Expanding the Class of Quadratic Control-Lyapunov Functions for Low-Thrust Trajectory Optimization

TL;DR

This paper introduces a novel eigendecomposition method for parameterizing full positive-definite matrices in quadratic Control-Lyapunov Functions, enhancing low-thrust trajectory optimization by achieving more optimal solutions than diagonal matrices.

Contribution

It proposes a new eigendecomposition approach for full matrices in CLFs and applies particle swarm optimization to improve low-thrust trajectory solutions.

Findings

Full matrices yield more optimal trajectories than diagonal matrices.

The eigendecomposition method guarantees positive-definiteness easily.

Improvements are significant for large orbital changes.

Abstract

Control laws derived from Control-Lyapunov Functions (CLFs) offer an efficient way for generating near-optimal many-revolution low-thrust trajectories. A common approach to constructing CLFs is to consider the family of quadratic functions using a diagonal weighting matrix. In this paper, we explore the advantages of using a larger family of quadratic functions. More specifically, we consider positive-definite weighting matrices with non-zero off-diagonal elements (hereafter referred to as "full" matrices). We propose a novel eigendecomposition method for parameterizing $N$-dimensional weighting matrices that is easy to implement and guarantees positive-definiteness of the weighting matrices. We use particle swarm optimization, which is a stochastic optimization algorithm, to optimize the parameters and generate near-optimal minimum-time low-thrust trajectories. Solutions obtained using a full positive-definite matrix are compared to the results from the (standard) diagonal weighting matrix for a number of benchmark problems. Results demonstrate that improvements in optimality are achieved, especially for maneuvers with large changes in orbital elements.