Pedro Tavares, Paulo Tabuada, ‡
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The control torque is obtained by electromagnetic interaction with the geomagnetic field (Wertz, 1995),
 (2)
where  is the control magnetic moment generated by the satellite coils and will be referred to as the control variable throughout the paper.  is the geomagnetic field.
Equation (2) shows that the control torque is always perpendicular to the geomagnetic field, pointing out the non-controllability of electromagnetic actuation. The direction parallel to the geomagnetic field is not controllable, but the geomagnetic field changes along the orbit. This implies that, e.g., yaw is not controllable over the poles but only a quarter of orbit later, i.e. approximately over the equator. Those characteristics must be adequately explored to regulate appropriately the satellite’s attitude. A time-varying predictive algorithm to determine the control moment, which takes advantage of the geomagnetic field changes, is proposed as a solution to this control problem. Motivation Using the satellite total energy as a Lyapunov candidate function (Wisniewski, 1997) shows that its time derivative is given by:  (3)
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