An algebraic approach based on symmetry relations of state space matrices for decoupling the dynamics of a large space structure

In order to design an effective control policy for the 10 m diameter primary mirror of the Grantecan Telescope, the coupling among the segments that compose the mirror through the structure that supports it must be treated. We present the mathematical demonstration of two important results in order to deal with the mirror dynamics decoupling. Those are: 1) the multiplicity of the eigenvalue corresponding to the mirror segments decreases as the number of structure modes considered increases, and 2) the vectorial subspace composed of the eigenvectors corresponding to the structure and the segments coupled with it is orthogonal to the one composed of the eigenvectors corresponding to the rest of segments.