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Dynamic Mode Decomposition with Control Liouville Operators

Joel A. Rosenfeld, Rushikesh Kamalapurkar

Abstract

This paper builds the theoretical foundations for dynamic mode decomposition (DMD) of control-affine dynamical systems by leveraging the theory of vector-valued reproducing kernel Hilbert spaces (RKHSs). Specifically, control Liouville operators and control occupation kernels are introduced to separate the drift dynamics from the input dynamics. A given feedback controller is represented through a multiplication operator and a composition of the control Liouville operator and the multiplication operator is used to express the nonlinear closed-loop system as a linear total derivative operator on RKHSs. A spectral decomposition of a finite-rank representation of the total derivative operator yields a DMD of the closed-loop system. The DMD generates a model that can be used to predict the trajectories of the closed-loop system. For a large class of systems, the total derivative operator is shown to be compact provided the domain and the range RKHSs are selected appropriately. The sequence of models, resulting from increasing-rank finite-rank representations of the compact total derivative operator, are shown to converge to the true system dynamics, provided sufficiently rich data are available. Numerical experiments are included to demonstrate the efficacy of the developed technique.

Dynamic Mode Decomposition with Control Liouville Operators

Abstract

This paper builds the theoretical foundations for dynamic mode decomposition (DMD) of control-affine dynamical systems by leveraging the theory of vector-valued reproducing kernel Hilbert spaces (RKHSs). Specifically, control Liouville operators and control occupation kernels are introduced to separate the drift dynamics from the input dynamics. A given feedback controller is represented through a multiplication operator and a composition of the control Liouville operator and the multiplication operator is used to express the nonlinear closed-loop system as a linear total derivative operator on RKHSs. A spectral decomposition of a finite-rank representation of the total derivative operator yields a DMD of the closed-loop system. The DMD generates a model that can be used to predict the trajectories of the closed-loop system. For a large class of systems, the total derivative operator is shown to be compact provided the domain and the range RKHSs are selected appropriately. The sequence of models, resulting from increasing-rank finite-rank representations of the compact total derivative operator, are shown to converge to the true system dynamics, provided sufficiently rich data are available. Numerical experiments are included to demonstrate the efficacy of the developed technique.

Paper Structure

This paper contains 32 sections, 15 theorems, 37 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Operators and Dynamic Mode Decomposition
  4. The Eigendecomposition Approach
  5. The Singular Value Decomposition Approach
  6. Related Work
  7. Vector-valued Reproducing Kernel Hilbert Spaces
  8. Closed Loop Nonlinear Systems as Operators over RKHSs
  9. Control Liouville Operators and Multiplication Operators
  10. Control Occupation Kernels
  11. Control Liouville Operators and Control Occupation Kernels
  12. Properties of Multiplication Operators
  13. Compact and Densely Defined Operators for DMD
  14. A Singular Value Decomposition Approach to DMD
  15. Existence of Bounded Multiplication Operators and Compact Differential Operators
  16. ...and 17 more sections

Key Result

Proposition 1

The span of the set $E := \{ K_{x,v} : v \in \mathcal{Y} \text{ and } x \in X \}$, is dense in $H$.

Figures (7)

  • Figure 1: A schematic diagram of the construction presented in Section \ref{['sec:OperatorRepresentation']}. The RKHSs are represented by filled circles. The squares at the endpoints of the dashed arrows passing through the circles indicate the domains and co-domains of the functions contained in the RKHSs. The thick arrows between the RKHSs indicate operators.
  • Figure 2: A schematic diagram of the finite-rank representation of the total derivative operator.
  • Figure 3: Relative error $\frac{\Vert F_{\mu}(x) - \hat{F}_{\mu}(x)\Vert_2}{\max_{x\in [-2,2]\times[-2,2]}\Vert F_{\mu}(x)\Vert_2}$ in the estimation of the vector field $F_\mu$ of the controlled Duffing oscillator as a function of $x$, obtained using SCLDMD (left) and CLDMD (right).
  • Figure 4: Error between predicted and true trajectories of the controlled duffing oscillator for the experiment in Section \ref{['subsec:duffing']}. The figure on the left is obtained using SCLDMD indirect prediction by solving \ref{['eq:convergent_closed_loop_model']} and the figure on the right is obtained using CLDMD indirect prediction by solving \ref{['eq:vector_field_reconstruction']}, both using the MATLAB®ode45 solver.
  • Figure 5: Predicted and true trajectories (left) and the corresponding prediction errors (right) of the controlled duffing oscillator for the experiment in Section \ref{['subsec:duffing']}. This result is obtained using CLDMD direct prediction \ref{['eq:finite_spectral_reconstruction']} with kernel parameter $\tilde{\rho} = 1e8$.
  • ...and 2 more figures

Theorems & Definitions (25)

  • Definition 1
  • Proposition 1
  • Definition 2
  • Definition 3
  • Definition 4: SCC.Rosenfeld.Russo.eatoappear
  • Definition 5: SCC.Rosenfeld.Kamalapurkar2021
  • Proposition 2: SCC.Rosenfeld.Kamalapurkar2021
  • Definition 6
  • Definition 7
  • Proposition 3: SCC.Rosenfeld.Kamalapurkar2021
  • ...and 15 more