Gröbner basis solutions to satellite trajectory control by pole placement

Controlling satellite trajectories is an important problem. In [12], an approach to the pole placement for the synthesis of a linear controller has been presented. It leads to solving five polynomial equations in nine unknown elements of the state space matrices of a compensator. This is an underconstrained system and therefore four of the unknown elements need to be considered as free parameters and set to some prior values to obtain a system of five equations in five unknowns. In [12], this system was solved for one chosen set of free parameters by Dixon resultants. In this work, we study and present Gröbner basis solutions to this problem of computation of a dynamic compensator for the satellite for different combinations of free input parameters. We show that the Gröbner basis method for solving systems of polynomial equations leads to very simple solutions for all combinations of free parameters. These solutions require to perform only the Gauss-Jordan elimination of a small matrix and computation of roots of a single variable polynomial. The maximum degree of this polynomial is not greater than six in general but for most combinations of the input free parameters its degree is even lower.