Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Dynamic Mode Decomposition of Control-affine Systems

Moad Abudia, Joel A. Rosenfeld, Rushikesh Kamalapurkar

TL;DR

This work develops a norm-convergent, finite-rank framework for dynamic mode decomposition of control-affine systems by embedding the dynamics in a vector-valued reproducing kernel Hilbert space and using control occupation kernels to separate drift from input effects. A finite-rank representation $P_{eta^M} A_{f,g} P_{d^M}$ of the control Liouville operator $A_{f,g}$ is constructed with Gram matrices $G_d$ and $G_eta$, and its singular values/functions are obtained from a matrix representation $[P_eta A_{f,g}]_d^eta = G_eta^{+} G_d$. This yields a data-driven spectral decomposition that identifies drift $f$ and control-effectiveness $g$ via $ olinebreak olinebreak F(x,u) olinebreak = olinebreak D G_eta^{+} eta(x) 1u$, and guarantees convergence $ olinebreak o F(x,u)$ as the data set grows. The method is validated on a Duffing oscillator, showing accurate trajectory predictions and precise vector-field deductions, thereby enabling reliable prediction of system responses to admissible controls in a principled, kernel-based setting.

Abstract

This paper builds on 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 used to separate the drift dynamics from the input dynamics. A provably convergent finite-rank estimation of a compact control Liouville operator is obtained, provided sufficiently rich data are available. A matrix representation of the finite-rank operator is used to construct a data-driven representation of its singular values, left singular functions, and right singular functions. The singular value decomposition is used to generate a data-driven model of the control-affine nonlinear system. The developed method generates a model that can be used to predict the trajectories of the system in response to any admissible control input. Numerical experiments are included to demonstrate the efficacy of the developed technique.

On Dynamic Mode Decomposition of Control-affine Systems

TL;DR

This work develops a norm-convergent, finite-rank framework for dynamic mode decomposition of control-affine systems by embedding the dynamics in a vector-valued reproducing kernel Hilbert space and using control occupation kernels to separate drift from input effects. A finite-rank representation of the control Liouville operator is constructed with Gram matrices and , and its singular values/functions are obtained from a matrix representation . This yields a data-driven spectral decomposition that identifies drift and control-effectiveness via , and guarantees convergence as the data set grows. The method is validated on a Duffing oscillator, showing accurate trajectory predictions and precise vector-field deductions, thereby enabling reliable prediction of system responses to admissible controls in a principled, kernel-based setting.

Abstract

This paper builds on 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 used to separate the drift dynamics from the input dynamics. A provably convergent finite-rank estimation of a compact control Liouville operator is obtained, provided sufficiently rich data are available. A matrix representation of the finite-rank operator is used to construct a data-driven representation of its singular values, left singular functions, and right singular functions. The singular value decomposition is used to generate a data-driven model of the control-affine nonlinear system. The developed method generates a model that can be used to predict the trajectories of the system in response to any admissible control input. Numerical experiments are included to demonstrate the efficacy of the developed technique.

Paper Structure

This paper contains 9 sections, 10 theorems, 24 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Statement
  4. Finite-rank Representation of the Control Liouville Operator
  5. Matrix Representation of the Finite-rank Operator
  6. Singular Functions of the Finite-rank Operator
  7. The SCLDMD Algorithm
  8. Numerical Experiments
  9. Conclusion

Key Result

Proposition 1

SCC.Abudia.Channagiri.eainprep If $H$ is a vvRKHS consisting of $\mathbb{R}^{m+1}$ (row) valued continuous functions over $\mathbb{R}^n$, $\gamma_u: [0,T] \to \mathbb{R}^n$ is a controlled trajectory with continuous controller $u:[0,T]\to\mathbb{R}^m$ satisfying, for all $t\in[0,T]$, $\frac{d^s\gamm

Figures (3)

  • Figure 1: The responses of the true Duffing oscillator and the identified duffing oscillator to the input $u(t)=\sin(t) + \cos(2t)$.
  • Figure 2: The difference in responses of the true Duffing oscillator and the identified duffing oscillator to the input $u(t)=\sin(t) + \cos(2t)$.
  • Figure 3: Error $\left\Vert f(x) - \hat{f}(x) \right\Vert$ as a function of $x$ (left) and error $\left\Vert g(x) - \hat{g}(x) \right\Vert$ as a function of $x$ (right).

Theorems & Definitions (23)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Definition 1
  • Proposition 4
  • proof
  • Definition 2
  • ...and 13 more