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

This letter 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 slew rates low relative to other popular alternatives.