Sum-of-Squares based computation of a Lyapunov function for proving stability of a satellite with electromagnetic actuation

This work focuses on the computation of a candidate Lyapunov function for a Low Earth Orbit satellite which is actuated using only magnetorquers. A satellite having only electromagnetic actuation is not controllable when the magnetic moment produced by the magnetorquers is parallel to the geomagnetic field. Further, the dynamics of the system are periodic due to the periodic nature of the geomagnetic field. Previously, a locally stable Proportional-Derivative control has been designed for such a satellite. In this work, we have found a polynomial candidate Lyapunov function for the resultant closed loop system using Sum-of-Squares (SoS) polynomials and Putinar’s Positivstellensatz. Unlike previous applications of SoS techniques on rigid bodies, the kinematics have been defined using unit quaternions. The unit quaternions have a well-known advantage of being a singularity free representation of attitude kinematics with only a single constraint. The unit quaternion constraint has been ensured using semialgebraic sets. Furthermore, special emphasis has been placed on the verification of the candidate Lyapunov function and we have simulated the closed loop system with the candidate Lyapunov function.