An Approach to Optimal Control of Electrodynamic Tethers in a Stochastically Varying Drag Environment

Electrodynamic Tethers (EDT s) offer a real option for zero-propellant orbital maneuvers in the near future. By controlling the electrical current through a long conductive cable aligned with the local vertical and in the presence of a magnetic field, the tether experiences an electrodynamic thrust. The local ionosphere provides the necessary electrons for the generation of an electrical current. Previous investigation has been focused on feed-forward or open-loop control schemes. Open-loop control methods are very susceptible to model error. The relevant models for an EDT system are the atmospheric density model and the magnetic field model. This paper will be concerned with errors in the atmospheric density model. The problem consists of two parts: solving the open loop non-linear optimal control problem and solving the associated linear feedback system to generate a control law. To solve the first part we assume the orbit remains nearly circular. We apply the method of averaging to the state dynamics to track secular changes only. The short period motion of the spacecraft drives the shape of the control. We vary the coefficients on a five term modified Fourier series describing the tether s alternating electrical control current. The series is modified by using square waves rather than sine and cosine waves. The open-loop control solution is then used as reference in the feedback problem. Solving the associated linear feedback system involves linearizing the state dynamics. Standard linearization yields a classic state-space structured system using state error to generate control corrections. We assume complete state feedback to simplify the solution. Treating the system as linear time invariant, we update the gain matrix once per orbit. Results indicate this strategy improves performance and reliability of a system with model errors and un-modeled disturbances, particularly for maneuvers that remain in the LEO regime for an extended time.
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