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On the Existence of Koopman Linear Embeddings for Controlled Nonlinear Systems

Xu Shang, Masih Haseli, Jorge Cortés, Yang Zheng

Abstract

Koopman linear representations have become a popular tool for control design of nonlinear systems, yet it remains unclear when such representations are exact. In this paper, we establish sufficient and necessary conditions under which a controlled nonlinear system admits an exact finite-dimensional Koopman linear representation, which we term Koopman linear embedding. We show that such a system must be transformable into a special control-affine preserved (CAP) structure, which enforces affine dependence of the state on the control input and isolates all nonlinearities into an autonomous subsystem. We further prove that this autonomous subsystem must itself admit a finite-dimensional Koopman linear model with a sufficiently-rich Koopman invariant subspace. Finally, we introduce a symbolic procedure to determine whether a given controlled nonlinear system admits the CAP structure, thereby elucidating whether Koopman approximation errors arise from intrinsic system dynamics or from the choice of lifting functions.

On the Existence of Koopman Linear Embeddings for Controlled Nonlinear Systems

Abstract

Koopman linear representations have become a popular tool for control design of nonlinear systems, yet it remains unclear when such representations are exact. In this paper, we establish sufficient and necessary conditions under which a controlled nonlinear system admits an exact finite-dimensional Koopman linear representation, which we term Koopman linear embedding. We show that such a system must be transformable into a special control-affine preserved (CAP) structure, which enforces affine dependence of the state on the control input and isolates all nonlinearities into an autonomous subsystem. We further prove that this autonomous subsystem must itself admit a finite-dimensional Koopman linear model with a sufficiently-rich Koopman invariant subspace. Finally, we introduce a symbolic procedure to determine whether a given controlled nonlinear system admits the CAP structure, thereby elucidating whether Koopman approximation errors arise from intrinsic system dynamics or from the choice of lifting functions.
Paper Structure (25 sections, 12 theorems, 109 equations, 1 figure, 2 algorithms)

This paper contains 25 sections, 12 theorems, 109 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Paper structure and notation
  4. Preliminaries and problem statement
  5. Koopman operator for autonomous systems
  6. Koopman Control Family and Koopman linear embedding
  7. Problem statement
  8. Koopman linear embedding and multi-step linear prediction
  9. Motivating example and informal characterizations
  10. $N$-step linear predictors and their existence
  11. Koopman linear embeddings
  12. Existence of $N$-step linear predictors
  13. Affine functions on general sets and their composition
  14. Linear predictors for 1D and 2D systems
  15. $N$-step linear predictors for general nonlinear systems
  16. ...and 10 more sections

Key Result

Proposition 1

Consider the nonlinear system eqn:nonlinear. Denote its system propagation at step $N$ as $x(N) := {f}_N(x, u_{0:N-1})$, where $x$ is the initial state and $u_{0:N-1}$ is the control sequence. Suppose that its input set $\mathcal{U}$ is compact and has non-empty interior, and define $\bar{\mathcal{U then, the system necessarily admits an $N$-step linear predictor.

Figures (1)

  • Figure 1: Visualization of the Koopman linear embedding for the slow-manifold system \ref{['eqn:slow-manifold']}. The blue surface represents the invariant manifold $z_3 = z_2^2$. The dark blue, light blue, and purple curves denote trajectories of the Koopman linear embedding \ref{['eqn:slow-manifold-linear']}, their projections on the original state space $x = Cz \in \mathbb{R}^2$, and the trajectories from the original nonlinear system \ref{['eqn:slow-manifold']}, respectively. The first two trajectories start from exact lifting of the initial conditions, $z_0 = \Phi(x_0)$, whereas the last trajectory originates from an inexact lifted initial condition.

Theorems & Definitions (27)

  • Definition 1: Koopman linear embedding
  • Definition 2: $N$-step linear predictor
  • Remark 1: Loss of affineness
  • Proposition 1: Exact linear predictors from uniform approximation
  • Theorem 1: Equivalence of $\infty$-step linear prediction and the CAP structure
  • Theorem 2: Equivalence of Koopman linear embedding and the CAP structure with Koopman closure
  • Remark 2: Connection to LTI systems and Koopman operator for autonomous systems
  • Remark 3: Affine systems
  • Example 1
  • Definition 3: Affine function on a set
  • ...and 17 more