On exact and approximate modeling of exponential time series

The authors consider the problem of identifying linear time-invariant models from frequency-response measurements. They depart from the standard setting of this problem by refraining from a priori assumptions on the input-output structure of the to-be-identified system. Without this structure, the optimal model which explains the exponential time series is characterized by an autonomous system. By a posteriori imposing the structure of input and output variables of the system, it is shown that the set of optimal (i.e., least complexity) input-output models can be derived in a straightforward way from a polynomial representation of the optimal autonomous system. The authors formulate this as a minimal complexity covering problem and provide a conceptual algorithm for the computation of its solution. The problem of approximate modeling of purely exponential data is also discussed. An equation error type of misfit function is considered, and a class of approximate input-output models which satisfy a uniform upper bound of the misfit criterion is characterized