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6.2 Proper Orthogonal Decomposition (POD) globalization. Recently, Varshney et al. [189] utilized ROMs to optimize multiscale systems. However, until now ROM-based optimization has been used only for small-scale optimal control or dynamic optimization problems, and its use for large-scale dynamic optimization problems, especially PSA, has not been explored. For PSA, very few studies involve model reduction or simplification. A compact representation developed by Chlendi et al. [49] using design of experiments, short-cut models suggested by Chung et al. [53], and a model simplification strategy chalked out by Zhang et al. [211], which relies on understanding detailed physics behind each operating step, are the only known articles. Our focus is to develop systematic model reduction strategies for PSA processes using POD. In subsequent sections we describe the concept of proper orthogonal decomposition and demonstrate construction of reduced-order models using POD. We also illustrate ROM con- struction with the help of an example of Burgers equation. 6.2 Proper Orthogonal Decomposition (POD) 6.2.1 Concept Proper orthogonal decomposition (POD), first introduced by Lumley [126] is now used in a wide variety of disciplines such as turbulence, image processing, signal analysis, data compression, oceanography, and process identification and control. The key idea of POD is to compute a set of orthonormal functions, called POD basis functions, such that they can describe the dynamic system under consideration with as few basis functions as possible. The set of basis functions is optimal in the sense that it captures and describes the dynamic behavior of the system with minimum number of basis functions. Given an ensemble of observations defined in a vector space, we seek to find a subspace, much smaller than the original vector space, such that the projection of all the observations on to that is maximal. The attractiveness of the POD technique lies in the fact that the basis functions are derived from the numerical solutions or experimental measurements of the system, thus exhibiting a local characteristic and ensuring Chapter 6. Reduced-order Modeling for Optimization 100PDF Image | Design and Operation of Pressure Swing Adsorption Processes
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