Learning Koopman Models From Data Under General Noise Conditions

TL;DR

This paper introduces a data-driven method for identifying Koopman models of nonlinear systems with inputs under various noise conditions, using deep encoders and multiple-shooting for efficient and accurate long-term predictions.

Contribution

It proposes a novel, statistically consistent approach combining deep state-space encoders and multiple-shooting to handle general noise in Koopman model identification.

Findings

Method is statistically consistent with increasing data

Efficient batch optimization enables long-term prediction accuracy

Validated on nonlinear benchmarks and quadcopter data

Abstract

This paper presents a novel identification approach of Koopman models of nonlinear systems with inputs under rather general noise conditions. The method uses deep state-space encoders based on the concept of state reconstructability and an efficient multiple-shooting formulation of the squared loss of the prediction error to estimate the dynamics and the lifted state only from input-output data. Furthermore, the Koopman model structure includes an innovation noise term that is used to handle process and measurement noise. It is shown that the proposed approach is statistically consistent (estimation error tends to zero when the number of data points goes to infinity) and computationally efficient due to the multiple-shooting formulation, by which the prediction error of the model can be calculated on multiple subsections of the data in parallel. The latter allows for efficient batch optimization of the network parameters and, at the same time, excellent long-term prediction capabilities of the obtained models. The performance of the approach is illustrated by nonlinear benchmark examples and experimental data from a Crazyflie 2.1 quadcopter.