IMA Journal of Applied Mathematics Advance Access originally published online on May 16, 2008
IMA Journal of Applied Mathematics 2008 73(3):449-476; doi:10.1093/imamat/hxn011
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Local identification of scalar hybrid models with tree structure
Institut für Mathematik I, Freie Universität Berlin, Arnimallee 2-6, 14195 Berlin, Germany
Bayer Technology Services GmbH, PT-M&T-CS, 51368 Leverkusen, Germany
Email: fiedler{at}mi.fu-berlin.de
Received on March 6, 2006; Accepted on May 14, 2007
Standard modelling approaches, e.g. 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 (SHMs) to resolve both issues. For simplicity, we consider reactor models which can be written as a tree-like composition of scalar input–output (i/o) functions uj. The vertices j of the finite tree structure represent known or unknown subprocesses of the overall process. Known processes are modelled by white-box functions uj; unknown processes are represented by black boxes uj. Oriented edges of the tree indicate composition of the i/o relations uj in a feedforward structure. The tree structure of a mixture of black and white boxes constitutes what we call an 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 i/o data clustered along a d-dimensional database of inputs. More precisely, information on partial derivatives of order up to 5 is required, in all directions, but only at base points within the d-dimensional database. The dimension d need not exceed the maximal input dimension of any individual black box in the SHM. Compared to the total input dimension n of the reactor, which may be much larger than d, this dimension reduction effectively avoids the curse of dimension: the complexity of our approach is polynomial in n and exponential in d, only, rather than exponential in n. Moreover, our unique identification of all black boxes accommodates a reliable global extrapolation, far beyond the original database, to input regions of full dimension. We illustrate our results with a model of an industrial continuous polymerization plant.
Keywords: Hilbert problem; singularity theory; parameter identification; transversality; normal form theory; hybrid model; chemical engineering; process flow sheet; process model; reaction model; chemical modelling; polymerization; process control.