Efficient Input-Constrained Impulsive Optimal Control of Linear Systems with Application to Spacecraft Relative Motion

TL;DR

This work addresses impulsive optimal control for linear time-varying systems with input magnitude constraints, a scenario common in spacecraft relative motion where actuators are delta-like yet bounded. It casts the problem in a dual semi-infinite programming framework, introducing magnitude constraints over time windows and showing the impulsive structure is preserved; the dual variables $\bm{\lambda}_f$ and $\{\sigma_k\}$ drive an efficient online solution, with input reconstruction recovering the impulse sequence. Key contributions include the magnitude-constrained impulsive formulation, proof of strong duality and reducibility, a discretization-based SIP solver, and an algorithm that reconstructs the optimal input from the dual solution. Validation on planar reconfiguration and the VISORS transfer demonstrates substantial computational efficiency (online-capable solver times) and meaningful Delta-V savings, highlighting practical impact for CubeSats and other space missions with tight resources and impulse limits.

Abstract

This work presents a novel algorithm for impulsive optimal control of linear time-varying systems with the inclusion of input magnitude constraints. Impulsive optimal control problems, where the optimal input solution is a sum of delta functions, are typically formulated as an optimization over a normed function space subject to integral equality constraints and can be efficiently solved for linear time-varying systems in their dual formulation. In this dual setting, the problem takes the form of a semi-infinite program which is readily solvable in online scenarios for constructing maneuver plans. This work augments the approach with the inclusion of magnitude constraints on the input over time windows of interest, which is shown to preserve the impulsive nature of the optimal solution and enable efficient solution procedures via semi-infinite programming. The resulting algorithm is demonstrated on the highly relevant problem of relative motion control of spacecraft in Low Earth Orbit (LEO) and compared to several other proposed solutions from the literature.

Figures (5)

• Figure 1: Planar Reconfiguration Demonstration.
• Figure 2: Planar Reconfiguration Relative Orbital Elements
• Figure 3: VISORS Reconfiguration Trajectory
• Figure 4: VISORS Reconfiguration Input Profile
• Figure 5: VISORS Reconfiguration Relative Orbital Elements

Theorems & Definitions (6)

• Theorem III.1
• proof
• Theorem III.2
• proof
• Lemma 1
• proof