell et al [5-6], have considered 2D heat transport problem with control input explicitly available in the Partial Differential Equation (PDE). The thermal diffusivity parameter has been taken as a known constant and several control strategies have been assessed with the modeling results of POD approach. In [6], the design has been discussed from the computational point of view. Another work focusing on 1D heat transport problem re-