On the Calculation of Operability Sets of Nonlinear High-Dimensional Processes

Abstract

The calculation of the steady state operability characteristics of nonlinear high-dimensional processes is a difficult task computationally as it involves the mapping of high-dimensional, constrained input sets through the nonlinear process model to a high-dimensional output set. The reverse mapping, from outputs to inputs, also of interest, is equally challenging. In this paper we propose the selection of a finite number of points as an adequately accurate representation of the overall input-output mapping of the detailed model. This approach is motivated by the well established design of experiments (DoE) [Montgomery, D. C. Design and Analysis of Experiments; Wiley: New York, 2005. Box, G. E. P.; Draper, N. R. Response Surfaces, Mixtures, and Ridge Analysis; Wiley: Hoboken, NJ, 2007. Box, G. E. P.; Hunter, J. S.; Hunter, W. G. Statistics for Experimenters: Design, Innovation and Discovery, 2nd ed.; John Wiley & Sons, Inc.: Hoboken, NJ, 2004.] methodology which has been successful in experimentally investigating similar relationships for processes that have not yet been modeled. For this reason, the proposed approach is called design of selective calculations (DoSC). The mapping of only selective points and the development of an interpolative response surface model for the output points enables the calculation of the desired operability sets with a much reduced number of calculations without any significant loss of accuracy. The applicability of the developed method is illustrated with two motivating examples and a plant-wide industrial process, the Tennessee Eastman challenge problem. Of particular interest is the demonstration that the response surface model with quadratic terms is rich enough to accurately describe even the complex phenomenon of input multiplicity. When input multiplicity exists, the image of the boundary of the input set does not fully describe the boundary of the image set. In such a case, mapping only the boundaries of the related sets is not sufficient. Besides its overall computational economy, the proposed method overcomes this important additional challenge.