A Koopman-Operator Control Optimization for Relative Motion in Space

TL;DR

This paper introduces a novel control optimization method using Koopman operator theory to linearize and solve nonlinear space rendezvous problems, enabling efficient energy-optimal control of satellite relative motion.

Contribution

It develops a high-order optimal control strategy based on Koopman operators, transforming nonlinear dynamics into a linear framework for space rendezvous applications.

Findings

Successfully applied to satellite rendezvous scenarios.

Achieved energy-efficient control solutions.

Demonstrated the effectiveness of polynomial-based Koopman representations.

Abstract

A high order optimal control strategy implemented in the Koopman operator framework is proposed in this work. The new technique exploits the Koopman representation of the solution of the equations of motion to develop an energy optimal inverse control methodology. The operator theory can reformulate a nonlinear dynamical system of finite dimension into a linear system with an infinite number of dimensions. As a results, the state of any nonlinear dynamics is represented as a linear combination of high-order orthogonal polynomials, which creates the state transition polynomial map of the solution. Since the optimal control technique can be reduced to a two-points boundary value problem, the Koopman map is used to connect the state and control variables in time, such that optimal values are obtained through map inversion and polynomial evaluation. The new technique is applied to rendezvous applications in space, where the relative motion between two satellites is modelled with a high-order polynomial series expansion of the Lagrangian of the system, such that the Clohessy-Wiltshire equations represent the reduction of the high-order model to a linear truncation.