Periodic H2 synthesis for spacecraft in elliptical orbits with atmospheric drag and J2 perturbations

The problem of the stationkeeping for a small spacecraft is studied and a solution based on periodic feedback control laws is considered. Linearized equations of the relative motion of the satellite near an eccentric reference orbit are derived in the presence of the second zonal gravitational harmonic J₂ and atmospheric drag perturbations. The obtained linear continuous-time model of the relative motion is T- periodic where T is the orbital period. After a discretization of the model, a state-feedback control law with performance requirement defined by the generalized H₂ operator norm may be computed by a linear matrix inequality-based algorithm. Illustrative nonlinear simulations show the efficiency of the approach based on the use of linearized spacecraft relative motion dynamics associated to systematic H₂ synthesis of stabilizing memoryless N-periodic state-feedback control laws.