Control strategies for stable orbits around phobos

This study compares three linear optimal controllers for stationkeeping with respect to reference orbits around Phobos. The dynamics of the Mars-Phobos system are constrained to the synodical plane of the circular restricted three-body problem (CRTBP) with Phobos modelled as an ellipsoid. The controllers rely on periodic orbits which permit the dynamics to be expressed as a linear system with periodic coefficients. A novel method of determining the necessary conditions for periodic orbits is formulated through nonlinear optimization techniques, where the function to be minimized is the vector norm of the difference between the initial and final conditions of the orbit. The optimization algorithm is a Nelder-Mead simplex and is shown to outperform any gradient-based methods as well as other techniques for determining such orbits. Two controllers, constant feedback and scheduled, are developed from the algebraic Riccati equation (ARE), which is solved at specific points on the reference orbits. These controllers are then compared to the optimal solution which uses the time-varying Riccati equation. At high orbits, the periodicity of the linearized system is very small and the controllers are nearly identical in performance. Closer orbits reveal increases in the periodicity of the dynamics, leading to an increase in performance of the time-varying Riccati equation-based controller over the scheduled and constant feedback gain cases