The Deep Input-Koopman Operator for Nonlinear Systems

Abstract

In this paper, we propose a method that exploit the Koopman operator theory to make the strongly nonlinear dynamical systems approximately represented in the linear framework based on deep neural network (DNN) which is data-driven and equation-free. On account of the conventional Koopman operator is incapable for actuated systems, we introduce the notion of input-Koopman operator for the systems incorporated with the effects of inputs and controls. We construct the controllability gramian for nonlinear systems that are represented in the finite-dimensional input-Koopman operators. Moreover, we illustrate the several relationship between the space of full state observable functions and the original local controllability.