Dimensionality Reduction with Koopman Generalized Eigenfunctions

TL;DR

This paper introduces a method to obtain lower-dimensional Koopman representations of nonlinear systems using generalized eigenfunctions, enabling linear control design for complex dynamics like quadrotor UAVs.

Contribution

It develops a novel approach to construct Koopman eigenfunctions that match the number of inputs, facilitating linearization control without high-dimensional approximations.

Findings

Successfully designed a linear quadratic (LQ) flight controller for a quadrotor UAV.

Validated the approach through numerical and hardware-in-the-loop simulations.

Achieved real-time implementation despite noise and sensor delays.

Abstract

This paper presents a methodology to achieve lower-dimensional Koopman quasi-linear representations of nonlinear system dynamics using Koopman generalized eigenfunctions. The proposed approach considers the analytically derived Koopman formulation of rigid body dynamics, but it can be extended to any data-driven or analytically derived generalized eigenfunction set. It achieves a representation for which the number of Koopman observables matches the number of inputs allowing for Koopman linearization control solutions rather than resorting to the least squares approximation method adopted in high dimensional Koopman formulations. Through a linear combination of Koopman generalized eigenfunctions a new set of Koopman generalized eigenfunction is constructed so that the zero order truncation approximate a Koopman eigenfunction which can be used to design linear control strategies to steer the dynamics of the original nonlinear system. The proposed methodology is tested by designing a linear quadratic (LQ) flight controller for a quadrotor UAV. Numerical and Hardware-in-the-loop (HIL) simulations validate the applicability and real-time implementability of the proposed approach in the presence of noise and sensor delays. The main advantage of the proposed method is the realization of a fully actuated Koopman based model which, in the case of the underactuated quadrotor system, allows to achieve trajectory tracking through a single linear control loop.