From Time Series to Affine Systems

TL;DR

This paper generalizes core behavioral systems theory results from linear to affine time-invariant systems, introducing new data-driven representations and a tailored persistence of excitation condition that reduces data requirements.

Contribution

It extends behavioral systems theory to affine systems, develops new data representations, and introduces a less conservative persistence of excitation condition.

Findings

New kernel, input-output, state-space, and finite-horizon representations for affine systems

A novel persistence of excitation condition specific to affine systems

A fundamental lemma that reduces data requirements for system identification

Abstract

The paper extends core results of behavioral systems theory from linear to affine time-invariant systems. We characterize the behavior of affine time-invariant systems via kernel, input-output, state-space, and finite-horizon data-driven representations, demonstrating a range of structural parallels with linear time-invariant systems. Building on these representations, we introduce a new persistence of excitation condition tailored to the model class of affine time-invariant systems. The condition yields a new fundamental lemma that parallels the classical result for linear systems while provably reducing data requirements. Our analysis highlights that excitation conditions must be adapted to the model class: overlooking structural differences may lead to unnecessarily conservative data requirements.