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From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma

Mircea Lazar

Abstract

The generalization of the Koopman operator to systems with control input and the derivation of a nonlinear fundamental lemma are two open problems that play a key role in the development of data-driven control methods for nonlinear systems. In this paper we derive a novel solution to these problems based on basis functions expansion in a product Hilbert space constructed as the tensor product between the Hilbert spaces of the state and input observable functions, respectively. We identify relaxed invariance conditions that guarantee existence of a bounded linear operator, i.e., the generalized Koopman operator, from the constructed product Hilbert space to the Hilbert space corresponding to the lifted state propagated forward in time. Compared to classical Koopman invariance conditions, measure preservation is not required. Moreover, we derive a nonlinear fundamental lemma by exploiting the constructed exact infinite-dimensional bilinear Koopman representation and Hankel operators. The effectiveness of the developed generalized Koopman embedding is illustrated on the Van der Pol oscillator and in predictive control of a soft-robotic manipulator model.

From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma

Abstract

The generalization of the Koopman operator to systems with control input and the derivation of a nonlinear fundamental lemma are two open problems that play a key role in the development of data-driven control methods for nonlinear systems. In this paper we derive a novel solution to these problems based on basis functions expansion in a product Hilbert space constructed as the tensor product between the Hilbert spaces of the state and input observable functions, respectively. We identify relaxed invariance conditions that guarantee existence of a bounded linear operator, i.e., the generalized Koopman operator, from the constructed product Hilbert space to the Hilbert space corresponding to the lifted state propagated forward in time. Compared to classical Koopman invariance conditions, measure preservation is not required. Moreover, we derive a nonlinear fundamental lemma by exploiting the constructed exact infinite-dimensional bilinear Koopman representation and Hankel operators. The effectiveness of the developed generalized Koopman embedding is illustrated on the Van der Pol oscillator and in predictive control of a soft-robotic manipulator model.

Paper Structure

This paper contains 18 sections, 12 theorems, 120 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The generalized Koopman operator
  4. Coordinate representation in Riesz bases
  5. Input-affine nonlinear systems
  6. Data-driven finite-dimensional approximation
  7. The nonlinear fundamental lemma: a bilinear-Koopman approach
  8. Koopman representations with lifted outputs
  9. The nonlinear fundamental lemma
  10. Illustrative examples
  11. Van der Pol Oscillator
  12. Standard joint-kernel EDMD - kernelized KIC Brunto_Koopmanu
  13. Khatri--Rao EDMD - Generalized Koopman
  14. Soft-robotic manipulator
  15. Conclusions
  16. ...and 3 more sections

Key Result

Proposition II.5

If $\{\psi_{i,x}\}_{i=1}^\infty$ is a Riesz basis for $\mathcal{H}_x$ with bounds $\alpha_x,\beta_x$ and $\{\psi_{i,u}\}_{i=1}^\infty$ is a Riesz basis for $\mathcal{H}_u$ with bounds $\alpha_u,\beta_u$, then the product functions form a Riesz basis for $\mathcal{H}$ with bounds $\alpha_x\alpha_u \leq \beta_x\beta_u$.

Figures (8)

  • Figure 1: Illustration of the Koopman operator modeling approach for dynamical systems with control input.
  • Figure 2: Illustration of the envisioned generalized Koopman operator architecture for systems with control input.
  • Figure 3: Generalized Koopman operator learning architecture: (i) State and input at time $t$ are lifted via Encoders; (ii) Tensor product of the resulting lifted state and input vectors; (iii) Koopman linear operator advances the lifted state to $t+1$; (iv) State at time $t+1$ is retrieved using a Decoder.
  • Figure 4: Kernelized KIC method Brunto_Koopmanu: Absolute feature error 3D plot averaged over 100 test trajectories.
  • Figure 5: Generalized Koopman: Absolute feature error 3D plot averaged over 100 test trajectories.
  • ...and 3 more figures

Theorems & Definitions (34)

  • Definition II.1
  • Definition II.4
  • Proposition II.5
  • Theorem III.1
  • proof
  • Remark III.2: Relation with the Koopman operator
  • Remark III.3: Implications for $F$
  • Proposition III.4
  • proof
  • Corollary III.5
  • ...and 24 more