Convex Bound of Nonlinear Dynamical Errors for Covariance Steering

TL;DR

This paper introduces a convex formulation to bound nonlinear errors in covariance steering, improving controller performance in highly nonlinear environments like cislunar space.

Contribution

It presents a convex upper-bound on nonlinear errors using Taylor series, enabling efficient controller gain optimization in nonlinear settings.

Findings

The convex formulation effectively bounds nonlinear errors.

The method maintains Gaussian distribution better in simulations.

Improves covariance controller performance in nonlinear environments.

Abstract

Applying linear controllers to nonlinear systems requires the dynamical linearization about a reference. In highly nonlinear environments such as cislunar space, the region of validity for these linearizations varies widely and can negatively affect controller performance if not carefully formulated. This paper presents a formulation that minimizes the nonlinear errors experienced by linear covariance controllers. The formulation involves upper-bounding the remainder term from the linearization process using higher-order terms in a Taylor series expansion, and resolving it into a convex function. This can serve as a cost function for controller gain optimization, and its convex nature allows for efficient solutions through convex optimization. This formulation is then demonstrated and compared with the current methods within a halo orbit stationkeeping scenario. The results show that the formulation proposed in this paper maintains the Gaussianity of the distribution in nonlinear simulations more effectively, thereby allowing the linear covariance controller to perform more as intended in nonlinear environments.