Geometric Conditions for Lossless Convexification in Linear Optimal Control with Discrete-Valued Inputs: Real-Time Implementation for Spacecraft Rendezvous

TL;DR

This paper introduces a geometric condition-based lossless convexification method for real-time optimal control of spacecraft rendezvous with discrete thrusters, ensuring computational efficiency and exact control solutions.

Contribution

It extends theoretical results on system normality preservation and establishes geometric conditions for exact convex relaxation in linear systems with discrete inputs.

Findings

The framework preserves system normality during reformulation.

Under geometric conditions, the convex solution satisfies original constraints.

Numerical simulations confirm real-time applicability for spacecraft maneuvers.

Abstract

Optimal control problems with discrete-valued inputs are inherently challenging due to their mixed-integer nature, rendering them generally intractable for real-time, safety-critical aerospace applications. Lossless convexification offers a powerful alternative by reformulating these mixed-integer programs into computationally efficient convex programs. This paper develops a lossless convexification framework for the optimal control of linear time-varying systems with discrete-valued inputs. We extend existing theoretical results by demonstrating that system normality is preserved when reformulating Lagrange-form problems into Mayer-form via an epigraph transformation. Furthermore, we establish that under simple geometric conditions on the input set, the solution to the relaxed convex problem strictly satisfies the original non-convex input constraints. This framework enables the real-time computation of optimal discrete-valued controls without resorting to mixed-integer optimization. The proposed algorithm is validated on a spacecraft rendezvous maneuver utilizing discrete-valued reaction thrusters in an elliptical orbit. Numerical results from Monte Carlo simulations confirm that the algorithm consistently yields exact discrete-valued control inputs with computational timelines compatible with safety-critical, on-board applications.