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Two Roads to Koopman Operator Theory for Control: Infinite Input Sequences and Operator Families

Masih Haseli, Igor Mezić, Jorge Cortés

Abstract

The Koopman operator, originally defined for dynamical systems without input, has inspired many applications in control. Yet, the theoretical foundations underpinning this progress in control remain underdeveloped. This paper investigates the theoretical structure and connections between two extensions of Koopman theory to control: (i) Koopman operator via infinite input sequences and (ii) the Koopman control family. Although these frameworks encode system information in fundamentally different ways, we show that under certain conditions on the function spaces they operate on, they are equivalent. The equivalence is both in terms of the actions of the Koopman-based formulations in each framework as well as the function values on the system trajectories. Our analysis provides constructive tools to translate between the frameworks, offering a unified perspective for Koopman methods in control.

Two Roads to Koopman Operator Theory for Control: Infinite Input Sequences and Operator Families

Abstract

The Koopman operator, originally defined for dynamical systems without input, has inspired many applications in control. Yet, the theoretical foundations underpinning this progress in control remain underdeveloped. This paper investigates the theoretical structure and connections between two extensions of Koopman theory to control: (i) Koopman operator via infinite input sequences and (ii) the Koopman control family. Although these frameworks encode system information in fundamentally different ways, we show that under certain conditions on the function spaces they operate on, they are equivalent. The equivalence is both in terms of the actions of the Koopman-based formulations in each framework as well as the function values on the system trajectories. Our analysis provides constructive tools to translate between the frameworks, offering a unified perspective for Koopman methods in control.
Paper Structure (21 sections, 19 theorems, 53 equations, 4 figures, 1 table)

This paper contains 21 sections, 19 theorems, 53 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Koopman Operator Viewpoint of Dynamical Systems
  3. A Cautionary Tale of Naive Koopman Extensions to Control Systems
  4. Naive Approach 1: Simple Composition Operators
  5. Naive Approach 2: Treating the Input as a State
  6. Formal Extensions of Koopman Operator Theory to Control Systems
  7. Koopman Operator via Infinite Input Sequences
  8. Koopman Control Family (KCF)
  9. Motivation and Problem Statement
  10. Connections Between Dynamical Systems in Different Frameworks
  11. Augmented System and its Associated Koopman Operator
  12. All Introduced Input-Free Systems Capture the Behavior of the Control System
  13. Connecting the Function Spaces
  14. Connecting $\mathcal{F}^\infty$ and $\mathcal{F}^{\operatorname{aug}}$
  15. Connecting $\mathcal{F}$ and $\mathcal{F}^{\operatorname{aug}}$
  16. ...and 6 more sections

Key Result

Lemma 3.1

(Necessary Condition for Well-defined $\mathcal{K}^\text{input-aug}$): Let the operator $\mathcal{K}^\text{input-aug}: \mathcal{S} \to \mathcal{S}$ associated with open-loop system eq:control-system be well defined. Define the range of the dynamic map $\mathcal{T}$ in eq:control-system as follows Then, at least one of the following hold:

Figures (4)

  • Figure 1: Connections between the functions spaces. The action of $\mathcal{E}_{\mathcal{F}}^{\mathcal{F}^{\operatorname{aug}}}$ and $\mathcal{E}_{\mathcal{F}}^{\mathcal{F}^{\operatorname{aug}}_{\operatorname{CI}}}$ coincide even though the have different codomains.
  • Figure 2: Commutative diagram illustrating Theorem \ref{['t:operator-connection-infinite-augmented']}.
  • Figure 3: Commutative diagram illustrating Theorem \ref{['t:operator-connection-augmented-KCF']}. Here, $u^* \in \mathcal{U}$ is arbitrary.
  • Figure 4: Commutative diagram illustrating Theorem \ref{['t:operators-on-control-independent']} and Propositions \ref{['p:operator-connection-ci-inf-aug']} and \ref{['p:operator-connection-ci-aug-orig']}. Here, $u^* \in \mathcal{U}$ is arbitrary.

Theorems & Definitions (42)

  • Lemma 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Definition 4.1
  • Lemma 4.2
  • Remark 4.3
  • Lemma 4.4
  • Remark 4.5
  • Remark 4.6
  • ...and 32 more