Exact linear modeling using Ore algebras

TL;DR

This paper develops algorithmic methods for exact linear modeling of observed data using Ore algebras, addressing both constant and variable coefficient cases with applications to system identification.

Contribution

It introduces new algorithms for computing MPUM and VMPUM models for polynomial and polynomial-exponential signals, expanding the tools for system identification.

Findings

Algorithms successfully compute models for various signals

Structural properties of models are analyzed

Examples demonstrate practical applicability

Abstract

Linear exact modeling is a problem coming from system identification: Given a set of observed trajectories, the goal is 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 the operators with varying coefficients were addressed only recently. This question can be tackled either using Gr\"obner 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 to compute "most powerful unfalsified models" (MPUM) and their counterparts with variable coefficients (VMPUM) for polynomial and polynomial-exponential signals. We also study the structural properties of the resulting models, discuss computer algebraic techniques behind algorithms and provide several examples.