Local identification of scalar hybrid models with tree structure

Standard modeling approaches, for example in chemical engineering, suffer from two principal difficulties: the curse of dimension and a lack of extrapolability. We propose an approach via structured hybrid models to resolve both issues. For simplicity we consider reactor models which can be written as a tree-like composition of scalar input-output functions uj. The vertices j of the finite tree structure represent known or unknown sub-processes of the overall process. Known processes are modeled by white-box functions uj; unknown processes are represented by black boxes uj. Oriented edges of the tree indicate composition of the input-output relations uj in a feed forward structure. The tree structure of a mixture of black and white boxes constitutes what we call a structured hybrid model (SHM). Under certain assumptions on differentiability, genericity, and monotonicity, we provide an inductive algorithm which uniquely identifies all black boxes in the SHM, up to a trivial scaling calibration between adjacent black boxes. Our result does not require any extra measurements interior to the SHM. Instead, we only require global, overall input-output data, clustered along a d-dimensional data base of inputs. The dimension d need not exceed the maximal input dimension of any individual black box in the SHM. Compared to the total input dimension of the reactor, which may be much higher than d, this dimension reduction effectively avoids the curse of dimension. Moreover, our unique identification of all black boxes accomodates a reliable global extrapolation, far beyond the original data base, to input-regions of full dimension. We illustrate our results with a model of an industrial continuous polymerization plant.