Linear Programming Approach to Relative-Orbit Control With Element-Wise Quantized Control

This paper considers the problem of close-proximity relative-orbit control of spacecraft with control quantization constraints, models the problem as a mixed-integer program, and reformulates the problem as a linear program. The reformulation uses linearized relative orbital element dynamics with a sum-of-absolute-values objective, and it is proved that optimal controls for the reformulated problem satisfy the quantization constraint, that is, the quantization constraints are convexified for the reformulated continuous-time optimal control problem. This problem is then discretized, converted to a finite-dimensional linear program, and solved using commercially available convex optimization solvers with polynomial-time convergence guarantees. Since the mathematical proofs of convexification are available only for continuous-time case, their validity for discrete-time is demonstrated with extensive simulations. To this end, Monte Carlo simulations indicate that quantization is achieved with high probability while keeping spacecraft slew rates low relative to other proposed approaches.