$\mu$ Control for Satellites Formation Flying

In this paper, a $\mu$ controller is designed for a satellite formation flying system around the Earth based on an uncertainty model derived from a nonlinear relative position equation. In this model, nonzero eccentricity and varying semimajor axis are included as parametric uncertainties. $J_2$ perturbation, atmospheric drag, and actuation and sensor noise are bounded by functional uncertainties. The $\mu$ controller design based on the nominal mission (an $800\mathrm{km}$ altitude circular reference orbit) is capable of achieving desired performance, is robust to uncertainties, and satisfies fuel consumption requirements even in a challenge nonnominal mission (a 0.1 eccentricity and $\mathrm{7,978}\mathrm{km}$ semimajor axis elliptic reference orbit) with the same control gain. In this nonnominal mission, the designed $\mu$ controller is able to keep formation with almost the same level of the $\mathrm{\Delta }V$ budget $(43.86\mathrm{m}∕\mathrm{s}∕\mathrm{year})$ as used in the nominal mission $(39.65\mathrm{m}∕\mathrm{s}∕\mathrm{year})$ . For comparison, linear quadratic regulator (LQR) and sliding mode controllers (SMC) are developed and extensively tuned to get the same $\mathrm{\Delta }V$ consumption as that of the designed $\mu$ controller for the nominal mission. However, as shown in the simulation, the designed linear robust controller (LQR) and nonlinear robust controller (SMC) have a serious $\mathrm{\Delta }V$ consumption penalty ( $1.72\mathrm{km}∕\mathrm{s}∕\mathrm{year}$ for SMC) or are unstable (for LQR) in the nonnominal mission.