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

The representation of a nonlinear dynamical system as infinite-dimensional linear operator over a Hilbert space of functions enables the study of the nonlinear system via pseudo-spectral analysis of the corresponding operator. In this letter, we develop a novel operator representation of discrete-time, control-affine nonlinear dynamical systems. We also demonstrate that this representation can be used to predict the behavior of the closed-loop system in response to a given feedback law. The representation is learned 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. The advantages of the developed method relative to existing techniques in the literature are also demonstrated.