Discrete-time LQ optimal control of satellite formations in elliptical orbits based on feedback linearization

The sampled-data representation of the relative motion is the foundation of the discrete-time LQ optimal control of spacecraft formations. The sampled-data description for the relative motion in circular orbits has been investigated in great detail. However, few are derived for the elliptical orbits. This paper will employ the discrete-time LQ optimal control theory to deal with the problem of relative orbit control of satellites in elliptic orbits. The formation vector is used to express the formation geometry, and nonlinear feedback is utilized to linearize the equations of the relative motion. An analytical state transition matrix is derived from the solutions of the linearized equations. Based on the state transition matrix, a sampled-data representation is presented for the linearized equations. The sampled-data representation is explicitly related to a sampling sequence of the true anomaly of the target satellite with constant length of the sampling intervals. In terms of the discrete-time model, a discrete-time LQ optimal controller is derived for the linearized equations. By combining the discrete-time LQ optimal control with the nonlinear feedback control, a digital controller is obtained for the satellite formations. Simulations are included to demonstrate the performance of the controller.