Exact linear modeling using Ore algebras

Abstract

Linear exact modeling is a problem coming from system identification: given a set of observed trajectories, the goal is to find a model (usually, a system of partial differential and/or difference equations) that explains the data as precisely as possible. The case of operators with constant coefficients is well studied and known in the systems theoretic literature, whereas operators with varying coefficients were addressed only recently. This question can be tackled either using Gröbner bases for modules over Ore algebras or by following the ideas from differential algebra and computing in commutative rings. In this paper, we present algorithmic methods for computing “most powerful unfalsified models” ( MPUM ) and their counterparts with variable coefficients ( V MPUM ) for polynomial and polynomial–exponential signals. We also study the structural properties of the resulting models, discuss computer algebraic techniques behind the algorithms and provide several examples.