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Dynamic Mode Decomposition of Control-Affine Nonlinear Systems using Discrete Control Liouville Operators

Zachary Morrison, Moad Abudia, Joel Rosenfeld, Rushikesh Kamalapurar

TL;DR

A novel operator representation of discrete-time, control-affine nonlinear dynamical systems that are affine in control is developed and it is demonstrated that this representation can be used to predict the behavior of the closed-loop system in response to a given feedback law.

Abstract

Representation of nonlinear dynamical systems as infinite-dimensional linear operators over Hilbert spaces enables analysis of nonlinear systems via pseudo-spectral operator analysis. In this paper, we provide a novel representation for discrete-time control-affine nonlinear dynamical systems as linear operators acting on a Hilbert space. We also demonstrate that this representation can be used to predict the behavior of the closed-loop system given a known feedback law using recorded snapshots of the system state resulting from arbitrary, potentially open-loop control inputs. We thereby extend the predictive capabilities of dynamic mode decomposition to discrete-time nonlinear systems that are affine in control. We validate the method using two numerical experiments by predicting the response of a controlled Duffing oscillator to a known feedback law, as well as demonstrating the advantage of the developed method relative to existing techniques in the literature.

Dynamic Mode Decomposition of Control-Affine Nonlinear Systems using Discrete Control Liouville Operators

TL;DR

A novel operator representation of discrete-time, control-affine nonlinear dynamical systems that are affine in control is developed and it is demonstrated that this representation can be used to predict the behavior of the closed-loop system in response to a given feedback law.

Abstract

Representation of nonlinear dynamical systems as infinite-dimensional linear operators over Hilbert spaces enables analysis of nonlinear systems via pseudo-spectral operator analysis. In this paper, we provide a novel representation for discrete-time control-affine nonlinear dynamical systems as linear operators acting on a Hilbert space. We also demonstrate that this representation can be used to predict the behavior of the closed-loop system given a known feedback law using recorded snapshots of the system state resulting from arbitrary, potentially open-loop control inputs. We thereby extend the predictive capabilities of dynamic mode decomposition to discrete-time nonlinear systems that are affine in control. We validate the method using two numerical experiments by predicting the response of a controlled Duffing oscillator to a known feedback law, as well as demonstrating the advantage of the developed method relative to existing techniques in the literature.
Paper Structure (13 sections, 5 theorems, 18 equations, 3 figures, 1 algorithm)

This paper contains 13 sections, 5 theorems, 18 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Problem Statement
  4. Operator representation of controlled discrete-time systems
  5. A Composition-like Kernel Propagation Operator
  6. Multiplication Operators
  7. The Discrete Control Liouville Operator
  8. Discrete-time control Liouville DMD
  9. Discrete Control Liouville Dynamic Mode Decomposition
  10. Convergence Properties of DCLDMD
  11. Numerical Experiments
  12. Discussion
  13. Conclusion

Key Result

Proposition 1

If $A \coloneqq \{\Tilde{K}_{x} : x \in X\}$, then $\text{span }A = \Tilde{H}$.

Figures (3)

  • Figure 1: A comparison of indirectly reconstructed trajectories $\hat{x}_{1}(t)$ and $\hat{x}_{2}(t)$ with the true trajectories $x_{1}(t)$ and $x_{2}(t)$ of the Duffing oscillator resulting from the linear feedback law $\mu$ in experiment \ref{['exp:Duffing']}.
  • Figure 2: A comparison of indirectly reconstructed trajectories $\hat{x}_{1}(t)$ and $\hat{x}_{2}(t)$ with the true trajectories $x_{1}(t)$ and $x_{2}(t)$ of the Duffing oscillator resulting from the nonlinear feedback law $\Bar{\mu}$ in experiment \ref{['exp:Duffing']}.
  • Figure 3: A comparison between the linear predictor developed in SCC.Korda.Mezic2018a and the indirect reconstruction via DCLDMD in experiment \ref{['exp:PredictionComp']}. Here, $\hat{x}_{i}(t)$, $x_{p,i}(t)$, and $x_{i}(t)$ represent the indirect reconstruction, the linear predictor, and the actual trajectories, respectively, where $i$ is a subscript denoting an element of the state.

Theorems & Definitions (14)

  • Definition 1
  • Proposition 1
  • proof
  • Definition 2
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Definition 3
  • Definition 4
  • ...and 4 more