Estimating Koopman Invariant Subspaces of Excited Systems Using Artificial Neural Networks

Abstract

In recent years, the Koopman operator was the topic of many extensive investigations in the nonlinear system identification community. Especially, when dealing with nonlinear systems no straight forward method is available to identify systems of this class. In modern data science a standard method is using artificial neural networks to extract models from data. This method is mainly used when there is a certain function behind the measured data, but little other information is available. This paper combines the Koopman framework and artificial neural networks to achieve a linear model for nonlinear systems. The structure of the network is similar to an autoencoder. The input part is the encoder which itself consists of two different parts. The first part propagates the measurements directly to the middle part. The second encoder introduces the state-space lifting which characterizes the Koopman framework. The middle layer of the network represents an estimation of a linear state-space system that acts on a Koopman operator invariant subspace. After this layer, the extended state-space must be decoded so that the outputs of the Koopman linear system are functions of the true states. The method is evaluated with a single pendulum and a nonlinear yeast glycolysis model. Additionally, we show the advantage of considering inputs as true inputs rather than additional states.