How to Be a Gray Box: The Art of Dynamic Semi-Physical Modeling

A general methodology for gray-box, or semi-physical, modeling is presented. This technique is intended to combine the best of two worlds: knowledge-based modeling , whereby mathematical equations are derived in order to describe a process, based on a physical (or chemical, biological, etc.) analysis, and black-box modeling, whereby a parameterized model is designed, whose parameters are estimated solely from measurements made on the process. The gray-box modeling technique is very valuable whenever a knowledge-based model exists, but is not fully satisfactory and cannot be improved by further analysis (or can only be improved at a very large computational cost). We describe the design methodology of a gray-box model, and illustrate it on a didactic example. We emphasize the importance of the choice of the discretization scheme used for transforming the differential equations of the knowledge-based model into a set of discrete-time recurrent equations. Finally, an application to a real, complex industrial process is presented.