Dynamic Mode Decomposition with Control Liouville Operators

TL;DR

This paper develops a theoretical framework for applying dynamic mode decomposition to control-affine systems using control Liouville operators within RKHSs, enabling trajectory prediction and system analysis.

Contribution

It introduces control Liouville operators and occupation kernels, providing a novel RKHS-based approach for DMD of controlled nonlinear systems.

Findings

The total derivative operator is compact under certain conditions.

Finite-rank models converge to true system dynamics with rich data.

Numerical experiments validate the method's effectiveness.

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.