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From Time Series to Affine Systems

A. Padoan, J. Eising, I. Markovsky

TL;DR

The paper extends behavioral systems theory to affine time-invariant systems, establishing complete kernel, IO, and state-space representations for affine behaviors and relating them to the associated difference systems. It introduces a model-class specific persistence of excitation, the generalized affine persistency of excitation (GAPE), and proves a corresponding affine fundamental lemma that enables finite-horizon trajectories to be reconstructed from data with reduced information requirements. By lifting affine systems to linear representations via homogenization (projective geometry), the work connects affine representations to lifted linear counterparts and clarifies when affine models admit nonempty representations. The results yield data-efficient, structurally justified methods for data-driven control and predictive control, with demonstrative advantages shown in a scalar-example and practical implications for affine local approximations of nonlinear dynamics.

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.

From Time Series to Affine Systems

TL;DR

The paper extends behavioral systems theory to affine time-invariant systems, establishing complete kernel, IO, and state-space representations for affine behaviors and relating them to the associated difference systems. It introduces a model-class specific persistence of excitation, the generalized affine persistency of excitation (GAPE), and proves a corresponding affine fundamental lemma that enables finite-horizon trajectories to be reconstructed from data with reduced information requirements. By lifting affine systems to linear representations via homogenization (projective geometry), the work connects affine representations to lifted linear counterparts and clarifies when affine models admit nonempty representations. The results yield data-efficient, structurally justified methods for data-driven control and predictive control, with demonstrative advantages shown in a scalar-example and practical implications for affine local approximations of nonlinear dynamics.

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.
Paper Structure (20 sections, 18 theorems, 145 equations, 1 figure)

This paper contains 20 sections, 18 theorems, 145 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A motivating application in data-driven control
  3. Affine systems
  4. Offset representations
  5. Kernel representations
  6. Input-output representations
  7. State-space representations
  8. System properties
  9. Controllability
  10. Integer invariants
  11. Fundamental lemma for affine systems
  12. Discussion
  13. Distinct model classes, distinct excitation conditions
  14. Data requirements of different excitation conditions
  15. The projective geometry of affine systems
  16. ...and 5 more sections

Key Result

theorem 1

berberich2022linear Consider the system eq:system_nonlinear_linearization_affine. Let $u_d \in (\mathbb{R}^m)^{\mathbf{T}}$, $x_d \in (\mathbb{R}^n)^{\mathbf{T}}$, $y_d \in (\mathbb{R}^p)^{\mathbf{T}}$ be an input/state/output trajectory of system eq:system_nonlinear_linearization_affine and let ${L Then every trajectory $\left[\right] \in \mathbb{R}^{(m+p)\mathbf{L}}$ of system eq:system_nonlinea

Figures (1)

  • Figure 1: An affine subspace $\textcolor{coolTeal}{\mathcal{B}} \subseteq \mathbb{R}^2$ (solid) is embedded into a linear subspace $\textcolor{hotPink}{\mathcal{B}_{lift}} \subseteq \mathbb{R}^3$ (shaded) through the map \ref{['eq:map_homogeneization']}, whose image intersects each equivalence class $[w] \in \mathbb{P}^2$ exactly once in the chart defined by $w_{3} = 1$ (dashed).

Theorems & Definitions (37)

  • theorem 1
  • theorem 2
  • theorem 3: Offset representations
  • example 1: An system without a constant offset trajectory
  • theorem 4: Affine kernel representations
  • theorem 5: Input-output partitions and difference systems
  • corollary 1: Affine input-output representations
  • theorem 6: Affine state-space representations
  • lemma 1: Controllability is translation-invariant
  • corollary 2: Controllability of affine systems
  • ...and 27 more