On the role of exact models in approximate modeling problems

The behavioral theory of dynamical system is used to address a deterministic system identification problem with a newly defined measure of misfit between data and linear time-invariant systems. An approximate model identification problem is formalized using this misfit criterium. In particular, Pareto optimal models are defined as feasible trade-offs between low complexity and low misfit models. The main result of this paper provides a complete characterization of bounded misfit and bounded complexity models. It is shown that this entire class of approximate models corresponds to the set of most powerful unfalsified models of reduced data sets. The reduced data sets are derived from Hankel norm approximations of the data. The main result therefore emphasizes the relevance of the exact modeling problem for the identification of approximate systems. The set of all Pareto optimal models is characterized as a simple consequence of this result.